diff options
Diffstat (limited to 'buch/papers')
328 files changed, 23874 insertions, 1701 deletions
diff --git a/buch/papers/0f1/images/konvergenzAiry.pdf b/buch/papers/0f1/images/konvergenzAiry.pdf Binary files differnew file mode 100644 index 0000000..206cd3a --- /dev/null +++ b/buch/papers/0f1/images/konvergenzAiry.pdf diff --git a/buch/papers/0f1/images/konvergenzNegativ.pdf b/buch/papers/0f1/images/konvergenzNegativ.pdf Binary files differnew file mode 100644 index 0000000..03b2ba1 --- /dev/null +++ b/buch/papers/0f1/images/konvergenzNegativ.pdf diff --git a/buch/papers/0f1/images/konvergenzPositiv.pdf b/buch/papers/0f1/images/konvergenzPositiv.pdf Binary files differnew file mode 100644 index 0000000..2e45129 --- /dev/null +++ b/buch/papers/0f1/images/konvergenzPositiv.pdf diff --git a/buch/papers/0f1/images/stabilitaet.pdf b/buch/papers/0f1/images/stabilitaet.pdf Binary files differnew file mode 100644 index 0000000..13dea39 --- /dev/null +++ b/buch/papers/0f1/images/stabilitaet.pdf diff --git a/buch/papers/0f1/listings/kettenbruchIterativ.c b/buch/papers/0f1/listings/kettenbruchIterativ.c new file mode 100644 index 0000000..d897b8f --- /dev/null +++ b/buch/papers/0f1/listings/kettenbruchIterativ.c @@ -0,0 +1,53 @@ +/**
+ * @brief Calculates the Hypergeometric Function 0F1(;b;z)
+ * @param b0 in 0F1(;b0;z)
+ * @param z in 0F1(;b0;z)
+ * @param n number of itertions (precision)
+ * @return Result
+ */
+static double fractionRekursion0f1(const double c, const double z, unsigned int n)
+{
+ //declaration
+ double a = 0.0;
+ double b = 0.0;
+ double Ak = 0.0;
+ double Bk = 0.0;
+ double Ak_1 = 0.0;
+ double Bk_1 = 0.0;
+ double Ak_2 = 0.0;
+ double Bk_2 = 0.0;
+
+ for (unsigned int k = 0; k <= n; ++k)
+ {
+ if (k == 0)
+ {
+ a = 1.0; //a0
+ //recursion fomula for A0, B0
+ Ak = a;
+ Bk = 1.0;
+ }
+ else if (k == 1)
+ {
+ a = 1.0; //a1
+ b = z/c; //b1
+ //recursion fomula for A1, B1
+ Ak = a * Ak_1 + b * 1.0;
+ Bk = a * Bk_1;
+ }
+ else
+ {
+ a = 1 + (z / (k * ((k - 1) + c)));//ak
+ b = -(z / (k * ((k - 1) + c))); //bk
+ //recursion fomula for Ak, Bk
+ Ak = a * Ak_1 + b * Ak_2;
+ Bk = a * Bk_1 + b * Bk_2;
+ }
+ //save old values
+ Ak_2 = Ak_1;
+ Bk_2 = Bk_1;
+ Ak_1 = Ak;
+ Bk_1 = Bk;
+ }
+ //approximation fraction
+ return Ak/Bk;
+}
diff --git a/buch/papers/0f1/listings/kettenbruchRekursion.c b/buch/papers/0f1/listings/kettenbruchRekursion.c new file mode 100644 index 0000000..3caaf43 --- /dev/null +++ b/buch/papers/0f1/listings/kettenbruchRekursion.c @@ -0,0 +1,27 @@ +/**
+ * @brief Calculates the Hypergeometric Function 0F1(;c;z)
+ * @param c in 0F1(;c;z)
+ * @param z in 0F1(;c;z)
+ * @param k number of itertions (precision)
+ * @return Result
+ */
+static double fractionIter0f1(const double c, const double z, unsigned int k)
+{
+ //declaration
+ double a = 0.0;
+ double b = 0.0;
+ double abk = 0.0;
+ double temp = 0.0;
+
+ for (; k > 0; --k)
+ {
+ abk = z / (k * ((k - 1) + c)); //abk = ak, bk
+
+ a = k > 1 ? (1 + abk) : 1; //a0, a1
+ b = k > 1 ? -abk : abk; //b1
+
+ temp = b / (a + temp); //bk / (ak + last result)
+ }
+
+ return a + temp; //a0 + temp
+}
\ No newline at end of file diff --git a/buch/papers/0f1/listings/potenzreihe.c b/buch/papers/0f1/listings/potenzreihe.c new file mode 100644 index 0000000..23fdfea --- /dev/null +++ b/buch/papers/0f1/listings/potenzreihe.c @@ -0,0 +1,69 @@ +#include <math.h>
+
+/**
+ * @brief Calculates pochhammer
+ * @param (a+n-1)!
+ * @return Result
+ */
+static double pochhammer(const double x, double n)
+{
+ double temp = x;
+
+ if (n > 0)
+ {
+ while (n > 1)
+ {
+ temp *= (x + n - 1);
+ --n;
+ }
+
+ return temp;
+ }
+ else
+ {
+ return 1;
+ }
+}
+
+/**
+ * @brief Calculates the Factorial
+ * @param n!
+ * @return Result
+ */
+static double fac(int n)
+{
+ double temp = n;
+
+ if (n > 0)
+ {
+ while (n > 1)
+ {
+ --n;
+ temp *= n;
+ }
+ return temp;
+ }
+ else
+ {
+ return 1;
+ }
+}
+
+/**
+ * @brief Calculates the Hypergeometric Function 0F1(;b;z)
+ * @param c in 0F1(;c;z)
+ * @param z in 0F1(;c;z)
+ * @param n number of itertions (precision)
+ * @return Result
+ */
+static double powerseries(const double c, const double z, unsigned int n)
+{
+ double temp = 0.0;
+
+ for (unsigned int k = 0; k < n; ++k)
+ {
+ temp += pow(z, k) / (factorial(k) * pochhammer(c, k));
+ }
+
+ return temp;
+}
\ No newline at end of file diff --git a/buch/papers/0f1/main.tex b/buch/papers/0f1/main.tex index 264ad56..0b1020f 100644 --- a/buch/papers/0f1/main.tex +++ b/buch/papers/0f1/main.tex @@ -1,36 +1,24 @@ -% -% main.tex -- Paper zum Thema <0f1> -% -% (c) 2020 Hochschule Rapperswil -% -\chapter{Thema\label{chapter:0f1}} -\lhead{Thema} -\begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/0f1/teil0.tex} -\input{papers/0f1/teil1.tex} -\input{papers/0f1/teil2.tex} -\input{papers/0f1/teil3.tex} - -\printbibliography[heading=subbibliography] -\end{refsection} +%
+% main.tex -- Paper zum Thema <0f1>
+%
+% (c) 2020 Hochschule Rapperswil
+%
+%
+
+
+
+\chapter{Algorithmus zur Berechnung von $\mathstrut_0F_1$\label{chapter:0f1}}
+\lhead{Algorithmus zur Berechnung von $\mathstrut_0F_1$}
+\begin{refsection}
+\chapterauthor{Fabian Dünki}
+
+
+
+
+\input{papers/0f1/teil0.tex}
+\input{papers/0f1/teil1.tex}
+\input{papers/0f1/teil2.tex}
+\input{papers/0f1/teil3.tex}
+
+\printbibliography[heading=subbibliography]
+\end{refsection}
diff --git a/buch/papers/0f1/references.bib b/buch/papers/0f1/references.bib index fb9cd8b..47555da 100644 --- a/buch/papers/0f1/references.bib +++ b/buch/papers/0f1/references.bib @@ -4,32 +4,82 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{0f1:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{0f1:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{0f1:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@online{0f1:library-gsl, + title = {GNU Scientific Library}, + url ={https://www.gnu.org/software/gsl/}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} } +@online{0f1:wiki-airyFunktion, + title = {Airy-Funktion}, + url ={https://de.wikipedia.org/wiki/Airy-Funktion}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} +} + +@online{0f1:wiki-kettenbruch, + title = {Kettenbruch}, + url ={https://de.wikipedia.org/wiki/Kettenbruch}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {25} +} + +@online{0f1:double, + title = {C - Data Types}, + url ={https://www.tutorialspoint.com/cprogramming/c_data_types.htm}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} +} + +@online{0f1:wolfram-0f1, + title = {Hypergeometric 0F1}, + url ={https://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Hypergeometric0F1}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} +} + +@online{0f1:wiki-fraction, + title = {Gauss continued fraction}, + url ={https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} +} + +@online{0f1:code, + title = {Vollständiger C-Code}, + url ={https://github.com/AndreasFMueller/SeminarSpezielleFunktionen/tree/master/buch/papers/0f1/listings}, + date = {2022-07-07}, + year = {2022}, + month = {7}, + day = {7} +} + +@book{0f1:SeminarNumerik, + title = {Mathematisches Seminar Numerik}, + author = {Andreas Müller et al}, + publisher = {Andreas Müller}, + year = {2022}, +} + +@article{0f1:kettenbrueche, + author = { Benjamin Bouhafs-Keller }, + title = { Kettenbrüche }, + journal = { Mathematisches Seminar Numerik }, + year = 2020, + volume = 13, + pages = {363--376}, + url = {https://github.com/AndreasFMueller/SeminarNumerik} +} diff --git a/buch/papers/0f1/teil0.tex b/buch/papers/0f1/teil0.tex index 9087808..adccac7 100644 --- a/buch/papers/0f1/teil0.tex +++ b/buch/papers/0f1/teil0.tex @@ -1,22 +1,15 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{0f1:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{0f1:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - +%
+% einleitung.tex -- Einleitung
+%
+% (c) 2022 Fabian Dünki, Hochschule Rapperswil
+%
+\section{Ausgangslage\label{0f1:section:ausgangslage}}
+\rhead{Ausgangslage}
+Die Hypergeometrische Funktion $\mathstrut_0F_1$ wird in vielen Funktionen als Basisfunktion benutzt,
+zum Beispiel um die Airy Funktion zu berechnen.
+In der GNU Scientific Library \cite{0f1:library-gsl}
+ist die Funktion $\mathstrut_0F_1$ vorhanden.
+Allerdings wirft die Funktion, bei negativen Übergabenwerten wie zum Beispiel \verb+gsl_sf_hyperg_0F1(1, -1)+, eine Exception.
+Bei genauerer Untersuchung hat sich gezeigt, dass die Funktion je nach Betriebssystem funktioniert oder eben nicht.
+So kann die Funktion unter Windows fehlerfrei aufgerufen werden, beim Mac OS und Linux sind negative Übergabeparameter im Moment nicht möglich.
+Ziel dieser Arbeit war es zu evaluieren, ob es mit einfachen mathematischen Operationen möglich ist, die Hypergeometrische Funktion $\mathstrut_0F_1$ zu implementieren.
diff --git a/buch/papers/0f1/teil1.tex b/buch/papers/0f1/teil1.tex index aca84d2..2ca9647 100644 --- a/buch/papers/0f1/teil1.tex +++ b/buch/papers/0f1/teil1.tex @@ -1,55 +1,101 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{0f1:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{0f1:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{0f1:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{0f1:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{0f1:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - +%
+% teil1.tex -- Mathematischer Hintergrund
+%
+% (c) 2022 Fabian Dünki, Hochschule Rapperswil
+%
+\section{Mathematischer Hintergrund
+\label{0f1:section:mathHintergrund}}
+\rhead{Mathematischer Hintergrund}
+Basierend auf den Herleitungen des vorhergehenden Kapitels \ref{buch:rekursion:section:hypergeometrische-funktion}, werden im nachfolgenden Abschnitt nochmals die Resultate
+beschrieben.
+
+\subsection{Hypergeometrische Funktion
+\label{0f1:subsection:hypergeometrisch}}
+Als Grundlage der umgesetzten Algorithmen dient die Hypergeometrische Funktion $\mathstrut_0F_1$. Diese ist eine Unterfunktion der allgemein definierten Funktion $\mathstrut_pF_q$.
+
+\begin{definition}
+ \label{0f1:math:qFp:def}
+ Die hypergeometrische Funktion
+ $\mathstrut_pF_q$ ist definiert durch die Reihe
+ \[
+ \mathstrut_pF_q
+ \biggl(
+ \begin{matrix}
+ a_1,\dots,a_p\\
+ b_1,\dots,b_q
+ \end{matrix}
+ ;
+ x
+ \biggr)
+ =
+ \mathstrut_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;x)
+ =
+ \sum_{k=0}^\infty
+ \frac{(a_1)_k\cdots(a_p)_k}{(b_1)_k\cdots(b_q)_k}\frac{x^k}{k!}.
+ \]
+\end{definition}
+
+Angewendet auf die Funktion $\mathstrut_pF_q$ ergibt sich für $\mathstrut_0F_1$:
+
+\begin{equation}
+ \label{0f1:math:0f1:eq}
+ \mathstrut_0F_1
+ \biggl(
+ \begin{matrix}
+ \\
+ b_1
+ \end{matrix}
+ ;
+ x
+ \biggr)
+ =
+ \mathstrut_0F_1(;b_1;x)
+ =
+ \sum_{k=0}^\infty
+ \frac{x^k}{(b_1)_k \cdot k!}.
+\end{equation}
+
+
+
+
+\subsection{Airy Funktion
+\label{0f1:subsection:airy}}
+Die Airy-Funktion $Ai(x)$ und die verwandte Funktion $Bi(x)$ werden als Airy-Funktion bezeichnet. Sie werden zur Lösung verschiedener physikalischer Probleme benutzt, wie zum Beispiel zur Lösung der Schrödinger-Gleichung. \cite{0f1:wiki-airyFunktion}
+
+\begin{definition}
+ \label{0f1:airy:differentialgleichung:def}
+ Die Differentialgleichung
+ $y'' - xy = 0$
+ heisst die {\em Airy-Differentialgleichung}. \cite{0f1:wiki-airyFunktion}
+\end{definition}
+
+Die Airy Funktion lässt sich auf verschiedene Arten darstellen. \cite{0f1:wiki-airyFunktion}
+Als hypergeometrische Funktion berechnet, ergibt sich wie in Kapitel \ref{buch:differentialgleichungen:section:hypergeometrisch} hergeleitet, folgende Lösungen der Airy-Differentialgleichung zu den Anfangsbedingungen $A(0)=1$ und $A'(0)=0$, sowie $B(0)=0$ und $B'(0)=0$.
+
+\begin{align}
+\label{0f1:airy:hypergeometrisch:eq}
+Ai(x)
+=
+\sum_{k=0}^\infty
+\frac{1}{(\frac23)_k} \frac{1}{k!}\biggl(\frac{x^3}{9}\biggr)^k
+=
+\mathstrut_0F_1\biggl(
+\begin{matrix}\text{---}\\\frac23\end{matrix};\frac{x^3}{9}
+\biggr).
+\\
+Bi(x)
+=
+\sum_{k=0}^\infty
+\frac{1}{(\frac43)_k} \frac{1}{k!}\biggl(\frac{x^3}{9}\biggr)^k
+=
+x\cdot\mathstrut_0F_1\biggl(
+\begin{matrix}\text{---}\\\frac43\end{matrix};
+\frac{x^3}{9}
+\biggr).
+\qedhere
+\end{align}
+
+In diesem speziellem Fall wird die Airy Funktion $Ai(x)$ \eqref{0f1:airy:hypergeometrisch:eq}
+benutzt, um die Stabilität der Algorithmen zu $\mathstrut_0F_1$ zu überprüfen.
+
+
diff --git a/buch/papers/0f1/teil2.tex b/buch/papers/0f1/teil2.tex index 804d11b..9269961 100644 --- a/buch/papers/0f1/teil2.tex +++ b/buch/papers/0f1/teil2.tex @@ -1,40 +1,172 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{0f1:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{0f1:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - +%
+% teil2.tex -- Umsetzung in C Programmen
+%
+% (c) 2022 Fabian Dünki, Hochschule Rapperswil
+%
+\section{Umsetzung
+\label{0f1:section:teil2}}
+\rhead{Umsetzung}
+Zur Umsetzung wurden drei verschiedene Ansätze gewählt.\cite{0f1:code} Dabei wurde der Schwerpunkt auf die Funktionalität und eine gute Lesbarkeit des Codes gelegt.
+Die Unterprogramme wurde jeweils, wie die GNU Scientific Library, in C geschrieben. Die Zwischenresultate wurden vom Hauptprogramm in einem CSV-File gespeichert. Anschliessen wurde mit der Matplot-Libray in Python die Resultate geplottet.
+
+\subsection{Potenzreihe
+\label{0f1:subsection:potenzreihe}}
+Die naheliegendste Lösung ist die Programmierung der Potenzreihe. Allerdings ist ein Problem dieser Umsetzung \ref{0f1:listing:potenzreihe}, dass die Fakultät im Nenner schnell grosse Werte annimmt und so der Bruch gegen Null strebt. Spätesten ab $k=167$ stösst diese Umsetzung \eqref{0f1:umsetzung:0f1:eq} an ihre Grenzen, da die Fakultät von $168$ eine Bereichsüberschreitung des \textit{double} Bereiches darstellt. \cite{0f1:double}
+
+\begin{align}
+ \label{0f1:umsetzung:0f1:eq}
+ \mathstrut_0F_1(;c;z)
+ &=
+ \sum_{k=0}^\infty
+ \frac{z^k}{(c)_k \cdot k!}
+ &=
+ \frac{1}{c}
+ +\frac{z^1}{(c+1) \cdot 1}
+ + \cdots
+ + \frac{z^{20}}{c(c+1)(c+2)\cdots(c+19) \cdot 2.4 \cdot 10^{18}}
+\end{align}
+
+\lstinputlisting[style=C,float,caption={Potenzreihe.},label={0f1:listing:potenzreihe}, firstline=59]{papers/0f1/listings/potenzreihe.c}
+
+\subsection{Kettenbruch
+\label{0f1:subsection:kettenbruch}}
+Ein endlicher Kettenbruch ist ein Bruch der Form
+\begin{equation*}
+a_0 + \cfrac{b_1}{a_1+\cfrac{b_2}{a_2+\cfrac{b_3}{a_3+\cdots}}}
+\end{equation*}
+in welchem $a_0, a_1,\dots,a_n$ und $b_1,b_2,\dots,b_n$ ganze Zahlen darstellen.
+Die Kurzschreibweise für einen allgemeinen Kettenbruch ist
+\begin{equation*}
+ a_0 + \frac{a_1|}{|b_1} + \frac{a_2|}{|b_2} + \frac{a_3|}{|b_3} + \cdots
+\end{equation*}
+und ist somit verknüpfbar mit der Potenzreihe.
+\cite{0f1:wiki-kettenbruch}
+Angewendet auf die Funktion $\mathstrut_0F_1$ bedeutet dies\cite{0f1:wiki-fraction}:
+\begin{equation*}
+ \mathstrut_0F_1(;c;z) = 1 + \frac{z}{c\cdot1!} + \frac{z^2}{c(c+1)\cdot2!} + \frac{z^3}{c(c+1)(c+2)\cdot3!} + \cdots
+\end{equation*}
+Nach allen Umformungen ergibt sich folgender, irregulärer Kettenbruch \eqref{0f1:math:kettenbruch:0f1:eq}
+\begin{equation}
+ \label{0f1:math:kettenbruch:0f1:eq}
+ \mathstrut_0F_1(;c;z) = 1 + \cfrac{\cfrac{z}{c}}{1+\cfrac{-\cfrac{z}{2(c+1)}}{1+\cfrac{z}{2(c+1)}+\cfrac{-\cfrac{z}{3(c+2)}}{1+\cfrac{z}{5(c+4)} + \cdots}}},
+\end{equation}
+der als Code \ref{0f1:listing:kettenbruchIterativ} umgesetzt wurde.
+\cite{0f1:wolfram-0f1}
+
+\lstinputlisting[style=C,float,caption={Iterativ umgesetzter Kettenbruch.},label={0f1:listing:kettenbruchIterativ}, firstline=8]{papers/0f1/listings/kettenbruchIterativ.c}
+
+\subsection{Rekursionsformel
+\label{0f1:subsection:rekursionsformel}}
+Wesentlich stabiler zur Berechnung eines Kettenbruches ist die Rekursionsformel. Nachfolgend wird die verkürzte Herleitung vom Kettenbruch zur Rekursionsformel aufgezeigt. Eine vollständige Schritt für Schritt Herleitung ist im Seminarbuch Numerik, im Kapitel Kettenbrüche zu finden. \cite{0f1:kettenbrueche}
+
+\subsubsection{Herleitung}
+Ein Näherungsbruch in der Form
+\begin{align*}
+ \cfrac{A_k}{B_k} = a_k + \cfrac{b_{k + 1}}{a_{k + 1} + \cfrac{p}{q}}
+\end{align*}
+lässt sich zu
+\begin{align*}
+ \cfrac{A_k}{B_k} = \cfrac{b_{k+1}}{a_{k+1} + \cfrac{p}{q}} = \frac{b_{k+1} \cdot q}{a_{k+1} \cdot q + p}
+\end{align*}
+umformen.
+Dies lässt sich auch durch die folgende Matrizenschreibweise ausdrücken:
+\begin{equation*}
+ \begin{pmatrix}
+ A_k\\
+ B_k
+ \end{pmatrix}
+ = \begin{pmatrix}
+ b_{k+1} \cdot q\\
+ a_{k+1} \cdot q + p
+ \end{pmatrix}
+ =\begin{pmatrix}
+ 0& b_{k+1}\\
+ 1& a_{k+1}
+ \end{pmatrix}
+ \begin{pmatrix}
+ p \\
+ q
+ \end{pmatrix}.
+ %\label{0f1:math:rekursionsformel:herleitung}
+\end{equation*}
+Wendet man dies nun auf den Kettenbruch in der Form
+\begin{equation*}
+ \frac{A_k}{B_k} = a_0 + \cfrac{b_1}{a_1+\cfrac{b_2}{a_2+\cfrac{\cdots}{\cdots+\cfrac{b_{k-1}}{a_{k-1} + \cfrac{b_k}{a_k}}}}}
+\end{equation*}
+an, ergibt sich folgende Matrixdarstellungen:
+
+\begin{align*}
+ \begin{pmatrix}
+ A_k\\
+ B_k
+ \end{pmatrix}
+ &=
+ \begin{pmatrix}
+ 1& a_0\\
+ 0& 1
+ \end{pmatrix}
+ \begin{pmatrix}
+ 0& b_1\\
+ 1& a_1
+ \end{pmatrix}
+ \cdots
+ \begin{pmatrix}
+ 0& b_{k-1}\\
+ 1& a_{k-1}
+ \end{pmatrix}
+ \begin{pmatrix}
+ b_k\\
+ a_k
+ \end{pmatrix}
+\end{align*}
+Nach vollständiger Induktion ergibt sich für den Schritt $k$, die Matrix
+\begin{equation}
+ \label{0f1:math:matrix:ende:eq}
+ \begin{pmatrix}
+ A_{k}\\
+ B_{k}
+ \end{pmatrix}
+ =
+ \begin{pmatrix}
+ A_{k-2}& A_{k-1}\\
+ B_{k-2}& B_{k-1}
+ \end{pmatrix}
+ \begin{pmatrix}
+ b_k\\
+ a_k
+ \end{pmatrix}.
+\end{equation}
+Und Schlussendlich kann der Näherungsbruch
+\[
+\frac{Ak}{Bk}
+\]
+berechnet werden.
+
+
+\subsubsection{Lösung}
+Die Berechnung von $A_k, B_k$ \eqref{0f1:math:matrix:ende:eq} kann man auch ohne die Matrizenschreibweise aufschreiben: \cite{0f1:wiki-fraction}
+\begin{itemize}
+\item Startbedingungen:
+\begin{align*}
+A_{-1} &= 0 & A_0 &= a_0 \\
+B_{-1} &= 1 & B_0 &= 1
+\end{align*}
+\item Schritt $k\to k+1$:
+\[
+\begin{aligned}
+\label{0f1:math:loesung:eq}
+k &\rightarrow k + 1:
+&
+A_{k+1} &= A_{k-1} \cdot b_k + A_k \cdot a_k \\
+&&
+B_{k+1} &= B_{k-1} \cdot b_k + B_k \cdot a_k
+\end{aligned}
+\]
+\item
+Näherungsbruch: \qquad$\displaystyle\frac{A_k}{B_k}$
+\end{itemize}
+
+Ein grosser Vorteil dieser Umsetzung \ref{0f1:listing:kettenbruchRekursion} ist, dass im Vergleich zum Code \ref{0f1:listing:kettenbruchIterativ} eine Division gespart werden kann und somit weniger Rundungsfehler entstehen können.
+
+%Code
+\lstinputlisting[style=C,float,caption={Rekursionsformel für Kettenbruch.},label={0f1:listing:kettenbruchRekursion}, firstline=8]{papers/0f1/listings/kettenbruchRekursion.c}
\ No newline at end of file diff --git a/buch/papers/0f1/teil3.tex b/buch/papers/0f1/teil3.tex index 25472cb..2855e26 100644 --- a/buch/papers/0f1/teil3.tex +++ b/buch/papers/0f1/teil3.tex @@ -1,40 +1,64 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{0f1:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{0f1:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - +%
+% teil3.tex -- Resultate und Ausblick
+%
+% (c) 2022 Fabian Dünki, Hochschule Rapperswil
+%
+\section{Auswertung
+\label{0f1:section:teil3}}
+\rhead{Resultate}
+Im Verlauf des Seminares hat sich gezeigt,
+das ein einfacher mathematischer Algorithmus zu implementieren gar nicht so einfach ist.
+So haben alle drei umgesetzten Ansätze Probleme mit grossen negativen $z$ in der Funktion $\mathstrut_0F_1(;c;z)$.
+Ebenso kann festgestellt werden,dass je grösser der Wert $z$ in $\mathstrut_0F_1(;c;z)$ wird, desto mehr weichen die berechneten Resultate von den Erwarteten ab. \cite{0f1:wolfram-0f1}
+
+\subsection{Konvergenz
+\label{0f1:subsection:konvergenz}}
+Es zeigt sich in Abbildung \ref{0f1:ausblick:plot:airy:konvergenz}, dass schon nach drei Iterationen ($k = 3$) die Funktionen schon genaue Resultate im Bereich von -2 bis 2 liefert. Ebenso kann festgestellt werden, dass der Kettenbruch schneller konvergiert und im positiven Bereich sogar mit der Referenzfunktion $Ai(x)$ übereinstimmt. Da die Rekursionsformel \ref{0f1:listing:kettenbruchRekursion} eine Abwandlung des Kettenbruches ist, verhalten sich die Funktionen in diesem Fall gleich.
+
+Erst wenn mehrere Durchläufe gemacht werden, um die Genauigkeit zu verbessern, ist der Kettenbruch den anderen zwei Algorithmen, bezüglich Konvergenz überlegen.
+Interessant ist auch, dass die Rekursionsformel nahezu gleich schnell wie die Potenzreihe konvergiert, aber sich danach einschwingt. Dieses Verhalten ist auch bei grösseren $z$ zu beobachten, allerdings ist dann die Differenz zwischen dem ersten lokalen Minimum von k bis zum Abbruch kleiner.
+\ref{0f1:ausblick:plot:konvergenz:positiv}
+Dieses Phänomen ist auf die Lösung der Rekursionsformel zurück zu führen.\ref{0f1:math:loesung:eq} Da im Gegensatz die ganz kleinen Werte nicht zu einer Konvergenz wie beim Kettenbruch führen, sondern sich noch eine Zeit lang durch die Multiplikation aufschwingen.
+
+Ist $z$ negativ wie im Abbild \ref{0f1:ausblick:plot:konvergenz:negativ}, führt dies zu einer Gegenseitigen Kompensation von negativen und positiven Termen so bricht die Rekursionsformel hier zusammen mit der Potenzreihe ab.
+Die ansteigende Differenz mit anschliessender, ist aufgrund der sich alternierenden Termen mit wechselnden Vorzeichens zu erklären.
+
+\subsection{Stabilität
+\label{0f1:subsection:Stabilitaet}}
+Verändert sich der Wert von z in $\mathstrut_0F_1(;c;z)$ gegen grössere positive Werte, wie zum Beispiel $c = 800$ liefert die Kettenbruch-Funktion \ref{0f1:listing:kettenbruchIterativ} \verb+inf+ zurück. Dies könnte durch ein Abbruchkriterien abgefangen werden. Allerdings würde das, bei grossen Werten zulasten der Genauigkeit gehen. Trotzdem könnte, je nach Anwendung, auf ein paar Nachkommastellen verzichtet werden.
+
+Wohingegen die Potenzreihe \ref{0f1:listing:potenzreihe} das Problem hat, dass je mehr Terme berechnet werden, desto schneller wächst die Fakultät und irgendwann gibt es eine Bereichsüberschreitung von \verb+double+. Schlussendlich gibt das Unterprogramm das Resultat \verb+-nan(ind)+ zurück.
+Die Rekursionformel \ref{0f1:listing:kettenbruchRekursion} liefert für sehr grosse positive Werte die genausten Ergebnisse, verglichen mit der GNU Scientific Library. Wie schon vermutet ist die Rekursionsformel, im positivem Bereich, der stabilste Algorithmus. Um die Stabilität zu gewährleisten, muss wie in \ref{0f1:ausblick:plot:konvergenz:positiv} dargestellt, die Iterationstiefe $k$ genug gross gewählt werden.
+
+Im negativem Bereich sind alle gewählten und umgesetzten Ansätze instabil. Grund dafür ist die Fakultät im Nenner, was zum Phänomen der Auslöschung führt.\cite{0f1:SeminarNumerik} Schön zu beobachten ist dies in der Abbildung \ref{0f1:ausblick:plot:airy:stabilitaet} mit der Airy-Funktion als Test. So sind sowohl der Kettenbruch, als auch die Rekursionsformel bis ungefähr $\frac{-15^3}{9}$ stabil. Dies macht auch Sinn, da beide auf der gleichen mathematischen Grundlage basieren. Danach verhält sich allerdings die Instabilität unterschiedlich. Das unterschiedliche Verhalten kann damit erklärt werden, dass beim Kettenbruch jeweils eine zusätzliche Division stattfindet. Diese Unterschiede sind auch in Abbildung \ref{0f1:ausblick:plot:konvergenz:positiv} festzustellen.
+
+
+
+\begin{figure}
+ \centering
+ \includegraphics[width=0.8\textwidth]{papers/0f1/images/konvergenzAiry.pdf}
+ \caption{Konvergenz nach drei Iterationen, dargestellt anhand der Airy Funktion zu den Anfangsbedingungen $Ai(0)=1$ und $Ai'(0)=0$.
+ \label{0f1:ausblick:plot:airy:konvergenz}}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[width=0.8\textwidth]{papers/0f1/images/konvergenzPositiv.pdf}
+ \caption{Konvergenz: Logarithmisch dargestellte Differenz vom erwarteten Endresultat.
+ \label{0f1:ausblick:plot:konvergenz:positiv}}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[width=0.8\textwidth]{papers/0f1/images/konvergenzNegativ.pdf}
+ \caption{Konvergenz: Logarithmisch dargestellte Differenz vom erwarteten Endresultat.
+ \label{0f1:ausblick:plot:konvergenz:negativ}}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[width=1\textwidth]{papers/0f1/images/stabilitaet.pdf}
+ \caption{Stabilität der 3 Algorithmen verglichen mit der Referenz Funktion $Ai(x)$.
+ \label{0f1:ausblick:plot:airy:stabilitaet}}
+\end{figure}
+
diff --git a/buch/papers/common/addpapers.tex b/buch/papers/common/addpapers.tex index dd2b07a..eb353d7 100644 --- a/buch/papers/common/addpapers.tex +++ b/buch/papers/common/addpapers.tex @@ -3,7 +3,6 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\input{papers/000template/main.tex} \input{papers/lambertw/main.tex} \input{papers/fm/main.tex} \input{papers/parzyl/main.tex} diff --git a/buch/papers/common/paperlist b/buch/papers/common/paperlist index d4e5c20..f607279 100644 --- a/buch/papers/common/paperlist +++ b/buch/papers/common/paperlist @@ -1,4 +1,3 @@ -000template lambertw fm parzyl diff --git a/buch/papers/dreieck/main.tex b/buch/papers/dreieck/main.tex index 75ba410..d7bc769 100644 --- a/buch/papers/dreieck/main.tex +++ b/buch/papers/dreieck/main.tex @@ -3,19 +3,21 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Dreieckstest und Beta-Funktion\label{chapter:dreieck}} -\lhead{Dreieckstest und Beta-Funktion} +\chapter{$\int P(t) e^{-t^2} \,dt$ in geschlossener Form? +\label{chapter:dreieck}} +\lhead{Integrierbarkeit in geschlossener Form} \begin{refsection} \chapterauthor{Andreas Müller} \noindent -Mit dem Dreieckstest kann man feststellen, wie gut ein Geruchs- -oder Geschmackstester verschiedene Gerüche oder Geschmäcker -unterscheiden kann. -Seine wahrscheinlichkeitstheoretische Erklärung benötigt die Beta-Funktion, -man kann die Beta-Funktion als durchaus als die mathematische Grundlage -der Weindegustation -bezeichnen. +Der Risch-Algorithmus erlaubt, eine definitive Antwort darauf zu geben, +\index{Risch-Algorithmus}% +\index{elementare Stammfunktion}% +ob eine elementare Funktion eine Stammfunktion in geschlossener Form hat. +Der Algorithmus ist jedoch ziemlich kompliziert. +In diesem Kapitel soll ein spezieller Fall mit Hilfe der Theorie der +orthogonale Polynome, speziell der Hermite-Polynome, behandelt werden, +wie er in der Arbeit \cite{dreieck:polint} untersucht wurde. \input{papers/dreieck/teil0.tex} \input{papers/dreieck/teil1.tex} diff --git a/buch/papers/dreieck/references.bib b/buch/papers/dreieck/references.bib index d2bbe08..47bd865 100644 --- a/buch/papers/dreieck/references.bib +++ b/buch/papers/dreieck/references.bib @@ -4,32 +4,12 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{dreieck:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@article{dreieck:polint, + author = { George Stoica }, + title = { Polynomials and Integration in Finite Terms }, + journal = { Amer. Math. Monthly }, + volume = 129, + year = 2022, + number = 1, + pages = {80--81} } - -@book{dreieck:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{dreieck:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} -} - diff --git a/buch/papers/dreieck/teil0.tex b/buch/papers/dreieck/teil0.tex index bcf2cf8..f9affe7 100644 --- a/buch/papers/dreieck/teil0.tex +++ b/buch/papers/dreieck/teil0.tex @@ -3,7 +3,48 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Testprinzip\label{dreieck:section:testprinzip}} -\rhead{Testprinzip} +\section{Problemstellung\label{dreieck:section:problemstellung}} +\rhead{Problemstellung} +Es ist bekannt, dass das Fehlerintegral +\[ +\frac{1}{\sqrt{2\pi}\sigma} \int_{-\infty}^x e^{-\frac{t^2}{2\sigma}}\,dt +\] +nicht in geschlossener Form dargestellt werden kann. +Mit der in Kapitel~\ref{buch:chapter:integral} skizzierten Theorie von +Liouville und dem Risch-Algorithmus kann dies strengt gezeigt werden. +Andererseits gibt es durchaus Integranden, die $e^{-t^2}$ enthalten, +für die eine Stammfunktion in geschlossener Form gefunden werden kann. +Zum Beispiel folgt aus der Ableitung +\[ +\frac{d}{dt} e^{-t^2} += +-2te^{-t^2} +\] +die Stammfunktion +\[ +\int te^{-t^2}\,dt += +-\frac12 e^{-t^2}. +\] +Leitet man $e^{-t^2}$ zweimal ab, erhält man +\[ +\frac{d^2}{dt^2} e^{-t^2} += +(4t^2-2) e^{-t^2} +\qquad\Rightarrow\qquad +\int (t^2-{\textstyle\frac12}) e^{-t^2}\,dt += +{\textstyle\frac14} +e^{-t^2}. +\] +Es gibt also viele weitere Polynome $P(t)$, für die der Integrand +$P(t)e^{-t^2}$ eine Stammfunktion in geschlossener Form hat. +Damit stellt sich jetzt das folgende allgemeine Problem. + +\begin{problem} +\label{dreieck:problem} +Für welche Polynome $P(t)$ hat der Integrand $P(t)e^{-t^2}$ +eine elementare Stammfunktion? +\end{problem} diff --git a/buch/papers/dreieck/teil1.tex b/buch/papers/dreieck/teil1.tex index 4abe2e1..45c1a23 100644 --- a/buch/papers/dreieck/teil1.tex +++ b/buch/papers/dreieck/teil1.tex @@ -3,9 +3,92 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Ordnungsstatistik und Beta-Funktion -\label{dreieck:section:ordnungsstatistik}} -\rhead{} +\section{Hermite-Polynome +\label{dreieck:section:hermite-polynome}} +\rhead{Hermite-Polyome} +In Abschnitt~\ref{dreieck:section:problemstellung} hat sich schon angedeutet, +dass die Polynome, die man durch Ableiten von $e^{-t^2}$ erhalten +kann, bezüglich des gestellten Problems besondere Eigenschaften +haben. +Zunächst halten wir fest, dass die Ableitung einer Funktion der Form +$P(t)e^{-t^2}$ mit einem Polynom $P(t)$ +\begin{equation} +\frac{d}{dt} P(t)e^{-t^2} += +P'(t)e^{-t^2} -2tP(t)e^{-t^2} += +(P'(t)-2tP(t)) e^{-t^2} +\label{dreieck:eqn:ableitung} +\end{equation} +ist. +Insbesondere hat die Ableitung wieder die Form $Q(t)e^{-t^2}$ +mit einem Polynome $Q(t)$, welches man auch als +\[ +Q(t) += +e^{t^2}\frac{d}{dt}P(t)e^{-t^2} +\] +erhalten kann. +Die Polynome, die man aus der Funktion $H_0(t)=e^{-t^2}$ durch +Ableiten erhalten kann, wurden bereits in +Abschnitt~\ref{buch:orthogonalitaet:section:rodrigues} +bis auf ein Vorzeichen hergeleitet, sie heissen die Hermite-Polynome +\index{Hermite-Polynome}% +und es gilt +\[ +H_n(t) += +(-1)^n +e^{t^2} \frac{d^n}{dt^n} e^{-t^2}. +\] +Das Vorzeichen dient dazu sicherzustellen, dass der Leitkoeffizient +immer $1$ ist. +Das Polynom $H_n(t)$ hat den Grad $n$. + +In Abschnitt wurde auch gezeigt, dass die Polynome $H_n(t)$ +bezüglich des Skalarproduktes +\[ +\langle f,g\rangle_{w} += +\int_{-\infty}^\infty f(t)g(t)e^{-t^2}\,dt, +\qquad +w(t)=e^{-t^2}, +\] +orthogonal sind. +Ausserdem folgt aus \eqref{dreieck:eqn:ableitung} +die Rekursionsbeziehung +\begin{equation} +H_{n}(t) += +2tH_{n-1}(t) +- +H_{n-1}'(t) +\label{dreieck:eqn:rekursion} +\end{equation} +für $n>0$. + +Im Hinblick auf die Problemstellung ist jetzt die Frage interessant, +ob die Integranden $H_n(t)e^{-t^2}$ eine Stammfunktion in geschlossener +Form haben. +Mit Hilfe der Rekursionsbeziehung~\eqref{dreieck:eqn:rekursion} +kann man für $n>0$ unmittelbar verifizieren, dass +\begin{align*} +\int H_n(t)e^{-t^2}\,dt +&= +\int \bigl( 2tH_{n-1}(t) - H'_{n-1}(t)\bigr)e^{-t^2}\,dt +\\ +&= +-\int \bigl( \exp'(-t^2) H_{n-1}(t) + H'_{n-1}(t)\bigr)e^{-t^2}\,dt +\\ +&= +-\int \bigl( e^{-t^2}H_{n-1}(t)\bigr)' \,dt += +-e^{-t^2}H_{n-1}(t) +\end{align*} +ist. +Für $n>0$ hat also $H_n(t)e^{-t^2}$ eine elementare Stammfunktion. +Die Hermite-Polynome sind also Lösungen für das +Problem~\ref{dreieck:problem}. diff --git a/buch/papers/dreieck/teil2.tex b/buch/papers/dreieck/teil2.tex index 83ea3cb..8e89f6a 100644 --- a/buch/papers/dreieck/teil2.tex +++ b/buch/papers/dreieck/teil2.tex @@ -3,7 +3,113 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Wahrscheinlichkeiten im Dreieckstest -\label{dreieck:section:wahrscheinlichkeiten}} -\rhead{Wahrscheinlichkeiten} +\section{Beliebige Polynome +\label{dreieck:section:beliebig}} +\rhead{Beliebige Polynome} +Im Abschnitt~\ref{dreieck:section:hermite-polynome} wurden die +Hermite-Polynome $H_n(t)$ mit $n>0$ als Lösungen des gestellten +Problems erkannt. +Eine Linearkombination von solchen Polynomen hat natürlich +ebenfalls eine elementare Stammfunktion. +Das Problem kann daher neu formuliert werden: + +\begin{problem} +\label{dreieck:problem2} +Welche Polynome $P(t)$ lassen sich aus den Hermite-Polynomen +$H_n(t)$ mit $n>0$ linear kombinieren? +\end{problem} + +Sei also +\[ +P(t) = p_0 + p_1t + \ldots + p_{n-1}t^{n-1} + p_nt^n +\] +ein beliebiges Polynom vom Grad $n$. +Eine elementare Stammfunktion von $P(t)e^{-t^2}$ existiert sicher, +wenn sich $P(t)$ aus den Funktionen $H_n(t)$ mit $n>0$ linear +kombinieren lässt. +Gesucht ist also zunächst eine Darstellung von $P(t)$ als Linearkombination +von Hermite-Polynomen. + +\begin{lemma} +Jedes Polynome $P(t)$ vom Grad $n$ lässt sich auf eindeutige Art und +Weise als Linearkombination +\begin{equation} +P(t) = a_0H_0(t) + a_1H_1(t) + \ldots + a_nH_n(t) += +\sum_{k=0}^n a_nH_n(t) +\label{dreieck:lemma} +\end{equation} +von Hermite-Polynomen schreiben. +\end{lemma} + +\begin{proof}[Beweis] +Zunächst halten wir fest, dass aus der +Rekursionsformel~\eqref{dreieck:eqn:rekursion} +folgt, dass der Leitkoeffizient bei jedem Rekursionsschnitt +mit $2$ multipliziert wird. +Der Leitkoeffizient von $H_n(t)$ ist also $2^n$. + +Wir führen den Beweis mit vollständiger Induktion. +Für $n=0$ ist $P(t)=p_0 = p_0 H_0(t)$ als Linearkombination von +Hermite-Polynomen darstellbar, dies ist die Induktionsverankerung. + +Wir nehmen jetzt im Sinne der Induktionsannahme an, +dass sich ein Polynom vom Grad $n-1$ als +Linearkombination der Polynome $H_0(t),\dots,H_{n-1}(t)$ schreiben +lässt und untersuchen ein Polynom $P(t)$ vom Grad $n$. +Da der Leitkoeffizient des Polynoms $H_n(t)$ ist $2^n$, ist zerlegen +wir +\[ +P(t) += +\underbrace{\biggl(P(t) - \frac{p_n}{2^n} H_n(t)\biggr)}_{\displaystyle = Q(t)} ++ +\frac{p_n}{2^n} H_n(t). +\] +Das Polynom $Q(t)$ hat Grad $n-1$, besitzt also nach Induktionsannahme +eine Darstellung +\[ +Q(t) = a_0H_0(t)+a_1H_1(t)+\ldots+a_{n-1}H_{n-1}(t) +\] +als Linearkombination der Polynome $H_0(t),\dots,H_{n-1}(t)$. +Somit ist +\[ +P(t) += a_0H_0(t)+a_1H_1(t)+\ldots+a_{n-1}H_{n-1}(t) + +\frac{p_n}{2^n} H_n(t) +\] +eine Darstellung von $P(t)$ als Linearkombination der Polynome +$H_0(t),\dots,H_n(t)$. +Damit ist der Induktionsschritt vollzogen und das Lemma für alle +$n$ bewiesen. +\end{proof} + +\begin{satz} +\label{dreieck:satz1} +Die Funktion $P(t)e^{-t^2}$ hat genau dann eine elementare Stammfunktion, +wenn in der Darstellung~\eqref{dreieck:lemma} +von $P(t)$ als Linearkombination von Hermite-Polynomen $a_0=0$ gilt. +\end{satz} + +\begin{proof}[Beweis] +Es ist +\begin{align*} +\int P(t)e^{-t^2}\,dt +&= +a_0\int e^{-t^2}\,dt ++ +\int +\sum_{k=1} a_kH_k(t)\,dt +\\ +&= +a_0 +\frac{\sqrt{\pi}}2 +\operatorname{erf}(t) ++ +\sum_{k=1} a_k\int H_k(t)\,dt. +\end{align*} +Da die Integrale in der Summe alle elementar darstellbar sind, +ist das Integral genau dann elementar, wenn $a_0=0$ ist. +\end{proof} + diff --git a/buch/papers/dreieck/teil3.tex b/buch/papers/dreieck/teil3.tex index e2dfd6b..c0c046a 100644 --- a/buch/papers/dreieck/teil3.tex +++ b/buch/papers/dreieck/teil3.tex @@ -3,8 +3,75 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Erweiterungen -\label{dreieck:section:erweiterungen}} -\rhead{Erweiterungen} +\section{Integralbedingung +\label{dreieck:section:integralbedingung}} +\rhead{Lösung} +Die Tatsache, dass die Hermite-Polynome orthogonal sind, erlaubt, das +Kriterium von Satz~\ref{dreieck:satz1} in einer besonders attraktiven +Integralform zu formulieren. + +Aus den Polynomen $H_n(t)$ lassen sich durch Normierung die +\index{orthogonale Polynome}% +\index{Polynome, orthogonale}% +orthonormierten Polynome +\[ +\tilde{H}_n(t) += +\frac{1}{\| H_n\|_w} H_n(t) +\qquad\text{mit}\quad +\|H_n\|_w^2 += +\int_{-\infty}^\infty H_n(t)e^{-t^2}\,dt +\] +bilden. +Da diese Polynome eine orthonormierte Basis des Vektorraums der Polynome +bilden, kann die gesuchte Zerlegung eines Polynoms $P(t)$ auch mit +Hilfe des Skalarproduktes gefunden werden: +\begin{align*} +P(t) +&= +\sum_{k=1}^n +\langle \tilde{H}_k, P\rangle_w +\tilde{H}_k(t) += +\sum_{k=1}^n +\biggl\langle \frac{H_k}{\|H_k\|_w}, P\biggr\rangle_w +\frac{H_k(t)}{\|H_k\|_w} += +\sum_{k=1}^n +\underbrace{ +\frac{ \langle H_k, P\rangle_w }{\|H_k\|_w^2} +}_{\displaystyle =a_k} +H_k(t). +\end{align*} +Die Darstellung von $P(t)$ als Linearkombination von Hermite-Polynomen +hat somit die Koeffizienten +\[ +a_k = \frac{\langle H_k,P\rangle_w}{\|H_k\|_w^2}. +\] +Aus dem Kriterium $a_0=0$ dafür, dass eine elementare Stammfunktion +von $P(t)e^{-t^2}$ existiert, wird daher die Bedingung, dass +$\langle H_0,P\rangle_w=0$ ist. +Da $H_0(t)=1$ ist, folgt als Bedingung +\[ +a_0 += +\langle H_0,P\rangle_w += +\int_{-\infty}^\infty P(t) e^{-t^2}\,dt += +0. +\] + +\begin{satz} +Ein Integrand der Form $P(t)e^{-t^2}$ mit einem Polynom $P(t)$ +hat genau dann eine elementare Stammfunktion, wenn +\[ +\int_{-\infty}^\infty P(t)e^{-t^2}\,dt = 0 +\] +ist. +\end{satz} + + diff --git a/buch/papers/ellfilter/Makefile.inc b/buch/papers/ellfilter/Makefile.inc index 8f20278..97e4089 100644 --- a/buch/papers/ellfilter/Makefile.inc +++ b/buch/papers/ellfilter/Makefile.inc @@ -3,12 +3,11 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-ellfilter = \ - papers/ellfilter/packages.tex \ - papers/ellfilter/main.tex \ - papers/ellfilter/references.bib \ - papers/ellfilter/teil0.tex \ - papers/ellfilter/teil1.tex \ - papers/ellfilter/teil2.tex \ - papers/ellfilter/teil3.tex - +dependencies-ellfilter = \ + papers/ellfilter/packages.tex \ + papers/ellfilter/main.tex \ + papers/ellfilter/references.bib \ + papers/ellfilter/einleitung.tex \ + papers/ellfilter/tschebyscheff.tex \ + papers/ellfilter/jacobi.tex \ + papers/ellfilter/elliptic.tex diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex new file mode 100644 index 0000000..37fd89f --- /dev/null +++ b/buch/papers/ellfilter/einleitung.tex @@ -0,0 +1,56 @@ +\section{Einleitung} + +% Lineare filter + +% Filter, Signalverarbeitung + + +Der womöglich wichtigste Filtertyp ist das Tiefpassfilter. +Dieses soll im Durchlassbereich unter der Grenzfrequenz $\Omega_p$ unverstärkt durchlassen und alle anderen Frequenzen vollständig auslöschen. + +% Bei der Implementierung von Filtern + +In der Elektrotechnik führen Schaltungen mit linearen Bauelementen wie Kondensatoren, Spulen und Widerständen immer zu linearen zeitinvarianten Systemen (LTI-System von englich \textit{time-invariant system}). +Die Übertragungsfunktion im Frequenzbereich $|H(\Omega)|$ eines solchen Systems ist dabei immer eine rationale Funktion, also eine Division von zwei Polynomen. +Die Polynome habe dabei immer reelle oder komplex-konjugierte Nullstellen. + + +\begin{equation} \label{ellfilter:eq:h_omega} + | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p} +\end{equation} + +$\Omega = 2 \pi f$ ist die analoge Frequenz + + +% Linear filter +Damit das Filter implementierbar und stabil ist, muss $H(\Omega)^2$ eine rationale Funktion sein, deren Nullstellen und Pole auf der linken Halbebene liegen. + +$N \in \mathbb{N} $ gibt dabei die Ordnung des Filters vor, also die maximale Anzahl Pole oder Nullstellen. + +Damit ein Filter die Passband Kondition erfüllt muss $|F_N(w)| \leq 1 \forall |w| \leq 1$ und für $|w| \geq 1$ sollte die Funktion möglichst schnell divergieren. +Eine einfaches Polynom, dass das erfüllt, erhalten wir wenn $F_N(w) = w^N$. +Tatsächlich erhalten wir damit das Butterworth Filter, wie in Abbildung \ref{ellfilter:fig:butterworth} ersichtlich. +\begin{figure} + \centering + \input{papers/ellfilter/python/F_N_butterworth.pgf} + \caption{$F_N$ für Butterworth filter. Der grüne Bereich definiert die erlaubten Werte für alle $F_N$-Funktionen.} + \label{ellfilter:fig:butterworth} +\end{figure} + +wenn $F_N(w)$ eine rationale Funktion ist, ist auch $H(\Omega)$ eine rationale Funktion und daher ein lineares Filter. %proof? + +\begin{align} + F_N(w) & = + \begin{cases} + w^N & \text{Butterworth} \\ + T_N(w) & \text{Tschebyscheff, Typ 1} \\ + [k_1 T_N (k^{-1} w^{-1})]^{-1} & \text{Tschebyscheff, Typ 2} \\ + R_N(w, \xi) & \text{Elliptisch (Cauer)} \\ + \end{cases} +\end{align} + +Mit der Ausnahme vom Butterworth filter sind alle Filter nach speziellen Funktionen benannt. +Alle diese Filter sind optimal für unterschiedliche Anwendungsgebiete. +Das Butterworth-Filter, zum Beispiel, ist maximal flach im Durchlassbereich. +Das Tschebyscheff-1 Filter sind maximal steil für eine definierte Welligkeit im Durchlassbereich, währendem es im Sperrbereich monoton abfallend ist. +Es scheint so als sind gewisse Eigenschaften dieser speziellen Funktionen verantwortlich für die Optimalität dieser Filter. diff --git a/buch/papers/ellfilter/elliptic.tex b/buch/papers/ellfilter/elliptic.tex new file mode 100644 index 0000000..88bfbfe --- /dev/null +++ b/buch/papers/ellfilter/elliptic.tex @@ -0,0 +1,92 @@ +\section{Elliptische rationale Funktionen} + +Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen +\begin{align} + R_N(\xi, w) &= \cd \left(N~f_1(\xi)~\cd^{-1}(w, 1/\xi), f_2(\xi)\right) \\ + &= \cd \left(N~\frac{K_1}{K}~\cd^{-1}(w, k), k_1)\right) , \quad k= 1/\xi, k_1 = 1/f(\xi) \\ + &= \cd \left(N~K_1~z , k_1 \right), \quad w= \cd(z K, k) +\end{align} + + +sieht ähnlich aus wie die trigonometrische Darstellung der Tschebyschef-Polynome \eqref{ellfilter:eq:chebychef_polynomials} +Anstelle vom Kosinus kommt hier die $\cd$-Funktion zum Einsatz. +Die Ordnungszahl $N$ kommt auch als Faktor for. +Zusätzlich werden noch zwei verschiedene elliptische Module $k$ und $k_1$ gebraucht. + + + +Sinus entspricht $\sn$ + +Damit die Nullstellen an ähnlichen Positionen zu liegen kommen wie bei den Tschebyscheff-Polynomen, muss die $\cd$-Funktion gewählt werden. + +Die $\cd^{-1}(w, k)$-Funktion ist um $K$ verschoben zur $\sn^{-1}(w, k)$-Funktion, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd}. +\begin{figure} + \centering + \input{papers/ellfilter/tikz/cd.tikz.tex} + \caption{ + $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$. + Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch. + } + \label{ellfilter:fig:cd} +\end{figure} +Auffallend ist, dass sich alle Nullstellen und Polstellen um $K$ verschoben haben. + +Durch das Konzept vom fundamentalen Rechteck, siehe Abbildung \ref{ellfilter:fig:fundamental_rectangle} können für alle inversen Jaccobi elliptischen Funktionen die Positionen der Null- und Polstellen anhand eines Diagramms ermittelt werden. +Der erste Buchstabe bestimmt die Position der Nullstelle und der zweite Buchstabe die Polstelle. +\begin{figure} + \centering + \input{papers/ellfilter/tikz/fundamental_rectangle.tikz.tex} + \caption{ + Fundamentales Rechteck der inversen Jaccobi elliptischen Funktionen. + } + \label{ellfilter:fig:fundamental_rectangle} +\end{figure} + +Auffallend an der $w = \sn(z, k)$-Funktion ist, dass sich $w$ auf der reellen Achse wie der Kosinus immer zwischen $-1$ und $1$ bewegt, während bei $\mathrm{Im(z) = K^\prime}$ die Werte zwischen $\pm 1/k$ und $\pm \infty$ verlaufen. +Die Funktion hat also Equirippel-Verhalten um $w=0$ und um $w=\pm \infty$. +Falls es möglich ist diese Werte abzufahren im Sti der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equirippel-Verhalten im Durchlass- und Sperrbereich aufweist. + + + +Analog zu Abbildung \ref{ellfilter:fig:arccos2} können wir auch bei den elliptisch rationalen Funktionen die komplexe $z$-Ebene betrachten, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd2}, um die besser zu verstehen. +\begin{figure} + \centering + \input{papers/ellfilter/tikz/cd2.tikz.tex} + \caption{ + $z_1$-Ebene der elliptischen rationalen Funktionen. + Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen passiert. + } + \label{ellfilter:fig:cd2} +\end{figure} +% Da die $\cd^{-1}$-Funktion + + + +\begin{figure} + \centering + \input{papers/ellfilter/python/F_N_elliptic.pgf} + \caption{$F_N$ für ein elliptischs filter.} + \label{ellfilter:fig:elliptic} +\end{figure} + +\subsection{Degree Equation} + +Der $\cd^{-1}$ Term muss so verzogen werden, dass die umgebene $\cd$-Funktion die Nullstellen und Pole trifft. +Dies trifft ein wenn die Degree Equation erfüllt ist. + +\begin{equation} + N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1} +\end{equation} + + +Leider ist das lösen dieser Gleichung nicht trivial. +Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut. + + +\subsection{Polynome?} + +Bei den Tschebyscheff-Polynomen haben wir gesehen, dass die Trigonometrische Formel zu einfachen Polynomen umgewandelt werden kann. +Im gegensatz zum $\cos^{-1}$ hat der $\cd^{-1}$ nicht nur Nullstellen sondern auch Pole. +Somit entstehen bei den elliptischen rationalen Funktionen, wie es der name auch deutet, rationale Funktionen, also ein Bruch von zwei Polynomen. + +Da Transformationen einer rationalen Funktionen mit Grundrechenarten, wie es in \eqref{ellfilter:eq:h_omega} der Fall ist, immer noch rationale Funktionen ergeben, stellt dies kein Problem für die Implementierung dar. diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex new file mode 100644 index 0000000..6a208fa --- /dev/null +++ b/buch/papers/ellfilter/jacobi.tex @@ -0,0 +1,189 @@ +\section{Jacobische elliptische Funktionen} + +%TODO $z$ or $u$ for parameter? + +Für das elliptische Filter wird statt der, für das Tschebyscheff-Filter benutzen Kreis-Trigonometrie die elliptischen Funktionen gebraucht. +Der Begriff elliptische Funktion wird für sehr viele Funktionen gebraucht, daher ist es hier wichtig zu erwähnen, dass es ausschliesslich um die Jacobischen elliptischen Funktionen geht. + +Im Wesentlichen erweitern die Jacobi elliptischen Funktionen die trigonometrische Funktionen für Ellipsen. +Zum Beispiel gibt es analog zum Sinus den elliptischen $\sn(z, k)$. +Im Gegensatz zum den trigonometrischen Funktionen haben die elliptischen Funktionen zwei parameter. +Zum einen gibt es den \textit{elliptische Modul} $k$, der die Exzentrizität der Ellipse parametrisiert. +Zum andern das Winkelargument $z$. +Im Kreis ist der Radius für alle Winkel konstant, bei Ellipsen ändert sich das. +Dies hat zur Folge, dass bei einer Ellipse die Kreisbodenstrecke nicht linear zum Winkel verläuft. +Darum kann hier nicht der gewohnte Winkel verwendet werden. +Das Winkelargument $z$ kann durch das elliptische Integral erster Art +\begin{equation} + z + = + F(\phi, k) + = + \int_{0}^{\phi} + \frac{ + d\theta + }{ + \sqrt{ + 1-k^2 \sin^2 \theta + } + } + = + \int_{0}^{\phi} + \frac{ + dt + }{ + \sqrt{ + (1-t^2)(1-k^2 t^2) + } + } %TODO which is right? are both functions from phi? +\end{equation} +mit dem Winkel $\phi$ in Verbindung liegt. + +Dabei wird das vollständige und unvollständige Elliptische integral unterschieden. +Beim vollständigen Integral +\begin{equation} + K(k) + = + \int_{0}^{\pi / 2} + \frac{ + d\theta + }{ + \sqrt{ + 1-k^2 \sin^2 \theta + } + } +\end{equation} +wird über ein viertel Ellipsenbogen integriert also bis $\phi=\pi/2$ und liefert das Winkelargument für eine Vierteldrehung. +Die Zahl wird oft auch abgekürzt mit $K = K(k)$ und ist für das elliptische Filter sehr relevant. +Alle elliptishen Funktionen sind somit $4K$-periodisch. + +Neben dem $\sn$ gibt es zwei weitere basis-elliptische Funktionen $\cn$ und $\dn$. +Dazu kommen noch weitere abgeleitete Funktionen, die durch Quotienten und Kehrwerte dieser Funktionen zustande kommen. +Insgesamt sind es die zwölf Funktionen +\begin{equation*} + \sn \quad + \ns \quad + \scelliptic \quad + \sd \quad + \cn \quad + \nc \quad + \cs \quad + \cd \quad + \dn \quad + \nd \quad + \ds \quad + \dc. +\end{equation*} + +Die Jacobischen elliptischen Funktionen können mit der inversen Funktion des kompletten elliptischen Integrals erster Art +\begin{equation} + \phi = F^{-1}(z, k) +\end{equation} +definiert werden. Dabei ist zu beachten dass nur das $z$ Argument der Funktion invertiert wird, also +\begin{equation} + z = F(\phi, k) + \Leftrightarrow + \phi = F^{-1}(z, k). +\end{equation} +Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral dargestellt werden: +\begin{equation} + \sin(\phi) + = + \sin \left( F^{-1}(z, k) \right) + = + \sn(z, k) + = + w +\end{equation} + +\begin{equation} + \phi + = + F^{-1}(z, k) + = + \sin^{-1} \big( \sn (z, k ) \big) + = + \sin^{-1} ( w ) +\end{equation} + +\begin{equation} + F(\phi, k) + = + z + = + F( \sin^{-1} \big( \sn (z, k ) \big) , k) + = + F( \sin^{-1} ( w ), k) +\end{equation} + +\begin{equation} + \sn^{-1}(w, k) + = + F(\phi, k), + \quad + \phi = \sin^{-1}(w) +\end{equation} + +\begin{align} + \sn^{-1}(w, k) + & = + \int_{0}^{\phi} + \frac{ + d\theta + }{ + \sqrt{ + 1-k^2 \sin^2 \theta + } + }, + \quad + \phi = \sin^{-1}(w) + \\ + & = + \int_{0}^{w} + \frac{ + dt + }{ + \sqrt{ + (1-t^2)(1-k^2 t^2) + } + } +\end{align} + +Beim $\cos^{-1}(x)$ haben wir gesehen, dass die analytische Fortsetzung bei $x < -1$ und $x > 1$ rechtwinklig in die Komplexen zahlen wandert. +Wenn man das gleiche mit $\sn^{-1}(w, k)$ macht, erkennt man zwei interessante Stellen. +Die erste ist die gleiche wie beim $\cos^{-1}(x)$ nämlich bei $t = \pm 1$. +Der erste Term unter der Wurzel wird dann negativ, während der zweite noch positiv ist, da $k \leq 1$. +\begin{equation} + \frac{ + 1 + }{ + \sqrt{ + (1-t^2)(1-k^2 t^2) + } + } + \in \mathbb{R} + \quad \forall \quad + -1 \leq t \leq 1 +\end{equation} +Die zweite stelle passiert wenn beide Faktoren unter der Wurzel negativ werden, was bei $t = 1/k$ der Fall ist. + + + + +Funktion in relle und komplexe Richtung periodisch + +In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime(k)$-periodisch. + + + +%TODO sn^{-1} grafik + +\begin{figure} + \centering + \input{papers/ellfilter/tikz/sn.tikz.tex} + \caption{ + $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$. + Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch. + } + % \label{ellfilter:fig:cd2} +\end{figure} diff --git a/buch/papers/ellfilter/main.tex b/buch/papers/ellfilter/main.tex index 26aaec1..c58dfa7 100644 --- a/buch/papers/ellfilter/main.tex +++ b/buch/papers/ellfilter/main.tex @@ -8,29 +8,10 @@ \begin{refsection} \chapterauthor{Nicolas Tobler} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/ellfilter/teil0.tex} -\input{papers/ellfilter/teil1.tex} -\input{papers/ellfilter/teil2.tex} -\input{papers/ellfilter/teil3.tex} +\input{papers/ellfilter/einleitung.tex} +\input{papers/ellfilter/tschebyscheff.tex} +\input{papers/ellfilter/jacobi.tex} +\input{papers/ellfilter/elliptic.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/ellfilter/packages.tex b/buch/papers/ellfilter/packages.tex index c94db34..9a550e2 100644 --- a/buch/papers/ellfilter/packages.tex +++ b/buch/papers/ellfilter/packages.tex @@ -8,3 +8,20 @@ % following example %\usepackage{packagename} +% \usepackage[dvipsnames]{xcolor} + +\usetikzlibrary{trees,shapes,decorations} + +\DeclareMathOperator{\sn}{\mathrm{sn}} +\DeclareMathOperator{\ns}{\mathrm{ns}} +\DeclareMathOperator{\scelliptic}{\mathrm{sc}} +\DeclareMathOperator{\sd}{\mathrm{sd}} +\DeclareMathOperator{\cn}{\mathrm{cn}} +\DeclareMathOperator{\nc}{\mathrm{nc}} +\DeclareMathOperator{\cs}{\mathrm{cs}} +\DeclareMathOperator{\cd}{\mathrm{cd}} +\DeclareMathOperator{\dn}{\mathrm{dn}} +\DeclareMathOperator{\nd}{\mathrm{nd}} +\DeclareMathOperator{\ds}{\mathrm{ds}} +\DeclareMathOperator{\dc}{\mathrm{dc}} + diff --git a/buch/papers/ellfilter/presentation/presentation.tex b/buch/papers/ellfilter/presentation/presentation.tex new file mode 100644 index 0000000..7fdb864 --- /dev/null +++ b/buch/papers/ellfilter/presentation/presentation.tex @@ -0,0 +1,413 @@ +\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer} + +% style +\mode<presentation>{ + \usetheme{Frankfurt} +} +%packages +\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-} +\usepackage[english]{babel} +\usepackage{graphicx} +\usepackage{array} + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\usepackage{ragged2e} + +\usepackage{bm} % bold math +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{multirow} % multi row in tables +\usepackage{booktabs} %toprule midrule bottomrue in tables +\usepackage{scrextend} +\usepackage{textgreek} +\usepackage[rgb]{xcolor} + +\usepackage{ marvosym } % \Lightning + +\usepackage{multimedia} % embedded videos + +\usepackage{tikz} +\usepackage{pgf} +\usepackage{pgfplots} + +\usepackage{algorithmic} + +%citations +\usepackage[style=verbose,backend=biber]{biblatex} +\addbibresource{references.bib} + + +%math font +\usefonttheme[onlymath]{serif} + +%Beamer Template modifications +%\definecolor{mainColor}{HTML}{0065A3} % HSR blue +\definecolor{mainColor}{HTML}{D72864} % OST pink +\definecolor{invColor}{HTML}{28d79b} % OST pink +\definecolor{dgreen}{HTML}{38ad36} % Dark green + +%\definecolor{mainColor}{HTML}{000000} % HSR blue +\setbeamercolor{palette primary}{bg=white,fg=mainColor} +\setbeamercolor{palette secondary}{bg=orange,fg=mainColor} +\setbeamercolor{palette tertiary}{bg=yellow,fg=red} +\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic +\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points) +\setbeamercolor{section in toc}{fg=black} % TOC sections +\setbeamertemplate{section in toc}[sections numbered] +\setbeamertemplate{subsection in toc}{% + \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par} + +\setbeamertemplate{itemize items}[circle] +\setbeamertemplate{description item}[circle] +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\beamertemplatenavigationsymbolsempty + +\setbeamercolor{footline}{fg=gray} +\setbeamertemplate{footline}{% + \hfill\usebeamertemplate***{navigation symbols} + \hspace{0.5cm} + \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm} +} + +\usepackage{caption} +\captionsetup{labelformat=empty} + +%Title Page +\title{Elliptische Filter} +\subtitle{Eine Anwendung der Jaccobi elliptischen Funktionen} +\author{Nicolas Tobler} +% \institute{OST Ostschweizer Fachhochschule} +% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}} +\date{\today} + +\input{../packages.tex} + +\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}% + +\newcommand*{\HL}{\textcolor{mainColor}} +\newcommand*{\RD}{\textcolor{red}} +\newcommand*{\BL}{\textcolor{blue}} +\newcommand*{\GN}{\textcolor{dgreen}} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + + +\makeatletter +\newcount\my@repeat@count +\newcommand{\myrepeat}[2]{% + \begingroup + \my@repeat@count=\z@ + \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}% + \endgroup +} +\makeatother + +\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings} + +\begin{document} + + \begin{frame} + \titlepage + \end{frame} + + \begin{frame} + \frametitle{Content} + \tableofcontents + \end{frame} + + \section{Linear Filter} + + \begin{frame} + \frametitle{Lineare Filter} + + + \begin{equation} + | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p} + \end{equation} + + \pause + + \begin{equation} + F_N(w) = w^N + \end{equation} + + \end{frame} + + \begin{frame} + \frametitle{Beispiel: Butterworth Filter} + + \begin{equation} + F_N(w) = w^N + \end{equation} + + \begin{center} + \input{../python/F_N_butterworth.pgf} + \end{center} + + \end{frame} + + + \begin{frame} + \frametitle{Arten von linearen filtern} + + \begin{align*} + F_N(w) & = + \begin{cases} + w^N & \text{Butterworth} \\ + T_N(w) & \text{Tschebyscheff, Typ 1} \\ + [k_1 T_N (k^{-1} w^{-1})]^{-1} & \text{Tschebyscheff, Typ 2} \\ + R_N(w,\xi) & \text{Elliptisch (Cauer)} \\ + \end{cases} + \end{align*} + + \end{frame} + + \section{Tschebycheff Filter} + + \begin{frame} + \frametitle{Tschebyscheff-Polynome} + + + \begin{columns} + \begin{column}[T]{0.35\textwidth} + + \begin{align*} + T_{0}(x)&=1\\ + T_{1}(x)&=x\\ + T_{2}(x)&=2x^{2}-1\\ + T_{3}(x)&=4x^{3}-3x\\ + T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x) + \end{align*} + + \end{column} + \begin{column}[T]{0.65\textwidth} + + \begin{center} + \resizebox{\textwidth}{!}{ + \input{../python/F_N_chebychev2.pgf} + } + \end{center} + + \end{column} + \end{columns} + + + + \end{frame} + + \begin{frame} + \frametitle{Tschebyscheff-Filter} + + \begin{equation*} + | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 T_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p} + \end{equation*} + + \begin{center} + \scalebox{0.9}{ + \input{../python/F_N_chebychev.pgf} + } + \end{center} + + \end{frame} + + + \begin{frame} + \frametitle{Tschebyscheff-Filter} + + Darstellung mit trigonometrischen Funktionen: + + \begin{align} \label{ellfilter:eq:chebychef_polynomials} + T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\ + &= \cos \left(N~z \right), \quad w= \cos(z) + \end{align} + + + \end{frame} + + \begin{frame} + \frametitle{Tschebyscheff-Filter} + + \begin{equation*} + z = \cos^{-1}(w) + \end{equation*} + + \begin{center} + \scalebox{0.85}{ + \input{../tikz/arccos.tikz.tex} + } + \end{center} + + \end{frame} + + \begin{frame} + \frametitle{Tschebyscheff-Filter} + + \begin{equation*} + z_1 = N~\cos^{-1}(w) + \end{equation*} + + \begin{center} + \scalebox{0.85}{ + \input{../tikz/arccos2.tikz.tex} + } + \end{center} + + \end{frame} + + + \section{Jaccobi elliptische Funktionen} + + \begin{frame} + \frametitle{Jaccobi elliptische Funktionen} + + + \begin{equation} + z + = + F(\phi, k) + = + \int_{0}^{\phi} + \frac{ + d\theta + }{ + \sqrt{ + 1-k^2 \sin^2 \theta + } + } + = + \int_{0}^{\phi} + \frac{ + dt + }{ + \sqrt{ + (1-t^2)(1-k^2 t^2) + } + } + \end{equation} + + \begin{equation} + K(k) + = + \int_{0}^{\pi / 2} + \frac{ + d\theta + }{ + \sqrt{ + 1-k^2 \sin^2 \theta + } + } + \end{equation} + + + + \end{frame} + + \begin{frame} + \frametitle{Jaccobi elliptische Funktionen} + + \begin{equation*} + z = \sn^{-1}(w, k) + \end{equation*} + + \begin{center} + \scalebox{0.7}{ + \input{../tikz/sn.tikz.tex} + } + \end{center} + + \end{frame} + + \begin{frame} + \frametitle{Fundamentales Rechteck} + + Nullstelle beim ersten Buchstabe, Polstelle beim zweiten Buchstabe + + \begin{center} + \scalebox{0.8}{ + \input{../tikz/fundamental_rectangle.tikz.tex} + } + \end{center} + + \end{frame} + + + \begin{frame} + \frametitle{Jaccobi elliptische Funktionen} + + \begin{equation*} + z = \cd^{-1}(w, k) + \end{equation*} + + \begin{center} + \scalebox{0.7}{ + \input{../tikz/cd.tikz.tex} + + } + \end{center} + + \end{frame} + + \begin{frame} + \frametitle{Periodizität in realer und imaginärer Richtung} + + \begin{center} + \input{../python/k.pgf} + \end{center} + + + \end{frame} + + \begin{frame} + \frametitle{Elliptisches Filter} + + \begin{equation*} + z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k) + \end{equation*} + + \begin{center} + \scalebox{0.8}{ + \input{../tikz/cd2.tikz.tex} + } + \end{center} + + \end{frame} + + \begin{frame} + \frametitle{Elliptisches Filter} + + \begin{columns} + + \begin{column}[T]{0.5\textwidth} + + \begin{center} + \resizebox{\textwidth}{!}{ + \input{../python/F_N_elliptic.pgf} + } + \end{center} + + \end{column} + \begin{column}[T]{0.5\textwidth} + + \begin{center} + \resizebox{\textwidth}{!}{ + \input{../python/elliptic.pgf} + } + \end{center} + + \end{column} + \end{columns} + + \end{frame} + + \begin{frame} + \frametitle{Gradgleichung} + + \begin{equation} + N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1} + \end{equation} + + \end{frame} + + \end{document} diff --git a/buch/papers/ellfilter/python/F_N_butterworth.pgf b/buch/papers/ellfilter/python/F_N_butterworth.pgf new file mode 100644 index 0000000..857e363 --- /dev/null +++ b/buch/papers/ellfilter/python/F_N_butterworth.pgf @@ -0,0 +1,1083 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{0.000000,0.501961,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.647059,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.630330in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.146442in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.146442in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.146442in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.146442in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.25}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.662555in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.662555in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.662555in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.50}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.178667in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.178667in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.178667in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.178667in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.694779in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.694779in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.694779in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.210892in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.210892in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.210892in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.210892in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.25}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.727004in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.727004in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.50}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.178667in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.987065in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.987065in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.987065in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=0.938840in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.425362in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{1.425362in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=1.377137in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.863658in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.863658in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{1.863658in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=1.815433in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.300082in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle F^2_N(w)\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.548970in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.549574in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.550580in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.551989in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.553800in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.556013in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.558629in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.561648in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.565069in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.568893in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.573119in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.577747in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.582778in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.588211in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.594047in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.600286in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.606927in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.613970in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.621416in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.629264in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.637515in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.646168in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.655224in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.664682in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.674543in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.684806in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.695471in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.706539in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.718010in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.729883in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.742159in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.754837in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.767917in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.781400in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.795285in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.809573in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.824264in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.839357in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.854852in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.870750in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.887050in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.903753in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.920858in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.938366in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.956276in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.974589in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.993304in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{1.012421in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{1.031941in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{1.051864in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{1.072189in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{1.092917in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{1.114047in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{1.135579in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{1.157514in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{1.179851in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{1.202591in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{1.225734in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{1.249279in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{1.273226in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{1.297576in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.322328in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.347483in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.373040in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.399000in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.452126in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.479294in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.506863in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.534835in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{1.563210in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{1.591987in}}% +\pgfpathlineto{\pgfqpoint{2.913736in}{1.621166in}}% +\pgfpathlineto{\pgfqpoint{2.945016in}{1.650748in}}% +\pgfpathlineto{\pgfqpoint{2.976295in}{1.680733in}}% +\pgfpathlineto{\pgfqpoint{3.007575in}{1.711120in}}% +\pgfpathlineto{\pgfqpoint{3.038854in}{1.741909in}}% +\pgfpathlineto{\pgfqpoint{3.070134in}{1.773101in}}% +\pgfpathlineto{\pgfqpoint{3.101413in}{1.804696in}}% +\pgfpathlineto{\pgfqpoint{3.132693in}{1.836692in}}% +\pgfpathlineto{\pgfqpoint{3.163972in}{1.869092in}}% +\pgfpathlineto{\pgfqpoint{3.195252in}{1.901894in}}% +\pgfpathlineto{\pgfqpoint{3.226531in}{1.935098in}}% +\pgfpathlineto{\pgfqpoint{3.257811in}{1.968705in}}% +\pgfpathlineto{\pgfqpoint{3.289091in}{2.002714in}}% +\pgfpathlineto{\pgfqpoint{3.320370in}{2.037126in}}% +\pgfpathlineto{\pgfqpoint{3.351650in}{2.071940in}}% +\pgfpathlineto{\pgfqpoint{3.382929in}{2.107156in}}% +\pgfpathlineto{\pgfqpoint{3.414209in}{2.142776in}}% +\pgfpathlineto{\pgfqpoint{3.445488in}{2.178797in}}% +\pgfpathlineto{\pgfqpoint{3.476768in}{2.215221in}}% +\pgfpathlineto{\pgfqpoint{3.508047in}{2.252048in}}% +\pgfpathlineto{\pgfqpoint{3.539327in}{2.289277in}}% +\pgfpathlineto{\pgfqpoint{3.561409in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.548770in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.548773in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.548781in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.548798in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.548829in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.548880in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.548958in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.549072in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.549231in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.549445in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.549727in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.550088in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.550544in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.551108in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.551796in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.552627in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.553618in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.554789in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.556160in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.557753in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.559591in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.561697in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.564096in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.566815in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.569880in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.573320in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.577165in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.581444in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.586189in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.591434in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.597211in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.603556in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.610505in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.618095in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.626364in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.635351in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.645098in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.655645in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.667035in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.679313in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.692523in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.706710in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.721923in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.738209in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.755618in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.774200in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{0.794006in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{0.815091in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{0.837506in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{0.861307in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{0.886550in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{0.913292in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{0.941592in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.971508in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{1.003102in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{1.036434in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{1.071567in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{1.108565in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{1.147494in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{1.188418in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.231405in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.276523in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.323841in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.373430in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.479708in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.536544in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.595942in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.657980in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{1.722735in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{1.790285in}}% +\pgfpathlineto{\pgfqpoint{2.913736in}{1.860708in}}% +\pgfpathlineto{\pgfqpoint{2.945016in}{1.934086in}}% +\pgfpathlineto{\pgfqpoint{2.976295in}{2.010499in}}% +\pgfpathlineto{\pgfqpoint{3.007575in}{2.090031in}}% +\pgfpathlineto{\pgfqpoint{3.038854in}{2.172766in}}% +\pgfpathlineto{\pgfqpoint{3.070134in}{2.258787in}}% +\pgfpathlineto{\pgfqpoint{3.090098in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.548770in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.548772in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.548774in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.548779in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.548788in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.548800in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.548820in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.548849in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.548890in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.548947in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.549025in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.549130in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.549268in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.549448in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.549678in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.549971in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.550339in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.550796in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.551358in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.552045in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.552878in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.553880in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.555077in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.556500in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.558181in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.560156in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.562466in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.565152in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.568265in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.571855in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.575980in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.580702in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.586087in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.592209in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.599146in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.606983in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.615811in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.625726in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.636835in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.649249in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.663089in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{0.678481in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{0.695564in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{0.714481in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{0.735388in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{0.758448in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{0.783835in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{0.811733in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.842338in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{0.875855in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{0.912502in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{0.952509in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{0.996118in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{1.043583in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{1.095172in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.151168in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.211867in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.277579in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.348630in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.508132in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.597316in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.693303in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.796505in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{1.907347in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{2.026275in}}% +\pgfpathlineto{\pgfqpoint{2.913736in}{2.153756in}}% +\pgfpathlineto{\pgfqpoint{2.945016in}{2.290274in}}% +\pgfpathlineto{\pgfqpoint{2.950492in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.548770in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.548771in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.548772in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.548775in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.548779in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.548786in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.548796in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.548810in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.548831in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.548861in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.548902in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.548959in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.549037in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.549140in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.549277in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.549456in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.549689in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.549987in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.550366in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.550845in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.551446in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.552193in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.553117in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.554251in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.555637in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.557321in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.559354in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.561799in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.564725in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.568210in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.572343in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.577226in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.582972in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.589709in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.597579in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.606742in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{0.617377in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{0.629681in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{0.643875in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{0.660200in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{0.678928in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{0.700353in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{0.724803in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.752636in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{0.784247in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{0.820066in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{0.860565in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{0.906258in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{0.957706in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{1.015520in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.080363in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.152955in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.234078in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.324575in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.537424in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.661827in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.799717in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.952328in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{2.120989in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{2.307124in}}% +\pgfpathlineto{\pgfqpoint{2.883786in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.800000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.800000,0.800000,0.800000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.800000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.727552in}{1.416153in}}% +\pgfpathlineto{\pgfqpoint{1.553360in}{1.416153in}}% +\pgfpathquadraticcurveto{\pgfqpoint{1.581138in}{1.416153in}}{\pgfqpoint{1.581138in}{1.443930in}}% +\pgfpathlineto{\pgfqpoint{1.581138in}{2.204733in}}% +\pgfpathquadraticcurveto{\pgfqpoint{1.581138in}{2.232510in}}{\pgfqpoint{1.553360in}{2.232510in}}% +\pgfpathlineto{\pgfqpoint{0.727552in}{2.232510in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.699774in}{2.232510in}}{\pgfqpoint{0.699774in}{2.204733in}}% +\pgfpathlineto{\pgfqpoint{0.699774in}{1.443930in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.699774in}{1.416153in}}{\pgfqpoint{0.727552in}{1.416153in}}% +\pgfpathlineto{\pgfqpoint{0.727552in}{1.416153in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.755330in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{0.894219in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{1.033108in}{2.128344in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.144219in,y=2.079733in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=1\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.755330in}{1.934671in}}% +\pgfpathlineto{\pgfqpoint{0.894219in}{1.934671in}}% +\pgfpathlineto{\pgfqpoint{1.033108in}{1.934671in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.144219in,y=1.886060in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=2\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.755330in}{1.740998in}}% +\pgfpathlineto{\pgfqpoint{0.894219in}{1.740998in}}% +\pgfpathlineto{\pgfqpoint{1.033108in}{1.740998in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.144219in,y=1.692387in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=3\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.755330in}{1.547325in}}% +\pgfpathlineto{\pgfqpoint{0.894219in}{1.547325in}}% +\pgfpathlineto{\pgfqpoint{1.033108in}{1.547325in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.144219in,y=1.498714in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=4\)}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/F_N_chebychev.pgf b/buch/papers/ellfilter/python/F_N_chebychev.pgf new file mode 100644 index 0000000..72d5834 --- /dev/null +++ b/buch/papers/ellfilter/python/F_N_chebychev.pgf @@ -0,0 +1,1066 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{0.000000,0.501961,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.647059,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.630330in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.146442in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.146442in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.146442in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.146442in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.25}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.662555in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.662555in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.662555in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.50}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.178667in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.178667in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.178667in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.178667in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.694779in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.694779in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.694779in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.210892in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.210892in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.210892in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.210892in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.25}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.727004in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.727004in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.50}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.178667in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.987065in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.987065in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{0.987065in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=0.938840in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.425362in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{1.425362in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=1.377137in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.863658in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{1.863658in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{1.863658in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=1.815433in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.630330in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.355638in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.300082in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle F^2_N(w)\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.548970in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.549574in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.550580in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.551989in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.553800in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.556013in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.558629in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.561648in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.565069in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.568893in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.573119in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.577747in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.582778in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.588211in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.594047in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.600286in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.606927in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.613970in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.621416in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.629264in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.637515in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.646168in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.655224in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.664682in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.674543in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.684806in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.695471in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.706539in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.718010in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.729883in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.742159in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.754837in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.767917in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.781400in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.795285in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.809573in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.824264in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.839357in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.854852in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.870750in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.887050in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.903753in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.920858in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.938366in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.956276in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.974589in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.993304in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{1.012421in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{1.031941in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{1.051864in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{1.072189in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{1.092917in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{1.114047in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{1.135579in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{1.157514in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{1.179851in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{1.202591in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{1.225734in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{1.249279in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{1.273226in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{1.297576in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.322328in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.347483in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.373040in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.399000in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.452126in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.479294in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.506863in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.534835in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{1.563210in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{1.591987in}}% +\pgfpathlineto{\pgfqpoint{2.913736in}{1.621166in}}% +\pgfpathlineto{\pgfqpoint{2.945016in}{1.650748in}}% +\pgfpathlineto{\pgfqpoint{2.976295in}{1.680733in}}% +\pgfpathlineto{\pgfqpoint{3.007575in}{1.711120in}}% +\pgfpathlineto{\pgfqpoint{3.038854in}{1.741909in}}% +\pgfpathlineto{\pgfqpoint{3.070134in}{1.773101in}}% +\pgfpathlineto{\pgfqpoint{3.101413in}{1.804696in}}% +\pgfpathlineto{\pgfqpoint{3.132693in}{1.836692in}}% +\pgfpathlineto{\pgfqpoint{3.163972in}{1.869092in}}% +\pgfpathlineto{\pgfqpoint{3.195252in}{1.901894in}}% +\pgfpathlineto{\pgfqpoint{3.226531in}{1.935098in}}% +\pgfpathlineto{\pgfqpoint{3.257811in}{1.968705in}}% +\pgfpathlineto{\pgfqpoint{3.289091in}{2.002714in}}% +\pgfpathlineto{\pgfqpoint{3.320370in}{2.037126in}}% +\pgfpathlineto{\pgfqpoint{3.351650in}{2.071940in}}% +\pgfpathlineto{\pgfqpoint{3.382929in}{2.107156in}}% +\pgfpathlineto{\pgfqpoint{3.414209in}{2.142776in}}% +\pgfpathlineto{\pgfqpoint{3.445488in}{2.178797in}}% +\pgfpathlineto{\pgfqpoint{3.476768in}{2.215221in}}% +\pgfpathlineto{\pgfqpoint{3.508047in}{2.252048in}}% +\pgfpathlineto{\pgfqpoint{3.539327in}{2.289277in}}% +\pgfpathlineto{\pgfqpoint{3.561409in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{1.424557in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{1.422145in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{1.418132in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{1.412530in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{1.405354in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{1.396623in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{1.386363in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{1.374602in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{1.361373in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{1.346715in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{1.330668in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{1.313281in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{1.294603in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{1.274690in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{1.253603in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{1.231405in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{1.208165in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{1.183956in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{1.158856in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{1.132948in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{1.106316in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{1.079053in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{1.051254in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{1.023019in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.994451in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.965659in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.936757in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.907863in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.879097in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.850586in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.822462in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.794859in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.767917in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.741781in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.716598in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.692523in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{0.669711in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{0.648326in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{0.628534in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{0.610505in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{0.594414in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{0.580441in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{0.568771in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{0.559591in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{0.553095in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{0.549479in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.548946in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{0.551703in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{0.557958in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{0.567929in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{0.581833in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{0.599896in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{0.622346in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{0.649414in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.681340in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{0.718364in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{0.760732in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{0.808696in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{0.862510in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{0.922433in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{0.988730in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{1.061668in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{1.141521in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.228566in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.323084in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.535689in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.654362in}}% +\pgfpathlineto{\pgfqpoint{2.788618in}{1.781678in}}% +\pgfpathlineto{\pgfqpoint{2.819897in}{1.917942in}}% +\pgfpathlineto{\pgfqpoint{2.851177in}{2.063463in}}% +\pgfpathlineto{\pgfqpoint{2.882457in}{2.218553in}}% +\pgfpathlineto{\pgfqpoint{2.900903in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{0.550579in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{0.555996in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{0.564979in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{0.577464in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{0.593357in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{0.612541in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{0.634873in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{0.660185in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{0.688287in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{0.718965in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{0.751984in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{0.787089in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.824004in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.862437in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.902078in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.942605in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.983681in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{1.024958in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{1.066081in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{1.106686in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{1.146406in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{1.184870in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{1.221710in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{1.256559in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{1.289056in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{1.318849in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{1.345598in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{1.368977in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{1.388677in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{1.404413in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{1.415923in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{1.422973in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{1.422924in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{1.415535in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{1.403113in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{1.385624in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{1.363088in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{1.335583in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{1.303245in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{1.266280in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{1.224962in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{1.179644in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{1.130759in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{1.078826in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{1.024455in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{0.968355in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{0.911337in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{0.854319in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{0.798335in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{0.744537in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{0.694207in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{0.648754in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{0.609730in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.578830in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{0.557901in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{0.548947in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{0.554140in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{0.575820in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{0.616509in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{0.678913in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{0.765934in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{0.880671in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{1.026434in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.206748in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.686256in}}% +\pgfpathlineto{\pgfqpoint{2.757338in}{1.993649in}}% +\pgfpathlineto{\pgfqpoint{2.785461in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.630330in}{0.548769in}}{\pgfqpoint{3.096674in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{0.661609in}{1.422146in}}% +\pgfpathlineto{\pgfqpoint{0.692889in}{1.412542in}}% +\pgfpathlineto{\pgfqpoint{0.724168in}{1.396682in}}% +\pgfpathlineto{\pgfqpoint{0.755448in}{1.374785in}}% +\pgfpathlineto{\pgfqpoint{0.786727in}{1.347155in}}% +\pgfpathlineto{\pgfqpoint{0.818007in}{1.314175in}}% +\pgfpathlineto{\pgfqpoint{0.849287in}{1.276306in}}% +\pgfpathlineto{\pgfqpoint{0.880566in}{1.234079in}}% +\pgfpathlineto{\pgfqpoint{0.911846in}{1.188091in}}% +\pgfpathlineto{\pgfqpoint{0.943125in}{1.138997in}}% +\pgfpathlineto{\pgfqpoint{0.974405in}{1.087504in}}% +\pgfpathlineto{\pgfqpoint{1.005684in}{1.034360in}}% +\pgfpathlineto{\pgfqpoint{1.036964in}{0.980345in}}% +\pgfpathlineto{\pgfqpoint{1.068243in}{0.926267in}}% +\pgfpathlineto{\pgfqpoint{1.099523in}{0.872943in}}% +\pgfpathlineto{\pgfqpoint{1.130802in}{0.821195in}}% +\pgfpathlineto{\pgfqpoint{1.162082in}{0.771836in}}% +\pgfpathlineto{\pgfqpoint{1.193361in}{0.725662in}}% +\pgfpathlineto{\pgfqpoint{1.224641in}{0.683436in}}% +\pgfpathlineto{\pgfqpoint{1.255921in}{0.645879in}}% +\pgfpathlineto{\pgfqpoint{1.287200in}{0.613660in}}% +\pgfpathlineto{\pgfqpoint{1.318480in}{0.587381in}}% +\pgfpathlineto{\pgfqpoint{1.349759in}{0.567570in}}% +\pgfpathlineto{\pgfqpoint{1.381039in}{0.554667in}}% +\pgfpathlineto{\pgfqpoint{1.412318in}{0.549018in}}% +\pgfpathlineto{\pgfqpoint{1.443598in}{0.550860in}}% +\pgfpathlineto{\pgfqpoint{1.474877in}{0.560318in}}% +\pgfpathlineto{\pgfqpoint{1.506157in}{0.577394in}}% +\pgfpathlineto{\pgfqpoint{1.537436in}{0.601962in}}% +\pgfpathlineto{\pgfqpoint{1.568716in}{0.633764in}}% +\pgfpathlineto{\pgfqpoint{1.599995in}{0.672404in}}% +\pgfpathlineto{\pgfqpoint{1.631275in}{0.717346in}}% +\pgfpathlineto{\pgfqpoint{1.662555in}{0.767917in}}% +\pgfpathlineto{\pgfqpoint{1.693834in}{0.823307in}}% +\pgfpathlineto{\pgfqpoint{1.725114in}{0.882572in}}% +\pgfpathlineto{\pgfqpoint{1.756393in}{0.944644in}}% +\pgfpathlineto{\pgfqpoint{1.787673in}{1.008337in}}% +\pgfpathlineto{\pgfqpoint{1.818952in}{1.072360in}}% +\pgfpathlineto{\pgfqpoint{1.850232in}{1.135334in}}% +\pgfpathlineto{\pgfqpoint{1.881511in}{1.195810in}}% +\pgfpathlineto{\pgfqpoint{1.912791in}{1.252288in}}% +\pgfpathlineto{\pgfqpoint{1.944070in}{1.303249in}}% +\pgfpathlineto{\pgfqpoint{1.975350in}{1.347179in}}% +\pgfpathlineto{\pgfqpoint{2.006629in}{1.382608in}}% +\pgfpathlineto{\pgfqpoint{2.037909in}{1.408144in}}% +\pgfpathlineto{\pgfqpoint{2.069189in}{1.422523in}}% +\pgfpathlineto{\pgfqpoint{2.100468in}{1.424652in}}% +\pgfpathlineto{\pgfqpoint{2.131748in}{1.413666in}}% +\pgfpathlineto{\pgfqpoint{2.163027in}{1.388989in}}% +\pgfpathlineto{\pgfqpoint{2.194307in}{1.350397in}}% +\pgfpathlineto{\pgfqpoint{2.225586in}{1.298092in}}% +\pgfpathlineto{\pgfqpoint{2.256866in}{1.232780in}}% +\pgfpathlineto{\pgfqpoint{2.288145in}{1.155757in}}% +\pgfpathlineto{\pgfqpoint{2.319425in}{1.069002in}}% +\pgfpathlineto{\pgfqpoint{2.350704in}{0.975275in}}% +\pgfpathlineto{\pgfqpoint{2.381984in}{0.878228in}}% +\pgfpathlineto{\pgfqpoint{2.413263in}{0.782522in}}% +\pgfpathlineto{\pgfqpoint{2.444543in}{0.693947in}}% +\pgfpathlineto{\pgfqpoint{2.475823in}{0.619562in}}% +\pgfpathlineto{\pgfqpoint{2.507102in}{0.567831in}}% +\pgfpathlineto{\pgfqpoint{2.538382in}{0.548781in}}% +\pgfpathlineto{\pgfqpoint{2.569661in}{0.574165in}}% +\pgfpathlineto{\pgfqpoint{2.600941in}{0.657630in}}% +\pgfpathlineto{\pgfqpoint{2.632220in}{0.814902in}}% +\pgfpathlineto{\pgfqpoint{2.663500in}{1.063985in}}% +\pgfpathlineto{\pgfqpoint{2.694779in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{2.726059in}{1.922215in}}% +\pgfpathlineto{\pgfqpoint{2.744758in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.630330in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.727004in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.800000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.800000,0.800000,0.800000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.800000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.803974in}{0.618213in}}% +\pgfpathlineto{\pgfqpoint{3.629782in}{0.618213in}}% +\pgfpathquadraticcurveto{\pgfqpoint{3.657560in}{0.618213in}}{\pgfqpoint{3.657560in}{0.645991in}}% +\pgfpathlineto{\pgfqpoint{3.657560in}{1.406793in}}% +\pgfpathquadraticcurveto{\pgfqpoint{3.657560in}{1.434571in}}{\pgfqpoint{3.629782in}{1.434571in}}% +\pgfpathlineto{\pgfqpoint{2.803974in}{1.434571in}}% +\pgfpathquadraticcurveto{\pgfqpoint{2.776196in}{1.434571in}}{\pgfqpoint{2.776196in}{1.406793in}}% +\pgfpathlineto{\pgfqpoint{2.776196in}{0.645991in}}% +\pgfpathquadraticcurveto{\pgfqpoint{2.776196in}{0.618213in}}{\pgfqpoint{2.803974in}{0.618213in}}% +\pgfpathlineto{\pgfqpoint{2.803974in}{0.618213in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.831751in}{1.330404in}}% +\pgfpathlineto{\pgfqpoint{2.970640in}{1.330404in}}% +\pgfpathlineto{\pgfqpoint{3.109529in}{1.330404in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.220640in,y=1.281793in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=1\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.831751in}{1.136732in}}% +\pgfpathlineto{\pgfqpoint{2.970640in}{1.136732in}}% +\pgfpathlineto{\pgfqpoint{3.109529in}{1.136732in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.220640in,y=1.088120in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=2\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.831751in}{0.943059in}}% +\pgfpathlineto{\pgfqpoint{2.970640in}{0.943059in}}% +\pgfpathlineto{\pgfqpoint{3.109529in}{0.943059in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.220640in,y=0.894448in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=3\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.831751in}{0.749386in}}% +\pgfpathlineto{\pgfqpoint{2.970640in}{0.749386in}}% +\pgfpathlineto{\pgfqpoint{3.109529in}{0.749386in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.220640in,y=0.700775in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=4\)}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/F_N_chebychev2.pgf b/buch/papers/ellfilter/python/F_N_chebychev2.pgf new file mode 100644 index 0000000..43ebb91 --- /dev/null +++ b/buch/papers/ellfilter/python/F_N_chebychev2.pgf @@ -0,0 +1,1023 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.500000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{5.500000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{5.500000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.012292in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.012292in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.012292in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.012292in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {\ensuremath{-}1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.998134in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.998134in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.998134in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.998134in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {\ensuremath{-}0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.983977in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.983977in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.983977in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.983977in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.969820in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.969820in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.969820in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.969820in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.955663in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{4.955663in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{4.955663in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.955663in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.983977in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {\ensuremath{-}2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.987065in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{0.987065in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{0.987065in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=0.938840in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {\ensuremath{-}1}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.425362in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{1.425362in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{1.425362in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.451287in, y=1.377137in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.863658in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{1.863658in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{1.863658in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.451287in, y=1.815433in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.451287in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.287707in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle T_N(w)\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.815123in}{0.538250in}}% +\pgfpathlineto{\pgfqpoint{0.867228in}{0.667673in}}% +\pgfpathlineto{\pgfqpoint{0.919332in}{0.789210in}}% +\pgfpathlineto{\pgfqpoint{0.971437in}{0.903055in}}% +\pgfpathlineto{\pgfqpoint{1.023541in}{1.009402in}}% +\pgfpathlineto{\pgfqpoint{1.075646in}{1.108444in}}% +\pgfpathlineto{\pgfqpoint{1.123409in}{1.192982in}}% +\pgfpathlineto{\pgfqpoint{1.171171in}{1.271695in}}% +\pgfpathlineto{\pgfqpoint{1.218934in}{1.344733in}}% +\pgfpathlineto{\pgfqpoint{1.266696in}{1.412244in}}% +\pgfpathlineto{\pgfqpoint{1.314459in}{1.474380in}}% +\pgfpathlineto{\pgfqpoint{1.362221in}{1.531289in}}% +\pgfpathlineto{\pgfqpoint{1.409984in}{1.583121in}}% +\pgfpathlineto{\pgfqpoint{1.453404in}{1.625960in}}% +\pgfpathlineto{\pgfqpoint{1.496825in}{1.664840in}}% +\pgfpathlineto{\pgfqpoint{1.540245in}{1.699871in}}% +\pgfpathlineto{\pgfqpoint{1.583666in}{1.731168in}}% +\pgfpathlineto{\pgfqpoint{1.627086in}{1.758841in}}% +\pgfpathlineto{\pgfqpoint{1.670507in}{1.783003in}}% +\pgfpathlineto{\pgfqpoint{1.713927in}{1.803767in}}% +\pgfpathlineto{\pgfqpoint{1.757348in}{1.821245in}}% +\pgfpathlineto{\pgfqpoint{1.800768in}{1.835549in}}% +\pgfpathlineto{\pgfqpoint{1.844189in}{1.846792in}}% +\pgfpathlineto{\pgfqpoint{1.887609in}{1.855086in}}% +\pgfpathlineto{\pgfqpoint{1.935372in}{1.860937in}}% +\pgfpathlineto{\pgfqpoint{1.983135in}{1.863505in}}% +\pgfpathlineto{\pgfqpoint{2.030897in}{1.862940in}}% +\pgfpathlineto{\pgfqpoint{2.078660in}{1.859391in}}% +\pgfpathlineto{\pgfqpoint{2.130764in}{1.852293in}}% +\pgfpathlineto{\pgfqpoint{2.182869in}{1.842015in}}% +\pgfpathlineto{\pgfqpoint{2.239316in}{1.827518in}}% +\pgfpathlineto{\pgfqpoint{2.295762in}{1.809766in}}% +\pgfpathlineto{\pgfqpoint{2.356551in}{1.787290in}}% +\pgfpathlineto{\pgfqpoint{2.421682in}{1.759685in}}% +\pgfpathlineto{\pgfqpoint{2.491154in}{1.726641in}}% +\pgfpathlineto{\pgfqpoint{2.564969in}{1.687966in}}% +\pgfpathlineto{\pgfqpoint{2.647468in}{1.641059in}}% +\pgfpathlineto{\pgfqpoint{2.738651in}{1.585589in}}% +\pgfpathlineto{\pgfqpoint{2.851545in}{1.513148in}}% +\pgfpathlineto{\pgfqpoint{3.064305in}{1.371911in}}% +\pgfpathlineto{\pgfqpoint{3.207593in}{1.278793in}}% +\pgfpathlineto{\pgfqpoint{3.307460in}{1.217378in}}% +\pgfpathlineto{\pgfqpoint{3.394301in}{1.167524in}}% +\pgfpathlineto{\pgfqpoint{3.472458in}{1.126261in}}% +\pgfpathlineto{\pgfqpoint{3.541931in}{1.093000in}}% +\pgfpathlineto{\pgfqpoint{3.607061in}{1.065165in}}% +\pgfpathlineto{\pgfqpoint{3.667850in}{1.042451in}}% +\pgfpathlineto{\pgfqpoint{3.724297in}{1.024458in}}% +\pgfpathlineto{\pgfqpoint{3.780743in}{1.009703in}}% +\pgfpathlineto{\pgfqpoint{3.832848in}{0.999169in}}% +\pgfpathlineto{\pgfqpoint{3.884953in}{0.991798in}}% +\pgfpathlineto{\pgfqpoint{3.932715in}{0.987985in}}% +\pgfpathlineto{\pgfqpoint{3.980478in}{0.987142in}}% +\pgfpathlineto{\pgfqpoint{4.028240in}{0.989420in}}% +\pgfpathlineto{\pgfqpoint{4.076003in}{0.994966in}}% +\pgfpathlineto{\pgfqpoint{4.119423in}{1.002971in}}% +\pgfpathlineto{\pgfqpoint{4.162844in}{1.013914in}}% +\pgfpathlineto{\pgfqpoint{4.206264in}{1.027907in}}% +\pgfpathlineto{\pgfqpoint{4.249685in}{1.045063in}}% +\pgfpathlineto{\pgfqpoint{4.293105in}{1.065493in}}% +\pgfpathlineto{\pgfqpoint{4.336526in}{1.089310in}}% +\pgfpathlineto{\pgfqpoint{4.379946in}{1.116627in}}% +\pgfpathlineto{\pgfqpoint{4.423367in}{1.147556in}}% +\pgfpathlineto{\pgfqpoint{4.466787in}{1.182209in}}% +\pgfpathlineto{\pgfqpoint{4.510208in}{1.220699in}}% +\pgfpathlineto{\pgfqpoint{4.553628in}{1.263138in}}% +\pgfpathlineto{\pgfqpoint{4.597049in}{1.309638in}}% +\pgfpathlineto{\pgfqpoint{4.644812in}{1.365612in}}% +\pgfpathlineto{\pgfqpoint{4.692574in}{1.426786in}}% +\pgfpathlineto{\pgfqpoint{4.740337in}{1.493310in}}% +\pgfpathlineto{\pgfqpoint{4.788099in}{1.565332in}}% +\pgfpathlineto{\pgfqpoint{4.835862in}{1.643002in}}% +\pgfpathlineto{\pgfqpoint{4.883624in}{1.726469in}}% +\pgfpathlineto{\pgfqpoint{4.931387in}{1.815884in}}% +\pgfpathlineto{\pgfqpoint{4.983491in}{1.920387in}}% +\pgfpathlineto{\pgfqpoint{5.035596in}{2.032338in}}% +\pgfpathlineto{\pgfqpoint{5.087701in}{2.151934in}}% +\pgfpathlineto{\pgfqpoint{5.139805in}{2.279368in}}% +\pgfpathlineto{\pgfqpoint{5.152831in}{2.312474in}}% +\pgfpathlineto{\pgfqpoint{5.152831in}{2.312474in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.963285in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{0.988805in}{2.065008in}}% +\pgfpathlineto{\pgfqpoint{1.014857in}{1.843281in}}% +\pgfpathlineto{\pgfqpoint{1.036568in}{1.682977in}}% +\pgfpathlineto{\pgfqpoint{1.058278in}{1.543280in}}% +\pgfpathlineto{\pgfqpoint{1.079988in}{1.422723in}}% +\pgfpathlineto{\pgfqpoint{1.101698in}{1.319903in}}% +\pgfpathlineto{\pgfqpoint{1.123409in}{1.233476in}}% +\pgfpathlineto{\pgfqpoint{1.140777in}{1.175276in}}% +\pgfpathlineto{\pgfqpoint{1.158145in}{1.126117in}}% +\pgfpathlineto{\pgfqpoint{1.175513in}{1.085395in}}% +\pgfpathlineto{\pgfqpoint{1.192881in}{1.052526in}}% +\pgfpathlineto{\pgfqpoint{1.210250in}{1.026952in}}% +\pgfpathlineto{\pgfqpoint{1.223276in}{1.012233in}}% +\pgfpathlineto{\pgfqpoint{1.236302in}{1.001095in}}% +\pgfpathlineto{\pgfqpoint{1.249328in}{0.993325in}}% +\pgfpathlineto{\pgfqpoint{1.262354in}{0.988718in}}% +\pgfpathlineto{\pgfqpoint{1.275380in}{0.987075in}}% +\pgfpathlineto{\pgfqpoint{1.288407in}{0.988202in}}% +\pgfpathlineto{\pgfqpoint{1.301433in}{0.991914in}}% +\pgfpathlineto{\pgfqpoint{1.318801in}{1.000572in}}% +\pgfpathlineto{\pgfqpoint{1.336169in}{1.013093in}}% +\pgfpathlineto{\pgfqpoint{1.353537in}{1.029086in}}% +\pgfpathlineto{\pgfqpoint{1.375248in}{1.053389in}}% +\pgfpathlineto{\pgfqpoint{1.401300in}{1.087971in}}% +\pgfpathlineto{\pgfqpoint{1.431694in}{1.134386in}}% +\pgfpathlineto{\pgfqpoint{1.466431in}{1.193454in}}% +\pgfpathlineto{\pgfqpoint{1.514193in}{1.281335in}}% +\pgfpathlineto{\pgfqpoint{1.648797in}{1.533319in}}% +\pgfpathlineto{\pgfqpoint{1.692217in}{1.606504in}}% +\pgfpathlineto{\pgfqpoint{1.731296in}{1.666194in}}% +\pgfpathlineto{\pgfqpoint{1.766032in}{1.713499in}}% +\pgfpathlineto{\pgfqpoint{1.796426in}{1.750004in}}% +\pgfpathlineto{\pgfqpoint{1.826821in}{1.781659in}}% +\pgfpathlineto{\pgfqpoint{1.852873in}{1.804782in}}% +\pgfpathlineto{\pgfqpoint{1.878925in}{1.824113in}}% +\pgfpathlineto{\pgfqpoint{1.904978in}{1.839604in}}% +\pgfpathlineto{\pgfqpoint{1.931030in}{1.851242in}}% +\pgfpathlineto{\pgfqpoint{1.957082in}{1.859041in}}% +\pgfpathlineto{\pgfqpoint{1.983135in}{1.863047in}}% +\pgfpathlineto{\pgfqpoint{2.009187in}{1.863329in}}% +\pgfpathlineto{\pgfqpoint{2.035239in}{1.859984in}}% +\pgfpathlineto{\pgfqpoint{2.061291in}{1.853128in}}% +\pgfpathlineto{\pgfqpoint{2.087344in}{1.842898in}}% +\pgfpathlineto{\pgfqpoint{2.113396in}{1.829452in}}% +\pgfpathlineto{\pgfqpoint{2.143790in}{1.809929in}}% +\pgfpathlineto{\pgfqpoint{2.174185in}{1.786560in}}% +\pgfpathlineto{\pgfqpoint{2.208921in}{1.755558in}}% +\pgfpathlineto{\pgfqpoint{2.243658in}{1.720467in}}% +\pgfpathlineto{\pgfqpoint{2.282736in}{1.676770in}}% +\pgfpathlineto{\pgfqpoint{2.330499in}{1.618420in}}% +\pgfpathlineto{\pgfqpoint{2.386945in}{1.544378in}}% +\pgfpathlineto{\pgfqpoint{2.491154in}{1.401257in}}% +\pgfpathlineto{\pgfqpoint{2.573653in}{1.290423in}}% +\pgfpathlineto{\pgfqpoint{2.630100in}{1.219842in}}% +\pgfpathlineto{\pgfqpoint{2.677863in}{1.165204in}}% +\pgfpathlineto{\pgfqpoint{2.716941in}{1.124786in}}% +\pgfpathlineto{\pgfqpoint{2.756020in}{1.088796in}}% +\pgfpathlineto{\pgfqpoint{2.790756in}{1.060903in}}% +\pgfpathlineto{\pgfqpoint{2.825492in}{1.037164in}}% +\pgfpathlineto{\pgfqpoint{2.855887in}{1.019988in}}% +\pgfpathlineto{\pgfqpoint{2.886281in}{1.006308in}}% +\pgfpathlineto{\pgfqpoint{2.916675in}{0.996229in}}% +\pgfpathlineto{\pgfqpoint{2.947070in}{0.989827in}}% +\pgfpathlineto{\pgfqpoint{2.973122in}{0.987304in}}% +\pgfpathlineto{\pgfqpoint{2.999174in}{0.987534in}}% +\pgfpathlineto{\pgfqpoint{3.025227in}{0.990514in}}% +\pgfpathlineto{\pgfqpoint{3.051279in}{0.996229in}}% +\pgfpathlineto{\pgfqpoint{3.081673in}{1.006308in}}% +\pgfpathlineto{\pgfqpoint{3.112068in}{1.019988in}}% +\pgfpathlineto{\pgfqpoint{3.142462in}{1.037164in}}% +\pgfpathlineto{\pgfqpoint{3.172856in}{1.057704in}}% +\pgfpathlineto{\pgfqpoint{3.207593in}{1.085092in}}% +\pgfpathlineto{\pgfqpoint{3.242329in}{1.116388in}}% +\pgfpathlineto{\pgfqpoint{3.281408in}{1.155866in}}% +\pgfpathlineto{\pgfqpoint{3.324828in}{1.204423in}}% +\pgfpathlineto{\pgfqpoint{3.372591in}{1.262618in}}% +\pgfpathlineto{\pgfqpoint{3.433379in}{1.342111in}}% +\pgfpathlineto{\pgfqpoint{3.533247in}{1.479174in}}% +\pgfpathlineto{\pgfqpoint{3.615746in}{1.590473in}}% +\pgfpathlineto{\pgfqpoint{3.667850in}{1.656113in}}% +\pgfpathlineto{\pgfqpoint{3.711271in}{1.706362in}}% +\pgfpathlineto{\pgfqpoint{3.750349in}{1.747148in}}% +\pgfpathlineto{\pgfqpoint{3.785086in}{1.779222in}}% +\pgfpathlineto{\pgfqpoint{3.815480in}{1.803631in}}% +\pgfpathlineto{\pgfqpoint{3.845874in}{1.824284in}}% +\pgfpathlineto{\pgfqpoint{3.876269in}{1.840876in}}% +\pgfpathlineto{\pgfqpoint{3.902321in}{1.851653in}}% +\pgfpathlineto{\pgfqpoint{3.928373in}{1.859081in}}% +\pgfpathlineto{\pgfqpoint{3.954425in}{1.863021in}}% +\pgfpathlineto{\pgfqpoint{3.980478in}{1.863350in}}% +\pgfpathlineto{\pgfqpoint{4.002188in}{1.860795in}}% +\pgfpathlineto{\pgfqpoint{4.023898in}{1.855619in}}% +\pgfpathlineto{\pgfqpoint{4.049951in}{1.845904in}}% +\pgfpathlineto{\pgfqpoint{4.076003in}{1.832340in}}% +\pgfpathlineto{\pgfqpoint{4.102055in}{1.814925in}}% +\pgfpathlineto{\pgfqpoint{4.128108in}{1.793690in}}% +\pgfpathlineto{\pgfqpoint{4.154160in}{1.768700in}}% +\pgfpathlineto{\pgfqpoint{4.184554in}{1.734938in}}% +\pgfpathlineto{\pgfqpoint{4.214949in}{1.696434in}}% +\pgfpathlineto{\pgfqpoint{4.249685in}{1.647018in}}% +\pgfpathlineto{\pgfqpoint{4.288763in}{1.585242in}}% +\pgfpathlineto{\pgfqpoint{4.332184in}{1.510192in}}% +\pgfpathlineto{\pgfqpoint{4.388631in}{1.405562in}}% +\pgfpathlineto{\pgfqpoint{4.505866in}{1.185791in}}% +\pgfpathlineto{\pgfqpoint{4.544944in}{1.120547in}}% +\pgfpathlineto{\pgfqpoint{4.575339in}{1.075851in}}% +\pgfpathlineto{\pgfqpoint{4.601391in}{1.043132in}}% +\pgfpathlineto{\pgfqpoint{4.623101in}{1.020680in}}% +\pgfpathlineto{\pgfqpoint{4.640469in}{1.006375in}}% +\pgfpathlineto{\pgfqpoint{4.657838in}{0.995735in}}% +\pgfpathlineto{\pgfqpoint{4.675206in}{0.989161in}}% +\pgfpathlineto{\pgfqpoint{4.688232in}{0.987152in}}% +\pgfpathlineto{\pgfqpoint{4.701258in}{0.987850in}}% +\pgfpathlineto{\pgfqpoint{4.714284in}{0.991448in}}% +\pgfpathlineto{\pgfqpoint{4.727310in}{0.998141in}}% +\pgfpathlineto{\pgfqpoint{4.740337in}{1.008133in}}% +\pgfpathlineto{\pgfqpoint{4.753363in}{1.021634in}}% +\pgfpathlineto{\pgfqpoint{4.766389in}{1.038861in}}% +\pgfpathlineto{\pgfqpoint{4.783757in}{1.068014in}}% +\pgfpathlineto{\pgfqpoint{4.801125in}{1.104738in}}% +\pgfpathlineto{\pgfqpoint{4.818494in}{1.149605in}}% +\pgfpathlineto{\pgfqpoint{4.835862in}{1.203207in}}% +\pgfpathlineto{\pgfqpoint{4.853230in}{1.266163in}}% +\pgfpathlineto{\pgfqpoint{4.870598in}{1.339113in}}% +\pgfpathlineto{\pgfqpoint{4.892308in}{1.445372in}}% +\pgfpathlineto{\pgfqpoint{4.914019in}{1.569642in}}% +\pgfpathlineto{\pgfqpoint{4.935729in}{1.713341in}}% +\pgfpathlineto{\pgfqpoint{4.957439in}{1.877948in}}% +\pgfpathlineto{\pgfqpoint{4.979149in}{2.065008in}}% +\pgfpathlineto{\pgfqpoint{5.004669in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{5.004669in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{4.732046in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.997762in}{0.534880in}}% +\pgfpathlineto{\pgfqpoint{1.010515in}{0.938420in}}% +\pgfpathlineto{\pgfqpoint{1.023541in}{1.256633in}}% +\pgfpathlineto{\pgfqpoint{1.036568in}{1.493946in}}% +\pgfpathlineto{\pgfqpoint{1.049594in}{1.662731in}}% +\pgfpathlineto{\pgfqpoint{1.058278in}{1.742696in}}% +\pgfpathlineto{\pgfqpoint{1.066962in}{1.800120in}}% +\pgfpathlineto{\pgfqpoint{1.075646in}{1.837754in}}% +\pgfpathlineto{\pgfqpoint{1.084330in}{1.858134in}}% +\pgfpathlineto{\pgfqpoint{1.088672in}{1.862591in}}% +\pgfpathlineto{\pgfqpoint{1.093014in}{1.863596in}}% +\pgfpathlineto{\pgfqpoint{1.097356in}{1.861410in}}% +\pgfpathlineto{\pgfqpoint{1.101698in}{1.856284in}}% +\pgfpathlineto{\pgfqpoint{1.110382in}{1.838165in}}% +\pgfpathlineto{\pgfqpoint{1.119067in}{1.811034in}}% +\pgfpathlineto{\pgfqpoint{1.132093in}{1.756980in}}% +\pgfpathlineto{\pgfqpoint{1.149461in}{1.667415in}}% +\pgfpathlineto{\pgfqpoint{1.179855in}{1.488530in}}% +\pgfpathlineto{\pgfqpoint{1.214592in}{1.289910in}}% +\pgfpathlineto{\pgfqpoint{1.236302in}{1.184400in}}% +\pgfpathlineto{\pgfqpoint{1.253670in}{1.114719in}}% +\pgfpathlineto{\pgfqpoint{1.266696in}{1.072074in}}% +\pgfpathlineto{\pgfqpoint{1.279722in}{1.038054in}}% +\pgfpathlineto{\pgfqpoint{1.292749in}{1.012779in}}% +\pgfpathlineto{\pgfqpoint{1.301433in}{1.000762in}}% +\pgfpathlineto{\pgfqpoint{1.310117in}{0.992549in}}% +\pgfpathlineto{\pgfqpoint{1.318801in}{0.988057in}}% +\pgfpathlineto{\pgfqpoint{1.327485in}{0.987177in}}% +\pgfpathlineto{\pgfqpoint{1.336169in}{0.989782in}}% +\pgfpathlineto{\pgfqpoint{1.344853in}{0.995723in}}% +\pgfpathlineto{\pgfqpoint{1.353537in}{1.004838in}}% +\pgfpathlineto{\pgfqpoint{1.366563in}{1.024074in}}% +\pgfpathlineto{\pgfqpoint{1.379590in}{1.049416in}}% +\pgfpathlineto{\pgfqpoint{1.396958in}{1.091524in}}% +\pgfpathlineto{\pgfqpoint{1.418668in}{1.155158in}}% +\pgfpathlineto{\pgfqpoint{1.444720in}{1.243239in}}% +\pgfpathlineto{\pgfqpoint{1.488141in}{1.403898in}}% +\pgfpathlineto{\pgfqpoint{1.531561in}{1.561511in}}% +\pgfpathlineto{\pgfqpoint{1.557614in}{1.646386in}}% +\pgfpathlineto{\pgfqpoint{1.579324in}{1.708574in}}% +\pgfpathlineto{\pgfqpoint{1.601034in}{1.761436in}}% +\pgfpathlineto{\pgfqpoint{1.618402in}{1.796276in}}% +\pgfpathlineto{\pgfqpoint{1.635771in}{1.824067in}}% +\pgfpathlineto{\pgfqpoint{1.648797in}{1.840132in}}% +\pgfpathlineto{\pgfqpoint{1.661823in}{1.852039in}}% +\pgfpathlineto{\pgfqpoint{1.674849in}{1.859776in}}% +\pgfpathlineto{\pgfqpoint{1.687875in}{1.863368in}}% +\pgfpathlineto{\pgfqpoint{1.700901in}{1.862878in}}% +\pgfpathlineto{\pgfqpoint{1.713927in}{1.858401in}}% +\pgfpathlineto{\pgfqpoint{1.726954in}{1.850065in}}% +\pgfpathlineto{\pgfqpoint{1.739980in}{1.838024in}}% +\pgfpathlineto{\pgfqpoint{1.757348in}{1.816524in}}% +\pgfpathlineto{\pgfqpoint{1.774716in}{1.789256in}}% +\pgfpathlineto{\pgfqpoint{1.796426in}{1.747911in}}% +\pgfpathlineto{\pgfqpoint{1.818137in}{1.699616in}}% +\pgfpathlineto{\pgfqpoint{1.844189in}{1.634299in}}% +\pgfpathlineto{\pgfqpoint{1.878925in}{1.538536in}}% +\pgfpathlineto{\pgfqpoint{1.983135in}{1.243956in}}% +\pgfpathlineto{\pgfqpoint{2.013529in}{1.169800in}}% +\pgfpathlineto{\pgfqpoint{2.039581in}{1.114392in}}% +\pgfpathlineto{\pgfqpoint{2.061291in}{1.074988in}}% +\pgfpathlineto{\pgfqpoint{2.083002in}{1.042391in}}% +\pgfpathlineto{\pgfqpoint{2.100370in}{1.021533in}}% +\pgfpathlineto{\pgfqpoint{2.117738in}{1.005506in}}% +\pgfpathlineto{\pgfqpoint{2.135106in}{0.994424in}}% +\pgfpathlineto{\pgfqpoint{2.148132in}{0.989394in}}% +\pgfpathlineto{\pgfqpoint{2.161159in}{0.987181in}}% +\pgfpathlineto{\pgfqpoint{2.174185in}{0.987773in}}% +\pgfpathlineto{\pgfqpoint{2.187211in}{0.991136in}}% +\pgfpathlineto{\pgfqpoint{2.200237in}{0.997224in}}% +\pgfpathlineto{\pgfqpoint{2.217605in}{1.009467in}}% +\pgfpathlineto{\pgfqpoint{2.234974in}{1.026248in}}% +\pgfpathlineto{\pgfqpoint{2.252342in}{1.047331in}}% +\pgfpathlineto{\pgfqpoint{2.274052in}{1.079308in}}% +\pgfpathlineto{\pgfqpoint{2.295762in}{1.116953in}}% +\pgfpathlineto{\pgfqpoint{2.321815in}{1.168623in}}% +\pgfpathlineto{\pgfqpoint{2.352209in}{1.236174in}}% +\pgfpathlineto{\pgfqpoint{2.391287in}{1.331016in}}% +\pgfpathlineto{\pgfqpoint{2.504181in}{1.611124in}}% +\pgfpathlineto{\pgfqpoint{2.534575in}{1.677221in}}% +\pgfpathlineto{\pgfqpoint{2.560627in}{1.727693in}}% +\pgfpathlineto{\pgfqpoint{2.586680in}{1.771344in}}% +\pgfpathlineto{\pgfqpoint{2.608390in}{1.801859in}}% +\pgfpathlineto{\pgfqpoint{2.630100in}{1.826589in}}% +\pgfpathlineto{\pgfqpoint{2.647468in}{1.841982in}}% +\pgfpathlineto{\pgfqpoint{2.664837in}{1.853327in}}% +\pgfpathlineto{\pgfqpoint{2.682205in}{1.860536in}}% +\pgfpathlineto{\pgfqpoint{2.699573in}{1.863558in}}% +\pgfpathlineto{\pgfqpoint{2.712599in}{1.863067in}}% +\pgfpathlineto{\pgfqpoint{2.725625in}{1.860222in}}% +\pgfpathlineto{\pgfqpoint{2.742993in}{1.852807in}}% +\pgfpathlineto{\pgfqpoint{2.760362in}{1.841331in}}% +\pgfpathlineto{\pgfqpoint{2.777730in}{1.825916in}}% +\pgfpathlineto{\pgfqpoint{2.795098in}{1.806716in}}% +\pgfpathlineto{\pgfqpoint{2.816808in}{1.777690in}}% +\pgfpathlineto{\pgfqpoint{2.838519in}{1.743493in}}% +\pgfpathlineto{\pgfqpoint{2.864571in}{1.696359in}}% +\pgfpathlineto{\pgfqpoint{2.894965in}{1.634247in}}% +\pgfpathlineto{\pgfqpoint{2.929702in}{1.556076in}}% +\pgfpathlineto{\pgfqpoint{2.986148in}{1.420053in}}% +\pgfpathlineto{\pgfqpoint{3.051279in}{1.264602in}}% +\pgfpathlineto{\pgfqpoint{3.086015in}{1.189019in}}% +\pgfpathlineto{\pgfqpoint{3.116410in}{1.130020in}}% +\pgfpathlineto{\pgfqpoint{3.142462in}{1.086119in}}% +\pgfpathlineto{\pgfqpoint{3.164172in}{1.054967in}}% +\pgfpathlineto{\pgfqpoint{3.185883in}{1.029260in}}% +\pgfpathlineto{\pgfqpoint{3.203251in}{1.012882in}}% +\pgfpathlineto{\pgfqpoint{3.220619in}{1.000409in}}% +\pgfpathlineto{\pgfqpoint{3.237987in}{0.991969in}}% +\pgfpathlineto{\pgfqpoint{3.255355in}{0.987657in}}% +\pgfpathlineto{\pgfqpoint{3.268382in}{0.987166in}}% +\pgfpathlineto{\pgfqpoint{3.281408in}{0.989038in}}% +\pgfpathlineto{\pgfqpoint{3.298776in}{0.995204in}}% +\pgfpathlineto{\pgfqpoint{3.316144in}{1.005522in}}% +\pgfpathlineto{\pgfqpoint{3.333512in}{1.019913in}}% +\pgfpathlineto{\pgfqpoint{3.350880in}{1.038258in}}% +\pgfpathlineto{\pgfqpoint{3.372591in}{1.066505in}}% +\pgfpathlineto{\pgfqpoint{3.394301in}{1.100293in}}% +\pgfpathlineto{\pgfqpoint{3.420353in}{1.147476in}}% +\pgfpathlineto{\pgfqpoint{3.446406in}{1.200975in}}% +\pgfpathlineto{\pgfqpoint{3.481142in}{1.280166in}}% +\pgfpathlineto{\pgfqpoint{3.528905in}{1.398487in}}% +\pgfpathlineto{\pgfqpoint{3.611404in}{1.604382in}}% +\pgfpathlineto{\pgfqpoint{3.646140in}{1.682101in}}% +\pgfpathlineto{\pgfqpoint{3.672192in}{1.733770in}}% +\pgfpathlineto{\pgfqpoint{3.698245in}{1.778285in}}% +\pgfpathlineto{\pgfqpoint{3.719955in}{1.809052in}}% +\pgfpathlineto{\pgfqpoint{3.737323in}{1.829084in}}% +\pgfpathlineto{\pgfqpoint{3.754691in}{1.844751in}}% +\pgfpathlineto{\pgfqpoint{3.772059in}{1.855828in}}% +\pgfpathlineto{\pgfqpoint{3.785086in}{1.861014in}}% +\pgfpathlineto{\pgfqpoint{3.798112in}{1.863458in}}% +\pgfpathlineto{\pgfqpoint{3.811138in}{1.863117in}}% +\pgfpathlineto{\pgfqpoint{3.824164in}{1.859967in}}% +\pgfpathlineto{\pgfqpoint{3.837190in}{1.853997in}}% +\pgfpathlineto{\pgfqpoint{3.850216in}{1.845218in}}% +\pgfpathlineto{\pgfqpoint{3.867584in}{1.829191in}}% +\pgfpathlineto{\pgfqpoint{3.884953in}{1.808333in}}% +\pgfpathlineto{\pgfqpoint{3.902321in}{1.782817in}}% +\pgfpathlineto{\pgfqpoint{3.924031in}{1.744730in}}% +\pgfpathlineto{\pgfqpoint{3.945741in}{1.700309in}}% +\pgfpathlineto{\pgfqpoint{3.971794in}{1.639665in}}% +\pgfpathlineto{\pgfqpoint{4.002188in}{1.560701in}}% +\pgfpathlineto{\pgfqpoint{4.045609in}{1.437971in}}% +\pgfpathlineto{\pgfqpoint{4.119423in}{1.227942in}}% +\pgfpathlineto{\pgfqpoint{4.149818in}{1.151107in}}% +\pgfpathlineto{\pgfqpoint{4.175870in}{1.093948in}}% +\pgfpathlineto{\pgfqpoint{4.197580in}{1.054139in}}% +\pgfpathlineto{\pgfqpoint{4.214949in}{1.028264in}}% +\pgfpathlineto{\pgfqpoint{4.232317in}{1.008285in}}% +\pgfpathlineto{\pgfqpoint{4.245343in}{0.997460in}}% +\pgfpathlineto{\pgfqpoint{4.258369in}{0.990394in}}% +\pgfpathlineto{\pgfqpoint{4.271395in}{0.987234in}}% +\pgfpathlineto{\pgfqpoint{4.284421in}{0.988096in}}% +\pgfpathlineto{\pgfqpoint{4.297447in}{0.993064in}}% +\pgfpathlineto{\pgfqpoint{4.310474in}{1.002190in}}% +\pgfpathlineto{\pgfqpoint{4.323500in}{1.015487in}}% +\pgfpathlineto{\pgfqpoint{4.336526in}{1.032928in}}% +\pgfpathlineto{\pgfqpoint{4.353894in}{1.062510in}}% +\pgfpathlineto{\pgfqpoint{4.371262in}{1.099057in}}% +\pgfpathlineto{\pgfqpoint{4.392973in}{1.153882in}}% +\pgfpathlineto{\pgfqpoint{4.414683in}{1.217773in}}% +\pgfpathlineto{\pgfqpoint{4.440735in}{1.304246in}}% +\pgfpathlineto{\pgfqpoint{4.479814in}{1.446826in}}% +\pgfpathlineto{\pgfqpoint{4.536260in}{1.652803in}}% +\pgfpathlineto{\pgfqpoint{4.562313in}{1.735052in}}% +\pgfpathlineto{\pgfqpoint{4.579681in}{1.781356in}}% +\pgfpathlineto{\pgfqpoint{4.597049in}{1.818848in}}% +\pgfpathlineto{\pgfqpoint{4.610075in}{1.840192in}}% +\pgfpathlineto{\pgfqpoint{4.623101in}{1.855001in}}% +\pgfpathlineto{\pgfqpoint{4.631785in}{1.860942in}}% +\pgfpathlineto{\pgfqpoint{4.640469in}{1.863546in}}% +\pgfpathlineto{\pgfqpoint{4.649154in}{1.862667in}}% +\pgfpathlineto{\pgfqpoint{4.657838in}{1.858175in}}% +\pgfpathlineto{\pgfqpoint{4.666522in}{1.849962in}}% +\pgfpathlineto{\pgfqpoint{4.675206in}{1.837945in}}% +\pgfpathlineto{\pgfqpoint{4.688232in}{1.812669in}}% +\pgfpathlineto{\pgfqpoint{4.701258in}{1.778650in}}% +\pgfpathlineto{\pgfqpoint{4.714284in}{1.736005in}}% +\pgfpathlineto{\pgfqpoint{4.731653in}{1.666324in}}% +\pgfpathlineto{\pgfqpoint{4.749021in}{1.583371in}}% +\pgfpathlineto{\pgfqpoint{4.770731in}{1.464716in}}% +\pgfpathlineto{\pgfqpoint{4.835862in}{1.093743in}}% +\pgfpathlineto{\pgfqpoint{4.848888in}{1.039690in}}% +\pgfpathlineto{\pgfqpoint{4.857572in}{1.012559in}}% +\pgfpathlineto{\pgfqpoint{4.866256in}{0.994439in}}% +\pgfpathlineto{\pgfqpoint{4.870598in}{0.989314in}}% +\pgfpathlineto{\pgfqpoint{4.874940in}{0.987128in}}% +\pgfpathlineto{\pgfqpoint{4.879282in}{0.988132in}}% +\pgfpathlineto{\pgfqpoint{4.883624in}{0.992590in}}% +\pgfpathlineto{\pgfqpoint{4.887966in}{1.000774in}}% +\pgfpathlineto{\pgfqpoint{4.896650in}{1.029477in}}% +\pgfpathlineto{\pgfqpoint{4.905335in}{1.076675in}}% +\pgfpathlineto{\pgfqpoint{4.914019in}{1.145012in}}% +\pgfpathlineto{\pgfqpoint{4.922703in}{1.237348in}}% +\pgfpathlineto{\pgfqpoint{4.931387in}{1.356777in}}% +\pgfpathlineto{\pgfqpoint{4.944413in}{1.594091in}}% +\pgfpathlineto{\pgfqpoint{4.957439in}{1.912304in}}% +\pgfpathlineto{\pgfqpoint{4.970193in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{4.970193in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{5.350000in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{5.350000in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.800000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.800000,0.800000,0.800000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.800000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.715177in}{1.609825in}}% +\pgfpathlineto{\pgfqpoint{1.610430in}{1.609825in}}% +\pgfpathquadraticcurveto{\pgfqpoint{1.638207in}{1.609825in}}{\pgfqpoint{1.638207in}{1.637603in}}% +\pgfpathlineto{\pgfqpoint{1.638207in}{2.204733in}}% +\pgfpathquadraticcurveto{\pgfqpoint{1.638207in}{2.232510in}}{\pgfqpoint{1.610430in}{2.232510in}}% +\pgfpathlineto{\pgfqpoint{0.715177in}{2.232510in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.687399in}{2.232510in}}{\pgfqpoint{0.687399in}{2.204733in}}% +\pgfpathlineto{\pgfqpoint{0.687399in}{1.637603in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.687399in}{1.609825in}}{\pgfqpoint{0.715177in}{1.609825in}}% +\pgfpathlineto{\pgfqpoint{0.715177in}{1.609825in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.742954in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{0.881843in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{1.020732in}{2.128344in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.131843in,y=2.079733in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=3\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.742954in}{1.934671in}}% +\pgfpathlineto{\pgfqpoint{0.881843in}{1.934671in}}% +\pgfpathlineto{\pgfqpoint{1.020732in}{1.934671in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.131843in,y=1.886060in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=6\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.742954in}{1.740998in}}% +\pgfpathlineto{\pgfqpoint{0.881843in}{1.740998in}}% +\pgfpathlineto{\pgfqpoint{1.020732in}{1.740998in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.131843in,y=1.692387in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=11\)}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/F_N_elliptic.pgf b/buch/papers/ellfilter/python/F_N_elliptic.pgf new file mode 100644 index 0000000..03084c6 --- /dev/null +++ b/buch/papers/ellfilter/python/F_N_elliptic.pgf @@ -0,0 +1,847 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{0.000000,0.501961,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{-174.068564in}}% +\pgfpathlineto{\pgfqpoint{2.247564in}{-174.068564in}}% +\pgfpathlineto{\pgfqpoint{2.247564in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{-174.068564in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.647059,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.247564in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{2.262704in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{2.262704in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{2.247564in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{2.247564in}{1.250043in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.262704in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{3.776737in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{3.776737in}{2.301962in}}% +\pgfpathlineto{\pgfqpoint{2.262704in}{2.301962in}}% +\pgfpathlineto{\pgfqpoint{2.262704in}{1.600680in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.733531in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.490547in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.490547in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.490547in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.490547in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.247564in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.247564in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.247564in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.247564in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.004580in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.004580in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.004580in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.004580in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.761597in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.761597in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.247564in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.348306in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{-4}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.899406in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.899406in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{0.899406in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.348306in, y=0.851181in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{-2}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.250043in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{1.250043in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.435112in, y=1.201818in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{0}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.600680in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{1.600680in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.435112in, y=1.552455in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{2}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{1.951318in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.951318in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{1.951318in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.435112in, y=1.903092in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{4}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.435112in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {10^{6}}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.292751in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle F^2_N(w)\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.739446in}{0.534880in}}% +\pgfpathlineto{\pgfqpoint{0.744132in}{0.623916in}}% +\pgfpathlineto{\pgfqpoint{0.750947in}{0.699506in}}% +\pgfpathlineto{\pgfqpoint{0.759276in}{0.759013in}}% +\pgfpathlineto{\pgfqpoint{0.769120in}{0.808295in}}% +\pgfpathlineto{\pgfqpoint{0.781235in}{0.852871in}}% +\pgfpathlineto{\pgfqpoint{0.794865in}{0.891083in}}% +\pgfpathlineto{\pgfqpoint{0.810009in}{0.924604in}}% +\pgfpathlineto{\pgfqpoint{0.827425in}{0.955729in}}% +\pgfpathlineto{\pgfqpoint{0.847112in}{0.984554in}}% +\pgfpathlineto{\pgfqpoint{0.869071in}{1.011252in}}% +\pgfpathlineto{\pgfqpoint{0.894059in}{1.036721in}}% +\pgfpathlineto{\pgfqpoint{0.922075in}{1.060823in}}% +\pgfpathlineto{\pgfqpoint{0.953878in}{1.084028in}}% +\pgfpathlineto{\pgfqpoint{0.989467in}{1.106127in}}% +\pgfpathlineto{\pgfqpoint{1.029598in}{1.127375in}}% +\pgfpathlineto{\pgfqpoint{1.075031in}{1.147865in}}% +\pgfpathlineto{\pgfqpoint{1.125764in}{1.167300in}}% +\pgfpathlineto{\pgfqpoint{1.182554in}{1.185675in}}% +\pgfpathlineto{\pgfqpoint{1.244645in}{1.202480in}}% +\pgfpathlineto{\pgfqpoint{1.312036in}{1.217494in}}% +\pgfpathlineto{\pgfqpoint{1.383214in}{1.230171in}}% +\pgfpathlineto{\pgfqpoint{1.455905in}{1.239991in}}% +\pgfpathlineto{\pgfqpoint{1.527083in}{1.246540in}}% +\pgfpathlineto{\pgfqpoint{1.594474in}{1.249707in}}% +\pgfpathlineto{\pgfqpoint{1.655808in}{1.249589in}}% +\pgfpathlineto{\pgfqpoint{1.711084in}{1.246442in}}% +\pgfpathlineto{\pgfqpoint{1.758788in}{1.240733in}}% +\pgfpathlineto{\pgfqpoint{1.800434in}{1.232740in}}% +\pgfpathlineto{\pgfqpoint{1.836780in}{1.222684in}}% +\pgfpathlineto{\pgfqpoint{1.867825in}{1.211013in}}% +\pgfpathlineto{\pgfqpoint{1.895085in}{1.197575in}}% +\pgfpathlineto{\pgfqpoint{1.919315in}{1.182199in}}% +\pgfpathlineto{\pgfqpoint{1.940517in}{1.165082in}}% +\pgfpathlineto{\pgfqpoint{1.959447in}{1.145758in}}% +\pgfpathlineto{\pgfqpoint{1.976106in}{1.124277in}}% +\pgfpathlineto{\pgfqpoint{1.991250in}{1.099472in}}% +\pgfpathlineto{\pgfqpoint{2.004122in}{1.072523in}}% +\pgfpathlineto{\pgfqpoint{2.015480in}{1.041896in}}% +\pgfpathlineto{\pgfqpoint{2.026081in}{1.004016in}}% +\pgfpathlineto{\pgfqpoint{2.035168in}{0.959254in}}% +\pgfpathlineto{\pgfqpoint{2.042740in}{0.905583in}}% +\pgfpathlineto{\pgfqpoint{2.048797in}{0.840043in}}% +\pgfpathlineto{\pgfqpoint{2.053341in}{0.758643in}}% +\pgfpathlineto{\pgfqpoint{2.056369in}{0.659102in}}% +\pgfpathlineto{\pgfqpoint{2.058129in}{0.534880in}}% +\pgfpathmoveto{\pgfqpoint{2.061041in}{0.534880in}}% +\pgfpathlineto{\pgfqpoint{2.064699in}{0.731366in}}% +\pgfpathlineto{\pgfqpoint{2.069999in}{0.841854in}}% +\pgfpathlineto{\pgfqpoint{2.076814in}{0.921040in}}% +\pgfpathlineto{\pgfqpoint{2.085143in}{0.984050in}}% +\pgfpathlineto{\pgfqpoint{2.095744in}{1.040507in}}% +\pgfpathlineto{\pgfqpoint{2.107859in}{1.088435in}}% +\pgfpathlineto{\pgfqpoint{2.121489in}{1.130355in}}% +\pgfpathlineto{\pgfqpoint{2.136633in}{1.167522in}}% +\pgfpathlineto{\pgfqpoint{2.153292in}{1.200289in}}% +\pgfpathlineto{\pgfqpoint{2.169193in}{1.224889in}}% +\pgfpathlineto{\pgfqpoint{2.182823in}{1.240496in}}% +\pgfpathlineto{\pgfqpoint{2.192666in}{1.247725in}}% +\pgfpathlineto{\pgfqpoint{2.200239in}{1.250017in}}% +\pgfpathlineto{\pgfqpoint{2.206296in}{1.248902in}}% +\pgfpathlineto{\pgfqpoint{2.211597in}{1.244804in}}% +\pgfpathlineto{\pgfqpoint{2.216897in}{1.236352in}}% +\pgfpathlineto{\pgfqpoint{2.222197in}{1.220917in}}% +\pgfpathlineto{\pgfqpoint{2.226741in}{1.197982in}}% +\pgfpathlineto{\pgfqpoint{2.231284in}{1.157051in}}% +\pgfpathlineto{\pgfqpoint{2.235070in}{1.089329in}}% +\pgfpathlineto{\pgfqpoint{2.237342in}{1.003949in}}% +\pgfpathlineto{\pgfqpoint{2.238856in}{0.869518in}}% +\pgfpathlineto{\pgfqpoint{2.239613in}{0.638914in}}% +\pgfpathlineto{\pgfqpoint{2.240370in}{0.794881in}}% +\pgfpathlineto{\pgfqpoint{2.243399in}{1.100517in}}% +\pgfpathlineto{\pgfqpoint{2.248700in}{1.280424in}}% +\pgfpathlineto{\pgfqpoint{2.266873in}{1.753784in}}% +\pgfpathlineto{\pgfqpoint{2.269144in}{1.924021in}}% +\pgfpathlineto{\pgfqpoint{2.270659in}{2.202839in}}% +\pgfpathlineto{\pgfqpoint{2.272930in}{1.848446in}}% +\pgfpathlineto{\pgfqpoint{2.276716in}{1.730165in}}% +\pgfpathlineto{\pgfqpoint{2.281260in}{1.672036in}}% +\pgfpathlineto{\pgfqpoint{2.286560in}{1.637950in}}% +\pgfpathlineto{\pgfqpoint{2.292618in}{1.617444in}}% +\pgfpathlineto{\pgfqpoint{2.298675in}{1.606779in}}% +\pgfpathlineto{\pgfqpoint{2.304733in}{1.601737in}}% +\pgfpathlineto{\pgfqpoint{2.311548in}{1.600286in}}% +\pgfpathlineto{\pgfqpoint{2.319120in}{1.602150in}}% +\pgfpathlineto{\pgfqpoint{2.328206in}{1.607676in}}% +\pgfpathlineto{\pgfqpoint{2.340322in}{1.618928in}}% +\pgfpathlineto{\pgfqpoint{2.355466in}{1.637536in}}% +\pgfpathlineto{\pgfqpoint{2.372881in}{1.664058in}}% +\pgfpathlineto{\pgfqpoint{2.391054in}{1.697587in}}% +\pgfpathlineto{\pgfqpoint{2.407713in}{1.734758in}}% +\pgfpathlineto{\pgfqpoint{2.422857in}{1.776122in}}% +\pgfpathlineto{\pgfqpoint{2.435729in}{1.820082in}}% +\pgfpathlineto{\pgfqpoint{2.447088in}{1.870149in}}% +\pgfpathlineto{\pgfqpoint{2.456174in}{1.923894in}}% +\pgfpathlineto{\pgfqpoint{2.463746in}{1.987030in}}% +\pgfpathlineto{\pgfqpoint{2.469804in}{2.064340in}}% +\pgfpathlineto{\pgfqpoint{2.474347in}{2.165039in}}% +\pgfpathlineto{\pgfqpoint{2.477435in}{2.315844in}}% +\pgfpathmoveto{\pgfqpoint{2.481180in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{2.484948in}{2.149178in}}% +\pgfpathlineto{\pgfqpoint{2.490248in}{2.050240in}}% +\pgfpathlineto{\pgfqpoint{2.497063in}{1.978983in}}% +\pgfpathlineto{\pgfqpoint{2.505392in}{1.923413in}}% +\pgfpathlineto{\pgfqpoint{2.515236in}{1.878185in}}% +\pgfpathlineto{\pgfqpoint{2.526594in}{1.840393in}}% +\pgfpathlineto{\pgfqpoint{2.539467in}{1.808260in}}% +\pgfpathlineto{\pgfqpoint{2.553854in}{1.780613in}}% +\pgfpathlineto{\pgfqpoint{2.569755in}{1.756622in}}% +\pgfpathlineto{\pgfqpoint{2.587928in}{1.734871in}}% +\pgfpathlineto{\pgfqpoint{2.608372in}{1.715370in}}% +\pgfpathlineto{\pgfqpoint{2.631089in}{1.698028in}}% +\pgfpathlineto{\pgfqpoint{2.656834in}{1.682284in}}% +\pgfpathlineto{\pgfqpoint{2.686365in}{1.667895in}}% +\pgfpathlineto{\pgfqpoint{2.720439in}{1.654789in}}% +\pgfpathlineto{\pgfqpoint{2.759814in}{1.642992in}}% +\pgfpathlineto{\pgfqpoint{2.806760in}{1.632261in}}% +\pgfpathlineto{\pgfqpoint{2.862036in}{1.622901in}}% +\pgfpathlineto{\pgfqpoint{2.928670in}{1.614877in}}% +\pgfpathlineto{\pgfqpoint{3.008934in}{1.608422in}}% +\pgfpathlineto{\pgfqpoint{3.108128in}{1.603650in}}% +\pgfpathlineto{\pgfqpoint{3.233824in}{1.600841in}}% +\pgfpathlineto{\pgfqpoint{3.396624in}{1.600449in}}% +\pgfpathlineto{\pgfqpoint{3.619242in}{1.603198in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.606074in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.606074in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.000000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.041667in}{-0.041667in}}{\pgfqpoint{0.041667in}{0.041667in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{-0.041667in}}% +\pgfpathcurveto{\pgfqpoint{0.011050in}{-0.041667in}}{\pgfqpoint{0.021649in}{-0.037276in}}{\pgfqpoint{0.029463in}{-0.029463in}}% +\pgfpathcurveto{\pgfqpoint{0.037276in}{-0.021649in}}{\pgfqpoint{0.041667in}{-0.011050in}}{\pgfqpoint{0.041667in}{0.000000in}}% +\pgfpathcurveto{\pgfqpoint{0.041667in}{0.011050in}}{\pgfqpoint{0.037276in}{0.021649in}}{\pgfqpoint{0.029463in}{0.029463in}}% +\pgfpathcurveto{\pgfqpoint{0.021649in}{0.037276in}}{\pgfqpoint{0.011050in}{0.041667in}}{\pgfqpoint{0.000000in}{0.041667in}}% +\pgfpathcurveto{\pgfqpoint{-0.011050in}{0.041667in}}{\pgfqpoint{-0.021649in}{0.037276in}}{\pgfqpoint{-0.029463in}{0.029463in}}% +\pgfpathcurveto{\pgfqpoint{-0.037276in}{0.021649in}}{\pgfqpoint{-0.041667in}{0.011050in}}{\pgfqpoint{-0.041667in}{0.000000in}}% +\pgfpathcurveto{\pgfqpoint{-0.041667in}{-0.011050in}}{\pgfqpoint{-0.037276in}{-0.021649in}}{\pgfqpoint{-0.029463in}{-0.029463in}}% +\pgfpathcurveto{\pgfqpoint{-0.021649in}{-0.037276in}}{\pgfqpoint{-0.011050in}{-0.041667in}}{\pgfqpoint{0.000000in}{-0.041667in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.041667in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.733531in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{2.050740in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{2.247564in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.733531in}{0.548769in}}{\pgfqpoint{3.028066in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.000000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.041667in}{-0.041667in}}{\pgfqpoint{0.041667in}{0.041667in}}{% +\pgfpathmoveto{\pgfqpoint{-0.041667in}{-0.041667in}}% +\pgfpathlineto{\pgfqpoint{0.041667in}{0.041667in}}% +\pgfpathmoveto{\pgfqpoint{-0.041667in}{0.041667in}}% +\pgfpathlineto{\pgfqpoint{0.041667in}{-0.041667in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.262704in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{2.482239in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.733531in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.733531in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.800000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.800000,0.800000,0.800000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.800000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.830753in}{1.997171in}}% +\pgfpathlineto{\pgfqpoint{2.157621in}{1.997171in}}% +\pgfpathquadraticcurveto{\pgfqpoint{2.185399in}{1.997171in}}{\pgfqpoint{2.185399in}{2.024949in}}% +\pgfpathlineto{\pgfqpoint{2.185399in}{2.204733in}}% +\pgfpathquadraticcurveto{\pgfqpoint{2.185399in}{2.232510in}}{\pgfqpoint{2.157621in}{2.232510in}}% +\pgfpathlineto{\pgfqpoint{0.830753in}{2.232510in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.802975in}{2.232510in}}{\pgfqpoint{0.802975in}{2.204733in}}% +\pgfpathlineto{\pgfqpoint{0.802975in}{2.024949in}}% +\pgfpathquadraticcurveto{\pgfqpoint{0.802975in}{1.997171in}}{\pgfqpoint{0.830753in}{1.997171in}}% +\pgfpathlineto{\pgfqpoint{0.830753in}{1.997171in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.858531in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{0.997420in}{2.128344in}}% +\pgfpathlineto{\pgfqpoint{1.136309in}{2.128344in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.247420in,y=2.079733in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle N=5, k=0.1\)}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/chebychef.py b/buch/papers/ellfilter/python/chebychef.py new file mode 100644 index 0000000..254ad4b --- /dev/null +++ b/buch/papers/ellfilter/python/chebychef.py @@ -0,0 +1,66 @@ +# %% + +import matplotlib.pyplot as plt +import scipy.signal +import numpy as np + + +order = 5 +passband_ripple_db = 1 +omega_c = 1000 + +a, b = scipy.signal.cheby1( + order, + passband_ripple_db, + omega_c, + btype='low', + analog=True, + output='ba', + fs=None, +) + +w, mag, phase = scipy.signal.bode((a, b), w=np.linspace(0,2000,256)) +f, axs = plt.subplots(2,1, sharex=True) +axs[0].plot(w, 10**(mag/20)) +axs[0].set_ylabel("$|H(\omega)| /$ db") +axs[0].grid(True, "both") +axs[1].plot(w, phase) +axs[1].set_ylabel(r"$arg H (\omega) / $ deg") +axs[1].grid(True, "both") +axs[1].set_xlim([0, 2000]) +axs[1].set_xlabel("$\omega$") +plt.show() + + +# %% Cheychev filter F_N plot + +w = np.linspace(-1.1,1.1, 1000) +plt.figure(figsize=(5.5,2.5)) +for N in [3,6,11]: + # F_N = np.cos(N * np.arccos(w)) + F_N = scipy.special.eval_chebyt(N, w) + plt.plot(w, F_N, label=f"$N={N}$") +plt.xlim([-1.2,1.2]) +plt.ylim([-2,2]) +plt.grid() +plt.xlabel("$w$") +plt.ylabel("$T_N(w)$") +plt.legend() +plt.tight_layout() +plt.savefig("F_N_chebychev2.pgf") +plt.show() + +# %% Build Chebychev polynomials + +N = 11 + +zeros = (np.arange(N)+0.5) * np.pi +zeros = np.cos(zeros/N) + +x = np.linspace(-1.2,1.2,1000) +y = np.prod(x[:, None] - zeros[None, :], axis=-1)*2**(N-1) + +plt.plot(x, y) +plt.ylim([-1,1]) +plt.grid() +plt.show() diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf new file mode 100644 index 0000000..31b77d4 --- /dev/null +++ b/buch/papers/ellfilter/python/elliptic.pgf @@ -0,0 +1,709 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{4.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{0.000000,0.501961,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.788459in}}% +\pgfpathlineto{\pgfqpoint{2.189776in}{1.788459in}}% +\pgfpathlineto{\pgfqpoint{2.189776in}{3.541645in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{3.541645in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{1.788459in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.647059,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.189776in}{0.724087in}}% +\pgfpathlineto{\pgfqpoint{2.205494in}{0.724087in}}% +\pgfpathlineto{\pgfqpoint{2.205494in}{1.788459in}}% +\pgfpathlineto{\pgfqpoint{2.189776in}{1.788459in}}% +\pgfpathlineto{\pgfqpoint{2.189776in}{0.724087in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.200000}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.200000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.205494in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.777315in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.777315in}{0.724087in}}% +\pgfpathlineto{\pgfqpoint{2.205494in}{0.724087in}}% +\pgfpathlineto{\pgfqpoint{2.205494in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.617954in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.403865in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{1.403865in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.403865in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.403865in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.189776in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.189776in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.189776in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.189776in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.975686in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.975686in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.975686in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.975686in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.761597in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.761597in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.189776in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.899406in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.899406in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{0.899406in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=0.851181in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.250043in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.250043in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{1.250043in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=1.201818in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.4}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.600680in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.600680in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{1.600680in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=1.552455in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.6}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{1.951318in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{1.951318in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{1.951318in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=1.903092in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.8}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.617954in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.343262in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.287707in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle |H(w)|\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.646254in}{2.300410in}}% +\pgfpathlineto{\pgfqpoint{0.674554in}{2.295805in}}% +\pgfpathlineto{\pgfqpoint{0.703640in}{2.287983in}}% +\pgfpathlineto{\pgfqpoint{0.734298in}{2.276528in}}% +\pgfpathlineto{\pgfqpoint{0.767315in}{2.260797in}}% +\pgfpathlineto{\pgfqpoint{0.802690in}{2.240472in}}% +\pgfpathlineto{\pgfqpoint{0.842781in}{2.213774in}}% +\pgfpathlineto{\pgfqpoint{0.889947in}{2.178486in}}% +\pgfpathlineto{\pgfqpoint{0.952050in}{2.127836in}}% +\pgfpathlineto{\pgfqpoint{1.147791in}{1.965399in}}% +\pgfpathlineto{\pgfqpoint{1.205963in}{1.922684in}}% +\pgfpathlineto{\pgfqpoint{1.257846in}{1.888382in}}% +\pgfpathlineto{\pgfqpoint{1.305012in}{1.860803in}}% +\pgfpathlineto{\pgfqpoint{1.349034in}{1.838524in}}% +\pgfpathlineto{\pgfqpoint{1.390698in}{1.820826in}}% +\pgfpathlineto{\pgfqpoint{1.430003in}{1.807434in}}% +\pgfpathlineto{\pgfqpoint{1.466950in}{1.798051in}}% +\pgfpathlineto{\pgfqpoint{1.501539in}{1.792361in}}% +\pgfpathlineto{\pgfqpoint{1.534555in}{1.790011in}}% +\pgfpathlineto{\pgfqpoint{1.566000in}{1.790875in}}% +\pgfpathlineto{\pgfqpoint{1.595872in}{1.794823in}}% +\pgfpathlineto{\pgfqpoint{1.624172in}{1.801709in}}% +\pgfpathlineto{\pgfqpoint{1.650899in}{1.811365in}}% +\pgfpathlineto{\pgfqpoint{1.676841in}{1.824030in}}% +\pgfpathlineto{\pgfqpoint{1.701996in}{1.839797in}}% +\pgfpathlineto{\pgfqpoint{1.726365in}{1.858757in}}% +\pgfpathlineto{\pgfqpoint{1.749949in}{1.880985in}}% +\pgfpathlineto{\pgfqpoint{1.773532in}{1.907483in}}% +\pgfpathlineto{\pgfqpoint{1.797115in}{1.938719in}}% +\pgfpathlineto{\pgfqpoint{1.820698in}{1.975139in}}% +\pgfpathlineto{\pgfqpoint{1.845068in}{2.018557in}}% +\pgfpathlineto{\pgfqpoint{1.871009in}{2.071217in}}% +\pgfpathlineto{\pgfqpoint{1.903240in}{2.144258in}}% +\pgfpathlineto{\pgfqpoint{1.949620in}{2.249449in}}% +\pgfpathlineto{\pgfqpoint{1.965342in}{2.277408in}}% +\pgfpathlineto{\pgfqpoint{1.977134in}{2.292569in}}% +\pgfpathlineto{\pgfqpoint{1.986567in}{2.299856in}}% +\pgfpathlineto{\pgfqpoint{1.993642in}{2.301922in}}% +\pgfpathlineto{\pgfqpoint{2.000717in}{2.300685in}}% +\pgfpathlineto{\pgfqpoint{2.007792in}{2.295839in}}% +\pgfpathlineto{\pgfqpoint{2.014867in}{2.287135in}}% +\pgfpathlineto{\pgfqpoint{2.023514in}{2.271025in}}% +\pgfpathlineto{\pgfqpoint{2.032947in}{2.246492in}}% +\pgfpathlineto{\pgfqpoint{2.043953in}{2.209014in}}% +\pgfpathlineto{\pgfqpoint{2.057317in}{2.152165in}}% +\pgfpathlineto{\pgfqpoint{2.076183in}{2.056775in}}% +\pgfpathlineto{\pgfqpoint{2.114702in}{1.858945in}}% +\pgfpathlineto{\pgfqpoint{2.126494in}{1.815249in}}% +\pgfpathlineto{\pgfqpoint{2.134355in}{1.796399in}}% +\pgfpathlineto{\pgfqpoint{2.139858in}{1.790308in}}% +\pgfpathlineto{\pgfqpoint{2.143002in}{1.790254in}}% +\pgfpathlineto{\pgfqpoint{2.146147in}{1.793264in}}% +\pgfpathlineto{\pgfqpoint{2.150077in}{1.802263in}}% +\pgfpathlineto{\pgfqpoint{2.154794in}{1.822852in}}% +\pgfpathlineto{\pgfqpoint{2.159510in}{1.857784in}}% +\pgfpathlineto{\pgfqpoint{2.165013in}{1.924261in}}% +\pgfpathlineto{\pgfqpoint{2.170516in}{2.030210in}}% +\pgfpathlineto{\pgfqpoint{2.181521in}{2.301670in}}% +\pgfpathlineto{\pgfqpoint{2.182308in}{2.299748in}}% +\pgfpathlineto{\pgfqpoint{2.183880in}{2.267565in}}% +\pgfpathlineto{\pgfqpoint{2.186238in}{2.135783in}}% +\pgfpathlineto{\pgfqpoint{2.192527in}{1.496420in}}% +\pgfpathlineto{\pgfqpoint{2.198816in}{1.001269in}}% +\pgfpathlineto{\pgfqpoint{2.205105in}{0.731898in}}% +\pgfpathlineto{\pgfqpoint{2.211393in}{0.583247in}}% +\pgfpathlineto{\pgfqpoint{2.213752in}{0.552138in}}% +\pgfpathlineto{\pgfqpoint{2.220827in}{0.630495in}}% +\pgfpathlineto{\pgfqpoint{2.227902in}{0.675607in}}% +\pgfpathlineto{\pgfqpoint{2.234977in}{0.701566in}}% +\pgfpathlineto{\pgfqpoint{2.241266in}{0.714626in}}% +\pgfpathlineto{\pgfqpoint{2.247554in}{0.721456in}}% +\pgfpathlineto{\pgfqpoint{2.253843in}{0.723972in}}% +\pgfpathlineto{\pgfqpoint{2.260918in}{0.723210in}}% +\pgfpathlineto{\pgfqpoint{2.269565in}{0.718770in}}% +\pgfpathlineto{\pgfqpoint{2.281357in}{0.708826in}}% +\pgfpathlineto{\pgfqpoint{2.300224in}{0.688160in}}% +\pgfpathlineto{\pgfqpoint{2.385123in}{0.590361in}}% +\pgfpathlineto{\pgfqpoint{2.417354in}{0.559882in}}% +\pgfpathlineto{\pgfqpoint{2.430717in}{0.549063in}}% +\pgfpathlineto{\pgfqpoint{2.463734in}{0.574407in}}% +\pgfpathlineto{\pgfqpoint{2.498323in}{0.597083in}}% +\pgfpathlineto{\pgfqpoint{2.535270in}{0.617584in}}% +\pgfpathlineto{\pgfqpoint{2.574575in}{0.635868in}}% +\pgfpathlineto{\pgfqpoint{2.617811in}{0.652529in}}% +\pgfpathlineto{\pgfqpoint{2.664977in}{0.667359in}}% +\pgfpathlineto{\pgfqpoint{2.717646in}{0.680619in}}% +\pgfpathlineto{\pgfqpoint{2.776604in}{0.692216in}}% +\pgfpathlineto{\pgfqpoint{2.843424in}{0.702155in}}% +\pgfpathlineto{\pgfqpoint{2.920462in}{0.710422in}}% +\pgfpathlineto{\pgfqpoint{3.010864in}{0.716921in}}% +\pgfpathlineto{\pgfqpoint{3.118561in}{0.721464in}}% +\pgfpathlineto{\pgfqpoint{3.250627in}{0.723829in}}% +\pgfpathlineto{\pgfqpoint{3.419640in}{0.723619in}}% +\pgfpathlineto{\pgfqpoint{3.651542in}{0.720038in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.717600in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.717600in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py new file mode 100644 index 0000000..b3336a1 --- /dev/null +++ b/buch/papers/ellfilter/python/elliptic.py @@ -0,0 +1,356 @@ + +# %% + +import scipy.special +import scipyx as spx +import numpy as np +import matplotlib.pyplot as plt +from matplotlib.patches import Rectangle + +import plot_params + +def last_color(): + return plt.gca().lines[-1].get_color() + +# define elliptic functions + +def ell_int(k): + """ Calculate K(k) """ + m = k**2 + return scipy.special.ellipk(m) + +def sn(z, k): + return spx.ellipj(z, k**2)[0] + +def cn(z, k): + return spx.ellipj(z, k**2)[1] + +def dn(z, k): + return spx.ellipj(z, k**2)[2] + +def cd(z, k): + sn, cn, dn, ph = spx.ellipj(z, k**2) + return cn / dn + +# https://mathworld.wolfram.com/JacobiEllipticFunctions.html eq 3-8 + +def sn_inv(z, k): + m = k**2 + return scipy.special.ellipkinc(np.arcsin(z), m) + +def cn_inv(z, k): + m = k**2 + return scipy.special.ellipkinc(np.arccos(z), m) + +def dn_inv(z, k): + m = k**2 + x = np.sqrt((1-z**2) / k**2) + return scipy.special.ellipkinc(np.arcsin(x), m) + +def cd_inv(z, k): + m = k**2 + x = np.sqrt(((m - 1) * z**2) / (m*z**2 - 1)) + return scipy.special.ellipkinc(np.arccos(x), m) + + +k = 0.8 +z = 0.5 + +assert np.allclose(sn_inv(sn(z ,k), k), z) +assert np.allclose(cn_inv(cn(z ,k), k), z) +assert np.allclose(dn_inv(dn(z ,k), k), z) +assert np.allclose(cd_inv(cd(z ,k), k), z) + + +# %% Buttwerworth filter F_N plot + +w = np.linspace(0,1.5, 100) +plt.figure(figsize=(4,2.5)) + +for N in range(1,5): + F_N = w**N + plt.plot(w, F_N**2, label=f"$N={N}$") +plt.gca().add_patch(Rectangle( + (0, 0), + 1, 1, + fc ='green', + alpha=0.2, + lw = 10, +)) +plt.gca().add_patch(Rectangle( + (1, 1), + 0.5, 1, + fc ='orange', + alpha=0.2, + lw = 10, +)) +plt.xlim([0,1.5]) +plt.ylim([0,2]) +plt.grid() +plt.xlabel("$w$") +plt.ylabel("$F^2_N(w)$") +plt.legend() +plt.tight_layout() +plt.savefig("F_N_butterworth.pgf") +plt.show() + +# %% Cheychev filter F_N plot + +w = np.linspace(0,1.5, 100) + +plt.figure(figsize=(4,2.5)) +for N in range(1,5): + # F_N = np.cos(N * np.arccos(w)) + F_N = scipy.special.eval_chebyt(N, w) + plt.plot(w, F_N**2, label=f"$N={N}$") +plt.gca().add_patch(Rectangle( + (0, 0), + 1, 1, + fc ='green', + alpha=0.2, + lw = 10, +)) +plt.gca().add_patch(Rectangle( + (1, 1), + 0.5, 1, + fc ='orange', + alpha=0.2, + lw = 10, +)) +plt.xlim([0,1.5]) +plt.ylim([0,2]) +plt.grid() +plt.xlabel("$w$") +plt.ylabel("$F^2_N(w)$") +plt.legend() +plt.tight_layout() +plt.savefig("F_N_chebychev.pgf") +plt.show() + + +# %% plot arcsin + +def lattice(a1, b1, c1, a2, b2, c2): + r1 = np.logspace(a1, b1, c1) + x1 = np.concatenate((-np.flip(r1), [0], r1), axis=0) + x1 = x1.astype(np.complex128) + r2 = np.logspace(a2, b2, c2) + x2 = np.concatenate((-np.flip(r2), [0], r2), axis=0) + x2 = x2.astype(np.complex128) + x = (x1[:, None] + (x2[None, :] * 1j)) + return x + +plt.figure(figsize=(12,12)) +y = np.arcsin(lattice(-1,6,1000, -1,5,10)) +plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5) +y = np.arcsin(lattice(-1,6,10, -1,5,100)).T +plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5) +y = np.arcsin(lattice(-1,6,10, -1,5,10)) +plt.plot(np.real(y), np.imag(y), ".", color="red", lw=0.5) +plt.show() + +# %% plot cd^-1 TODO complex cd^-1 missing + + +r = np.logspace(-1,8, 50) + + + +x = np.concatenate((-np.flip(r), [0], r), axis=0) +y = cd_inv(x, 0.99) + +plt.figure(figsize=(12,12)) +plt.plot(np.real(y), np.imag(y), "-") +plt.show() + +# %%plot cd +plt.figure(figsize=(10,6)) +z = np.linspace(-4,4, 500) +for k in [0, 0.9, 0.99, 0.999, 0.99999]: + w = cd(z*ell_int(k), k) + plt.plot(z, w, label=f"$k={k}$") +plt.grid() +plt.legend() +# plt.xlim([-4,4]) +plt.xlabel("$u$") +plt.ylabel("$cd(uK, k)$") +plt.show() + +# %% Test ???? + +N = 5 +k = 0.9 +k1 = k**N + +assert np.allclose(k**(-N), k1**(-1)) + +K = ell_int(k) +Kp = ell_int(np.sqrt(1-k**2)) + +K1 = ell_int(k1) +Kp1 = ell_int(np.sqrt(1-k1**2)) + +print(Kp * (K1 / K) * N, Kp1) + + +# %% + + +k = 0.9 +k_prim = np.sqrt(1 - k**2) +K = ell_int(k) +Kp = ell_int(k_prim) + +print(K, Kp) + +zs = [ + 0 + (K + 0j) * np.linspace(0,1,25), + K + (Kp*1j) * np.linspace(0,1,25), + (K + Kp*1j) + (-K) * np.linspace(0,1,25), +] + + +for z in zs: + plt.plot(np.real(z), np.imag(z)) +plt.show() + + + +for z in zs: + w = cd(z, k) + plt.plot(np.real(w), np.imag(w)) +plt.show() + + + + + +# %% + +for i in range(10): + x = np.linspace(i*1,i*1+1,10, dtype=np.complex64) + w = np.arccos(x) + + x2 = np.cos(w) + x4 = np.cos(w+ 2*np.pi) + x3 = np.cos(np.conj(w)) + + assert np.allclose(x2, x4, rtol=0.001, atol=1e-5) + + assert np.allclose(x2, x3) + assert np.allclose(x2, x, rtol=0.001, atol=1e-5) + + plt.plot(np.real(w), np.imag(w), ".-") + +for i in range(10): + x = -np.linspace(i*1,i*1+1,100, dtype=np.complex64) + w = np.arccos(x) + plt.plot(np.real(w), np.imag(w), ".-") + +plt.grid() +plt.show() + + + + +# %% + +plt.plot(omega, np.abs(G)) +plt.show() + + +def cd_inv(u, m): + return K(1/2) - F(np.arcsin()) + +def K(m): + return scipy.special.ellipk(m) + +def L(n, xi): + return 1 #TODO + +def R(n, xi, x): + cn(n*K(1/L(n, xi))/K(1/xi) * cd_inv(x, 1/xi, 1/L(n, xi))) + +epsilon = 0.1 +n = 3 +omega = np.linspace(0, np.pi, 1000) +omega_0 = 1 +xi = 1.1 + +G = 1 / np.sqrt(1 + epsilon**2 * R(n, xi, omega/omega_0)**2) + + +plt.plot(omega, np.abs(G)) +plt.show() + + + +# %% Chebychef + +epsilon = 0.5 +omega = np.linspace(0, np.pi, 1000) +omega_0 = 1 +n = 4 + +def chebychef_poly(n, x): + x = x.astype(np.complex64) + y = np.cos(n* np.arccos(x)) + return np.real(y) + +F_omega = chebychef_poly + +for n in (1,2,3,4): + plt.plot(omega, F_omega(n, omega/omega_0)**2) +plt.ylim([0,5]) +plt.xlim([0,1.5]) +plt.grid() +plt.show() + +for n in (1,2,3,4): + G = 1 / np.sqrt(1 + epsilon**2 * F_omega(n, omega/omega_0)**2) + plt.plot(omega, np.abs(G)) +plt.grid() +plt.show() + + + + +# %% + + +k = np.concatenate(([0.00001,0.0001,0.001], np.linspace(0,1,101)[1:-1], [0.999,0.9999, 0.99999]), axis=0) +K = ell_int(k) +K_prime = ell_int(np.sqrt(1-k**2)) + + +f, axs = plt.subplots(1,2, figsize=(5,2.5)) +axs[0].plot(k, K, linewidth=0.1) +axs[0].text(k[30], K[30]+0.1, f"$K$") +axs[0].plot(k, K_prime, linewidth=0.1) +axs[0].text(k[30], K_prime[30]+0.1, f"$K^\prime$") +axs[0].set_xlim([0,1]) +axs[0].set_ylim([0,4]) +axs[0].set_xlabel("$k$") + +axs[1].axvline(x=np.pi/2, color="gray", linewidth=0.5) +axs[1].axhline(y=np.pi/2, color="gray", linewidth=0.5) +axs[1].text(0.1, np.pi/2 + 0.1, "$\pi/2$") +axs[1].text(np.pi/2+0.1, 0.1, "$\pi/2$") +axs[1].plot(K, K_prime, linewidth=1) + +k = np.array([0.1,0.2,0.4,0.6,0.9,0.99]) +K = ell_int(k) +K_prime = ell_int(np.sqrt(1-k**2)) + +axs[1].plot(K, K_prime, '.', color=last_color(), markersize=2) +for x, y, n in zip(K, K_prime, k): + axs[1].text(x+0.1, y+0.1, f"$k={n:.2f}$", rotation_mode="anchor") +axs[1].set_ylabel("$K^\prime$") +axs[1].set_xlabel("$K$") +axs[1].set_xlim([0,6]) +axs[1].set_ylim([0,5]) +plt.tight_layout() +plt.savefig("k.pgf") +plt.show() + +print(K[0], K[-1]) diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py new file mode 100644 index 0000000..29c6f47 --- /dev/null +++ b/buch/papers/ellfilter/python/elliptic2.py @@ -0,0 +1,149 @@ +# %% + +import matplotlib.pyplot as plt +import scipy.signal +import numpy as np +import matplotlib +from matplotlib.patches import Rectangle + +import plot_params + +def ellip_filter(N): + + order = N + passband_ripple_db = 3 + stopband_attenuation_db = 20 + omega_c = 1 + + a, b = scipy.signal.ellip( + order, + passband_ripple_db, + stopband_attenuation_db, + omega_c, + btype='low', + analog=True, + output='ba', + fs=None + ) + + w, mag_db, phase = scipy.signal.bode((a, b), w=np.linspace(0*omega_c,2*omega_c, 4000)) + + mag = 10**(mag_db/20) + + passband_ripple = 10**(-passband_ripple_db/20) + epsilon2 = (1/passband_ripple)**2 - 1 + + FN2 = ((1/mag**2) - 1) + + return w/omega_c, FN2 / epsilon2, mag, a, b + + +plt.figure(figsize=(4,2.5)) + +for N in [5]: + w, FN2, mag, a, b = ellip_filter(N) + plt.semilogy(w, FN2, label=f"$N={N}, k=0.1$", linewidth=1) + +plt.gca().add_patch(Rectangle( + (0, 0), + 1, 1, + fc ='green', + alpha=0.2, + lw = 10, +)) +plt.gca().add_patch(Rectangle( + (1, 1), + 0.01, 1e2-1, + fc ='orange', + alpha=0.2, + lw = 10, +)) + +plt.gca().add_patch(Rectangle( + (1.01, 100), + 1, 1e6, + fc ='red', + alpha=0.2, + lw = 10, +)) + +zeros = [0,0.87,1] +poles = [1.01,1.155] + +import matplotlib.transforms +plt.plot( # mark errors as vertical bars + zeros, + np.zeros_like(zeros), + "o", + mfc='none', + color='black', + transform=matplotlib.transforms.blended_transform_factory( + plt.gca().transData, + plt.gca().transAxes, + ), +) +plt.plot( # mark errors as vertical bars + poles, + np.ones_like(poles), + "x", + mfc='none', + color='black', + transform=matplotlib.transforms.blended_transform_factory( + plt.gca().transData, + plt.gca().transAxes, + ), +) + +plt.xlim([0,2]) +plt.ylim([1e-4,1e6]) +plt.grid() +plt.xlabel("$w$") +plt.ylabel("$F^2_N(w)$") +plt.legend() +plt.tight_layout() +plt.savefig("F_N_elliptic.pgf") +plt.show() + + + +plt.figure(figsize=(4,2.5)) +plt.plot(w, mag, linewidth=1) + +plt.gca().add_patch(Rectangle( + (0, np.sqrt(2)/2), + 1, 1, + fc ='green', + alpha=0.2, + lw = 10, +)) +plt.gca().add_patch(Rectangle( + (1, 0.1), + 0.01, np.sqrt(2)/2 - 0.1, + fc ='orange', + alpha=0.2, + lw = 10, +)) + +plt.gca().add_patch(Rectangle( + (1.01, 0), + 1, 0.1, + fc ='red', + alpha=0.2, + lw = 10, +)) + +plt.grid() +plt.xlim([0,2]) +plt.ylim([0,1]) +plt.xlabel("$w$") +plt.ylabel("$|H(w)|$") +plt.tight_layout() +plt.savefig("elliptic.pgf") +plt.show() + +print("zeros", a) +print("poles", b) + + + + diff --git a/buch/papers/ellfilter/python/k.pgf b/buch/papers/ellfilter/python/k.pgf new file mode 100644 index 0000000..95d61d4 --- /dev/null +++ b/buch/papers/ellfilter/python/k.pgf @@ -0,0 +1,1157 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{5.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{5.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.316407in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.256930in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.256930in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.316407in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{0.316407in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.316407in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.801538in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.801538in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.25}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.286669in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.286669in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.50}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.771800in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.771800in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.256930in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.256930in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.286669in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.149740in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{0.987065in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.149740in, y=0.938840in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{1.425362in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.149740in, y=1.377137in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{1.863658in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.149740in, y=1.815433in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {3}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.316407in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.149740in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {4}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.316407in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.100375pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.316427in}{1.237243in}}% +\pgfpathlineto{\pgfqpoint{0.316601in}{1.237243in}}% +\pgfpathlineto{\pgfqpoint{0.318348in}{1.237244in}}% +\pgfpathlineto{\pgfqpoint{0.335813in}{1.237261in}}% +\pgfpathlineto{\pgfqpoint{0.355218in}{1.237312in}}% +\pgfpathlineto{\pgfqpoint{0.374623in}{1.237398in}}% +\pgfpathlineto{\pgfqpoint{0.394028in}{1.237519in}}% +\pgfpathlineto{\pgfqpoint{0.413434in}{1.237674in}}% +\pgfpathlineto{\pgfqpoint{0.432839in}{1.237864in}}% +\pgfpathlineto{\pgfqpoint{0.452244in}{1.238089in}}% +\pgfpathlineto{\pgfqpoint{0.471649in}{1.238349in}}% +\pgfpathlineto{\pgfqpoint{0.491054in}{1.238644in}}% +\pgfpathlineto{\pgfqpoint{0.510460in}{1.238974in}}% +\pgfpathlineto{\pgfqpoint{0.529865in}{1.239340in}}% +\pgfpathlineto{\pgfqpoint{0.549270in}{1.239742in}}% +\pgfpathlineto{\pgfqpoint{0.568675in}{1.240180in}}% +\pgfpathlineto{\pgfqpoint{0.588081in}{1.240655in}}% +\pgfpathlineto{\pgfqpoint{0.607486in}{1.241166in}}% +\pgfpathlineto{\pgfqpoint{0.626891in}{1.241714in}}% +\pgfpathlineto{\pgfqpoint{0.646296in}{1.242300in}}% +\pgfpathlineto{\pgfqpoint{0.665702in}{1.242924in}}% +\pgfpathlineto{\pgfqpoint{0.685107in}{1.243586in}}% +\pgfpathlineto{\pgfqpoint{0.704512in}{1.244287in}}% +\pgfpathlineto{\pgfqpoint{0.723917in}{1.245028in}}% +\pgfpathlineto{\pgfqpoint{0.743322in}{1.245809in}}% +\pgfpathlineto{\pgfqpoint{0.762728in}{1.246630in}}% +\pgfpathlineto{\pgfqpoint{0.782133in}{1.247492in}}% +\pgfpathlineto{\pgfqpoint{0.801538in}{1.248396in}}% +\pgfpathlineto{\pgfqpoint{0.820943in}{1.249343in}}% +\pgfpathlineto{\pgfqpoint{0.840349in}{1.250333in}}% +\pgfpathlineto{\pgfqpoint{0.859754in}{1.251367in}}% +\pgfpathlineto{\pgfqpoint{0.879159in}{1.252446in}}% +\pgfpathlineto{\pgfqpoint{0.898564in}{1.253571in}}% +\pgfpathlineto{\pgfqpoint{0.917969in}{1.254743in}}% +\pgfpathlineto{\pgfqpoint{0.937375in}{1.255962in}}% +\pgfpathlineto{\pgfqpoint{0.956780in}{1.257230in}}% +\pgfpathlineto{\pgfqpoint{0.976185in}{1.258548in}}% +\pgfpathlineto{\pgfqpoint{0.995590in}{1.259917in}}% +\pgfpathlineto{\pgfqpoint{1.014996in}{1.261339in}}% +\pgfpathlineto{\pgfqpoint{1.034401in}{1.262814in}}% +\pgfpathlineto{\pgfqpoint{1.053806in}{1.264344in}}% +\pgfpathlineto{\pgfqpoint{1.073211in}{1.265930in}}% +\pgfpathlineto{\pgfqpoint{1.092617in}{1.267575in}}% +\pgfpathlineto{\pgfqpoint{1.112022in}{1.269279in}}% +\pgfpathlineto{\pgfqpoint{1.131427in}{1.271045in}}% +\pgfpathlineto{\pgfqpoint{1.150832in}{1.272874in}}% +\pgfpathlineto{\pgfqpoint{1.170237in}{1.274768in}}% +\pgfpathlineto{\pgfqpoint{1.189643in}{1.276729in}}% +\pgfpathlineto{\pgfqpoint{1.209048in}{1.278760in}}% +\pgfpathlineto{\pgfqpoint{1.228453in}{1.280863in}}% +\pgfpathlineto{\pgfqpoint{1.247858in}{1.283040in}}% +\pgfpathlineto{\pgfqpoint{1.267264in}{1.285294in}}% +\pgfpathlineto{\pgfqpoint{1.286669in}{1.287627in}}% +\pgfpathlineto{\pgfqpoint{1.306074in}{1.290044in}}% +\pgfpathlineto{\pgfqpoint{1.325479in}{1.292546in}}% +\pgfpathlineto{\pgfqpoint{1.344884in}{1.295137in}}% +\pgfpathlineto{\pgfqpoint{1.364290in}{1.297822in}}% +\pgfpathlineto{\pgfqpoint{1.383695in}{1.300603in}}% +\pgfpathlineto{\pgfqpoint{1.403100in}{1.303485in}}% +\pgfpathlineto{\pgfqpoint{1.422505in}{1.306473in}}% +\pgfpathlineto{\pgfqpoint{1.441911in}{1.309570in}}% +\pgfpathlineto{\pgfqpoint{1.461316in}{1.312784in}}% +\pgfpathlineto{\pgfqpoint{1.480721in}{1.316118in}}% +\pgfpathlineto{\pgfqpoint{1.500126in}{1.319579in}}% +\pgfpathlineto{\pgfqpoint{1.519532in}{1.323174in}}% +\pgfpathlineto{\pgfqpoint{1.538937in}{1.326910in}}% +\pgfpathlineto{\pgfqpoint{1.558342in}{1.330793in}}% +\pgfpathlineto{\pgfqpoint{1.577747in}{1.334833in}}% +\pgfpathlineto{\pgfqpoint{1.597152in}{1.339039in}}% +\pgfpathlineto{\pgfqpoint{1.616558in}{1.343420in}}% +\pgfpathlineto{\pgfqpoint{1.635963in}{1.347988in}}% +\pgfpathlineto{\pgfqpoint{1.655368in}{1.352753in}}% +\pgfpathlineto{\pgfqpoint{1.674773in}{1.357730in}}% +\pgfpathlineto{\pgfqpoint{1.694179in}{1.362933in}}% +\pgfpathlineto{\pgfqpoint{1.713584in}{1.368377in}}% +\pgfpathlineto{\pgfqpoint{1.732989in}{1.374081in}}% +\pgfpathlineto{\pgfqpoint{1.752394in}{1.380064in}}% +\pgfpathlineto{\pgfqpoint{1.771800in}{1.386349in}}% +\pgfpathlineto{\pgfqpoint{1.791205in}{1.392961in}}% +\pgfpathlineto{\pgfqpoint{1.810610in}{1.399927in}}% +\pgfpathlineto{\pgfqpoint{1.830015in}{1.407281in}}% +\pgfpathlineto{\pgfqpoint{1.849420in}{1.415059in}}% +\pgfpathlineto{\pgfqpoint{1.868826in}{1.423303in}}% +\pgfpathlineto{\pgfqpoint{1.888231in}{1.432062in}}% +\pgfpathlineto{\pgfqpoint{1.907636in}{1.441392in}}% +\pgfpathlineto{\pgfqpoint{1.927041in}{1.451361in}}% +\pgfpathlineto{\pgfqpoint{1.946447in}{1.462048in}}% +\pgfpathlineto{\pgfqpoint{1.965852in}{1.473546in}}% +\pgfpathlineto{\pgfqpoint{1.985257in}{1.485971in}}% +\pgfpathlineto{\pgfqpoint{2.004662in}{1.499462in}}% +\pgfpathlineto{\pgfqpoint{2.024067in}{1.514194in}}% +\pgfpathlineto{\pgfqpoint{2.043473in}{1.530388in}}% +\pgfpathlineto{\pgfqpoint{2.062878in}{1.548326in}}% +\pgfpathlineto{\pgfqpoint{2.082283in}{1.568383in}}% +\pgfpathlineto{\pgfqpoint{2.101688in}{1.591069in}}% +\pgfpathlineto{\pgfqpoint{2.121094in}{1.617098in}}% +\pgfpathlineto{\pgfqpoint{2.140499in}{1.647519in}}% +\pgfpathlineto{\pgfqpoint{2.159904in}{1.683962in}}% +\pgfpathlineto{\pgfqpoint{2.179309in}{1.729164in}}% +\pgfpathlineto{\pgfqpoint{2.198715in}{1.788269in}}% +\pgfpathlineto{\pgfqpoint{2.218120in}{1.872854in}}% +\pgfpathlineto{\pgfqpoint{2.237525in}{2.019955in}}% +\pgfpathlineto{\pgfqpoint{2.247876in}{2.315844in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.316407in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.100375pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.454821in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{0.471649in}{2.265444in}}% +\pgfpathlineto{\pgfqpoint{0.491054in}{2.214262in}}% +\pgfpathlineto{\pgfqpoint{0.510460in}{2.168554in}}% +\pgfpathlineto{\pgfqpoint{0.529865in}{2.127278in}}% +\pgfpathlineto{\pgfqpoint{0.549270in}{2.089666in}}% +\pgfpathlineto{\pgfqpoint{0.568675in}{2.055132in}}% +\pgfpathlineto{\pgfqpoint{0.588081in}{2.023222in}}% +\pgfpathlineto{\pgfqpoint{0.607486in}{1.993575in}}% +\pgfpathlineto{\pgfqpoint{0.626891in}{1.965899in}}% +\pgfpathlineto{\pgfqpoint{0.646296in}{1.939958in}}% +\pgfpathlineto{\pgfqpoint{0.665702in}{1.915554in}}% +\pgfpathlineto{\pgfqpoint{0.685107in}{1.892522in}}% +\pgfpathlineto{\pgfqpoint{0.704512in}{1.870720in}}% +\pgfpathlineto{\pgfqpoint{0.723917in}{1.850031in}}% +\pgfpathlineto{\pgfqpoint{0.743322in}{1.830351in}}% +\pgfpathlineto{\pgfqpoint{0.762728in}{1.811591in}}% +\pgfpathlineto{\pgfqpoint{0.782133in}{1.793672in}}% +\pgfpathlineto{\pgfqpoint{0.801538in}{1.776528in}}% +\pgfpathlineto{\pgfqpoint{0.820943in}{1.760096in}}% +\pgfpathlineto{\pgfqpoint{0.840349in}{1.744324in}}% +\pgfpathlineto{\pgfqpoint{0.859754in}{1.729164in}}% +\pgfpathlineto{\pgfqpoint{0.879159in}{1.714573in}}% +\pgfpathlineto{\pgfqpoint{0.898564in}{1.700513in}}% +\pgfpathlineto{\pgfqpoint{0.917969in}{1.686948in}}% +\pgfpathlineto{\pgfqpoint{0.937375in}{1.673849in}}% +\pgfpathlineto{\pgfqpoint{0.956780in}{1.661185in}}% +\pgfpathlineto{\pgfqpoint{0.976185in}{1.648932in}}% +\pgfpathlineto{\pgfqpoint{0.995590in}{1.637065in}}% +\pgfpathlineto{\pgfqpoint{1.014996in}{1.625563in}}% +\pgfpathlineto{\pgfqpoint{1.034401in}{1.614406in}}% +\pgfpathlineto{\pgfqpoint{1.053806in}{1.603575in}}% +\pgfpathlineto{\pgfqpoint{1.073211in}{1.593053in}}% +\pgfpathlineto{\pgfqpoint{1.092617in}{1.582826in}}% +\pgfpathlineto{\pgfqpoint{1.112022in}{1.572877in}}% +\pgfpathlineto{\pgfqpoint{1.131427in}{1.563195in}}% +\pgfpathlineto{\pgfqpoint{1.150832in}{1.553766in}}% +\pgfpathlineto{\pgfqpoint{1.170237in}{1.544578in}}% +\pgfpathlineto{\pgfqpoint{1.189643in}{1.535621in}}% +\pgfpathlineto{\pgfqpoint{1.209048in}{1.526884in}}% +\pgfpathlineto{\pgfqpoint{1.228453in}{1.518359in}}% +\pgfpathlineto{\pgfqpoint{1.247858in}{1.510036in}}% +\pgfpathlineto{\pgfqpoint{1.267264in}{1.501906in}}% +\pgfpathlineto{\pgfqpoint{1.286669in}{1.493962in}}% +\pgfpathlineto{\pgfqpoint{1.306074in}{1.486197in}}% +\pgfpathlineto{\pgfqpoint{1.325479in}{1.478603in}}% +\pgfpathlineto{\pgfqpoint{1.344884in}{1.471174in}}% +\pgfpathlineto{\pgfqpoint{1.364290in}{1.463903in}}% +\pgfpathlineto{\pgfqpoint{1.383695in}{1.456785in}}% +\pgfpathlineto{\pgfqpoint{1.403100in}{1.449815in}}% +\pgfpathlineto{\pgfqpoint{1.422505in}{1.442986in}}% +\pgfpathlineto{\pgfqpoint{1.441911in}{1.436294in}}% +\pgfpathlineto{\pgfqpoint{1.461316in}{1.429735in}}% +\pgfpathlineto{\pgfqpoint{1.480721in}{1.423303in}}% +\pgfpathlineto{\pgfqpoint{1.500126in}{1.416995in}}% +\pgfpathlineto{\pgfqpoint{1.519532in}{1.410805in}}% +\pgfpathlineto{\pgfqpoint{1.538937in}{1.404732in}}% +\pgfpathlineto{\pgfqpoint{1.558342in}{1.398770in}}% +\pgfpathlineto{\pgfqpoint{1.577747in}{1.392916in}}% +\pgfpathlineto{\pgfqpoint{1.597152in}{1.387167in}}% +\pgfpathlineto{\pgfqpoint{1.616558in}{1.381520in}}% +\pgfpathlineto{\pgfqpoint{1.635963in}{1.375971in}}% +\pgfpathlineto{\pgfqpoint{1.655368in}{1.370518in}}% +\pgfpathlineto{\pgfqpoint{1.674773in}{1.365158in}}% +\pgfpathlineto{\pgfqpoint{1.694179in}{1.359888in}}% +\pgfpathlineto{\pgfqpoint{1.713584in}{1.354705in}}% +\pgfpathlineto{\pgfqpoint{1.732989in}{1.349607in}}% +\pgfpathlineto{\pgfqpoint{1.752394in}{1.344593in}}% +\pgfpathlineto{\pgfqpoint{1.771800in}{1.339658in}}% +\pgfpathlineto{\pgfqpoint{1.791205in}{1.334802in}}% +\pgfpathlineto{\pgfqpoint{1.810610in}{1.330022in}}% +\pgfpathlineto{\pgfqpoint{1.830015in}{1.325316in}}% +\pgfpathlineto{\pgfqpoint{1.849420in}{1.320682in}}% +\pgfpathlineto{\pgfqpoint{1.868826in}{1.316118in}}% +\pgfpathlineto{\pgfqpoint{1.888231in}{1.311623in}}% +\pgfpathlineto{\pgfqpoint{1.907636in}{1.307195in}}% +\pgfpathlineto{\pgfqpoint{1.927041in}{1.302831in}}% +\pgfpathlineto{\pgfqpoint{1.946447in}{1.298531in}}% +\pgfpathlineto{\pgfqpoint{1.965852in}{1.294293in}}% +\pgfpathlineto{\pgfqpoint{1.985257in}{1.290116in}}% +\pgfpathlineto{\pgfqpoint{2.004662in}{1.285997in}}% +\pgfpathlineto{\pgfqpoint{2.024067in}{1.281936in}}% +\pgfpathlineto{\pgfqpoint{2.043473in}{1.277931in}}% +\pgfpathlineto{\pgfqpoint{2.062878in}{1.273982in}}% +\pgfpathlineto{\pgfqpoint{2.082283in}{1.270085in}}% +\pgfpathlineto{\pgfqpoint{2.101688in}{1.266241in}}% +\pgfpathlineto{\pgfqpoint{2.121094in}{1.262449in}}% +\pgfpathlineto{\pgfqpoint{2.140499in}{1.258706in}}% +\pgfpathlineto{\pgfqpoint{2.159904in}{1.255013in}}% +\pgfpathlineto{\pgfqpoint{2.179309in}{1.251367in}}% +\pgfpathlineto{\pgfqpoint{2.198715in}{1.247768in}}% +\pgfpathlineto{\pgfqpoint{2.218120in}{1.244215in}}% +\pgfpathlineto{\pgfqpoint{2.237525in}{1.240707in}}% +\pgfpathlineto{\pgfqpoint{2.254990in}{1.237588in}}% +\pgfpathlineto{\pgfqpoint{2.256736in}{1.237278in}}% +\pgfpathlineto{\pgfqpoint{2.256911in}{1.237247in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.316407in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{0.316407in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.256930in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.256930in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.316407in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.256930in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.316407in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.256930in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.859754in,y=1.295197in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle K\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.859754in,y=1.772994in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle K^\prime\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.874885in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.874885in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{2.874885in}{0.548769in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.874885in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.521726in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.521726in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{4.168566in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.168566in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {4}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{4.815407in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.815407in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {6}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.845146in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle K\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{0.548769in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{0.899406in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=0.851181in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{1.250043in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=1.201818in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{1.600680in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=1.552455in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {3}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{1.951318in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=1.903092in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {4}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.874885in}{2.301955in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.708218in, y=2.253730in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {5}\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.652662in,y=1.425362in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle K^\prime\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.501875pt}% +\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.382912in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{3.382912in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.501875pt}% +\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.874885in}{1.099548in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{1.099548in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.383004in}{2.315844in}}% +\pgfpathlineto{\pgfqpoint{3.383027in}{2.264692in}}% +\pgfpathlineto{\pgfqpoint{3.383116in}{2.164019in}}% +\pgfpathlineto{\pgfqpoint{3.383230in}{2.086013in}}% +\pgfpathlineto{\pgfqpoint{3.383370in}{2.022353in}}% +\pgfpathlineto{\pgfqpoint{3.383536in}{1.968602in}}% +\pgfpathlineto{\pgfqpoint{3.383728in}{1.922109in}}% +\pgfpathlineto{\pgfqpoint{3.383946in}{1.881163in}}% +\pgfpathlineto{\pgfqpoint{3.384190in}{1.844597in}}% +\pgfpathlineto{\pgfqpoint{3.384460in}{1.811577in}}% +\pgfpathlineto{\pgfqpoint{3.384756in}{1.781487in}}% +\pgfpathlineto{\pgfqpoint{3.385079in}{1.753859in}}% +\pgfpathlineto{\pgfqpoint{3.385429in}{1.728331in}}% +\pgfpathlineto{\pgfqpoint{3.385807in}{1.704613in}}% +\pgfpathlineto{\pgfqpoint{3.386211in}{1.682473in}}% +\pgfpathlineto{\pgfqpoint{3.386644in}{1.661720in}}% +\pgfpathlineto{\pgfqpoint{3.387104in}{1.642197in}}% +\pgfpathlineto{\pgfqpoint{3.387593in}{1.623771in}}% +\pgfpathlineto{\pgfqpoint{3.388110in}{1.606330in}}% +\pgfpathlineto{\pgfqpoint{3.388657in}{1.589779in}}% +\pgfpathlineto{\pgfqpoint{3.389233in}{1.574035in}}% +\pgfpathlineto{\pgfqpoint{3.389839in}{1.559026in}}% +\pgfpathlineto{\pgfqpoint{3.390475in}{1.544692in}}% +\pgfpathlineto{\pgfqpoint{3.391142in}{1.530976in}}% +\pgfpathlineto{\pgfqpoint{3.391841in}{1.517831in}}% +\pgfpathlineto{\pgfqpoint{3.392571in}{1.505213in}}% +\pgfpathlineto{\pgfqpoint{3.393334in}{1.493085in}}% +\pgfpathlineto{\pgfqpoint{3.394130in}{1.481412in}}% +\pgfpathlineto{\pgfqpoint{3.394960in}{1.470164in}}% +\pgfpathlineto{\pgfqpoint{3.395825in}{1.459313in}}% +\pgfpathlineto{\pgfqpoint{3.396725in}{1.448833in}}% +\pgfpathlineto{\pgfqpoint{3.397661in}{1.438702in}}% +\pgfpathlineto{\pgfqpoint{3.398633in}{1.428899in}}% +\pgfpathlineto{\pgfqpoint{3.399643in}{1.419406in}}% +\pgfpathlineto{\pgfqpoint{3.400692in}{1.410204in}}% +\pgfpathlineto{\pgfqpoint{3.401781in}{1.401278in}}% +\pgfpathlineto{\pgfqpoint{3.402910in}{1.392614in}}% +\pgfpathlineto{\pgfqpoint{3.404081in}{1.384197in}}% +\pgfpathlineto{\pgfqpoint{3.405294in}{1.376014in}}% +\pgfpathlineto{\pgfqpoint{3.406552in}{1.368056in}}% +\pgfpathlineto{\pgfqpoint{3.407855in}{1.360310in}}% +\pgfpathlineto{\pgfqpoint{3.409204in}{1.352766in}}% +\pgfpathlineto{\pgfqpoint{3.410602in}{1.345416in}}% +\pgfpathlineto{\pgfqpoint{3.412049in}{1.338250in}}% +\pgfpathlineto{\pgfqpoint{3.413548in}{1.331261in}}% +\pgfpathlineto{\pgfqpoint{3.415099in}{1.324441in}}% +\pgfpathlineto{\pgfqpoint{3.416706in}{1.317782in}}% +\pgfpathlineto{\pgfqpoint{3.418369in}{1.311279in}}% +\pgfpathlineto{\pgfqpoint{3.420091in}{1.304923in}}% +\pgfpathlineto{\pgfqpoint{3.421874in}{1.298711in}}% +\pgfpathlineto{\pgfqpoint{3.423720in}{1.292636in}}% +\pgfpathlineto{\pgfqpoint{3.425632in}{1.286693in}}% +\pgfpathlineto{\pgfqpoint{3.427613in}{1.280876in}}% +\pgfpathlineto{\pgfqpoint{3.429665in}{1.275182in}}% +\pgfpathlineto{\pgfqpoint{3.431792in}{1.269606in}}% +\pgfpathlineto{\pgfqpoint{3.433997in}{1.264143in}}% +\pgfpathlineto{\pgfqpoint{3.436283in}{1.258789in}}% +\pgfpathlineto{\pgfqpoint{3.438654in}{1.253542in}}% +\pgfpathlineto{\pgfqpoint{3.441114in}{1.248396in}}% +\pgfpathlineto{\pgfqpoint{3.443668in}{1.243349in}}% +\pgfpathlineto{\pgfqpoint{3.446321in}{1.238398in}}% +\pgfpathlineto{\pgfqpoint{3.449077in}{1.233539in}}% +\pgfpathlineto{\pgfqpoint{3.451943in}{1.228770in}}% +\pgfpathlineto{\pgfqpoint{3.454924in}{1.224087in}}% +\pgfpathlineto{\pgfqpoint{3.458028in}{1.219487in}}% +\pgfpathlineto{\pgfqpoint{3.461261in}{1.214970in}}% +\pgfpathlineto{\pgfqpoint{3.464631in}{1.210531in}}% +\pgfpathlineto{\pgfqpoint{3.468147in}{1.206168in}}% +\pgfpathlineto{\pgfqpoint{3.471820in}{1.201880in}}% +\pgfpathlineto{\pgfqpoint{3.475659in}{1.197664in}}% +\pgfpathlineto{\pgfqpoint{3.479676in}{1.193518in}}% +\pgfpathlineto{\pgfqpoint{3.483885in}{1.189440in}}% +\pgfpathlineto{\pgfqpoint{3.488300in}{1.185428in}}% +\pgfpathlineto{\pgfqpoint{3.492938in}{1.181480in}}% +\pgfpathlineto{\pgfqpoint{3.497817in}{1.177595in}}% +\pgfpathlineto{\pgfqpoint{3.502957in}{1.173771in}}% +\pgfpathlineto{\pgfqpoint{3.508384in}{1.170006in}}% +\pgfpathlineto{\pgfqpoint{3.514123in}{1.166299in}}% +\pgfpathlineto{\pgfqpoint{3.520206in}{1.162648in}}% +\pgfpathlineto{\pgfqpoint{3.526670in}{1.159052in}}% +\pgfpathlineto{\pgfqpoint{3.533555in}{1.155509in}}% +\pgfpathlineto{\pgfqpoint{3.540911in}{1.152019in}}% +\pgfpathlineto{\pgfqpoint{3.548796in}{1.148579in}}% +\pgfpathlineto{\pgfqpoint{3.557281in}{1.145189in}}% +\pgfpathlineto{\pgfqpoint{3.566449in}{1.141846in}}% +\pgfpathlineto{\pgfqpoint{3.576405in}{1.138552in}}% +\pgfpathlineto{\pgfqpoint{3.587275in}{1.135303in}}% +\pgfpathlineto{\pgfqpoint{3.599224in}{1.132099in}}% +\pgfpathlineto{\pgfqpoint{3.612461in}{1.128939in}}% +\pgfpathlineto{\pgfqpoint{3.627261in}{1.125822in}}% +\pgfpathlineto{\pgfqpoint{3.644002in}{1.122747in}}% +\pgfpathlineto{\pgfqpoint{3.663208in}{1.119713in}}% +\pgfpathlineto{\pgfqpoint{3.685656in}{1.116719in}}% +\pgfpathlineto{\pgfqpoint{3.712547in}{1.113764in}}% +\pgfpathlineto{\pgfqpoint{3.745902in}{1.110847in}}% +\pgfpathlineto{\pgfqpoint{3.789516in}{1.107968in}}% +\pgfpathlineto{\pgfqpoint{3.851932in}{1.105126in}}% +\pgfpathlineto{\pgfqpoint{3.960478in}{1.102320in}}% +\pgfpathlineto{\pgfqpoint{4.328852in}{1.099824in}}% +\pgfpathlineto{\pgfqpoint{4.700641in}{1.099576in}}% +\pgfpathlineto{\pgfqpoint{4.829296in}{1.099567in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}% +\pgfusepath{clip}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.006944in}{-0.006944in}}{\pgfqpoint{0.006944in}{0.006944in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{-0.006944in}}% +\pgfpathcurveto{\pgfqpoint{0.001842in}{-0.006944in}}{\pgfqpoint{0.003608in}{-0.006213in}}{\pgfqpoint{0.004910in}{-0.004910in}}% +\pgfpathcurveto{\pgfqpoint{0.006213in}{-0.003608in}}{\pgfqpoint{0.006944in}{-0.001842in}}{\pgfqpoint{0.006944in}{0.000000in}}% +\pgfpathcurveto{\pgfqpoint{0.006944in}{0.001842in}}{\pgfqpoint{0.006213in}{0.003608in}}{\pgfqpoint{0.004910in}{0.004910in}}% +\pgfpathcurveto{\pgfqpoint{0.003608in}{0.006213in}}{\pgfqpoint{0.001842in}{0.006944in}}{\pgfqpoint{0.000000in}{0.006944in}}% +\pgfpathcurveto{\pgfqpoint{-0.001842in}{0.006944in}}{\pgfqpoint{-0.003608in}{0.006213in}}{\pgfqpoint{-0.004910in}{0.004910in}}% +\pgfpathcurveto{\pgfqpoint{-0.006213in}{0.003608in}}{\pgfqpoint{-0.006944in}{0.001842in}}{\pgfqpoint{-0.006944in}{0.000000in}}% +\pgfpathcurveto{\pgfqpoint{-0.006944in}{-0.001842in}}{\pgfqpoint{-0.006213in}{-0.003608in}}{\pgfqpoint{-0.004910in}{-0.004910in}}% +\pgfpathcurveto{\pgfqpoint{-0.003608in}{-0.006213in}}{\pgfqpoint{-0.001842in}{-0.006944in}}{\pgfqpoint{0.000000in}{-0.006944in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.006944in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.384190in}{1.844597in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.388110in}{1.606330in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.405294in}{1.376014in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.441114in}{1.248396in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.612461in}{1.128939in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.960478in}{1.102320in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.874885in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{2.874885in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.815407in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.874885in}{0.548769in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{0.548769in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.874885in}{2.301955in}}% +\pgfpathlineto{\pgfqpoint{4.815407in}{2.301955in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.907227in,y=1.134612in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \pi/2\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.415254in,y=0.583833in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \pi/2\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.416532in,y=1.879661in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.10\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.420452in,y=1.641394in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.20\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.437636in,y=1.411078in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.40\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.473456in,y=1.283460in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.60\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.644803in,y=1.164003in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.90\)}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.992820in,y=1.137383in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.99\)}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/ellfilter/python/plot_params.py b/buch/papers/ellfilter/python/plot_params.py new file mode 100644 index 0000000..4ddd1d8 --- /dev/null +++ b/buch/papers/ellfilter/python/plot_params.py @@ -0,0 +1,9 @@ +import matplotlib + +matplotlib.rcParams.update({ + "pgf.texsystem": "pdflatex", + 'font.family': 'serif', + 'font.size': 9, + 'text.usetex': True, + 'pgf.rcfonts': False, +}) diff --git a/buch/papers/ellfilter/references.bib b/buch/papers/ellfilter/references.bib index 81b3577..8f21971 100644 --- a/buch/papers/ellfilter/references.bib +++ b/buch/papers/ellfilter/references.bib @@ -4,32 +4,19 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{ellfilter:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{ellfilter:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{ellfilter:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@online{ellfilter:bib:orfanidis, + author = { Sophocles J. Orfanidis}, + title = { LECTURE NOTES ON ELLIPTIC FILTER DESIGN }, + year = 2006, + url = {https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf} } +% Schwalm +% https://en.wikipedia.org/wiki/Elliptic_rational_functions +% https://en.wikipedia.org/wiki/Rational_function +% https://en.wikipedia.org/wiki/Jacobi_elliptic_functions +% https://de.wikipedia.org/wiki/Elliptisches_Integral +% https://de.wikipedia.org/wiki/Tschebyschow-Polynom +% https://en.wikipedia.org/wiki/Chebyshev_filter +% https://mathworld.wolfram.com/JacobiEllipticFunctions.html +% https://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html diff --git a/buch/papers/ellfilter/teil0.tex b/buch/papers/ellfilter/teil0.tex deleted file mode 100644 index fd04ba9..0000000 --- a/buch/papers/ellfilter/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{ellfilter:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{ellfilter:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/ellfilter/teil1.tex b/buch/papers/ellfilter/teil1.tex deleted file mode 100644 index 7e62a2f..0000000 --- a/buch/papers/ellfilter/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{ellfilter:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{ellfilter:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{ellfilter:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{ellfilter:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{ellfilter:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/ellfilter/teil2.tex b/buch/papers/ellfilter/teil2.tex deleted file mode 100644 index 71fdc6d..0000000 --- a/buch/papers/ellfilter/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{ellfilter:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{ellfilter:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/ellfilter/teil3.tex b/buch/papers/ellfilter/teil3.tex deleted file mode 100644 index 79a5f3d..0000000 --- a/buch/papers/ellfilter/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{ellfilter:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{ellfilter:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/ellfilter/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex new file mode 100644 index 0000000..2772620 --- /dev/null +++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex @@ -0,0 +1,66 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z$}; + \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$}; + + \begin{scope}[xscale=0.6] + + \clip(-7.5,-2) rectangle (7.5,2); + + \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5); + \draw[thick, ->, orange] (1, 0) -- (0,0); + \draw[thick, ->, red] (2, 0) -- (1,0); + \draw[thick, ->, blue] (2,1.5) -- (2, 0); + + \foreach \i in {-2,...,1} { + \begin{scope}[opacity=0.5, xshift=\i*4cm] + \draw[->, orange] (-1, 0) -- (0,0); + \draw[->, darkgreen] (0, 0) -- (0,1.5); + \draw[->, darkgreen] (0, 0) -- (0,-1.5); + \draw[->, orange] (1, 0) -- (0,0); + \draw[->, red] (2, 0) -- (1,0); + \draw[->, blue] (2,1.5) -- (2, 0); + \draw[->, blue] (2,-1.5) -- (2, 0); + \draw[->, red] (2, 0) -- (3,0); + + \node[zero] at (1,0) {}; + \node[zero] at (3,0) {}; + \end{scope} + } + + \node[gray, anchor=north] at (-6,0) {$-3\pi$}; + \node[gray, anchor=north] at (-4,0) {$-2\pi$}; + \node[gray, anchor=north] at (-2,0) {$-\pi$}; + % \node[gray, anchor=north] at (0,0) {$0$}; + \node[gray, anchor=north] at (2,0) {$\pi$}; + \node[gray, anchor=north] at (4,0) {$2\pi$}; + \node[gray, anchor=north] at (6,0) {$3\pi$}; + + \node[gray, anchor=east] at (0,-1.5) {$-\infty$}; + % \node[gray, anchor=south east] at (0, 0) {$0$}; + \node[gray, anchor=east] at (0, 1.5) {$\infty$}; + + \end{scope} + + \begin{scope}[yshift=-2.5cm] + + \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$w$}; + + \draw[thick, ->, blue] (-4, 0) -- (-2, 0); + \draw[thick, ->, red] (-2, 0) -- (0, 0); + \draw[thick, ->, orange] (0, 0) -- (2, 0); + \draw[thick, ->, darkgreen] (2, 0) -- (4, 0); + + \node[anchor=south] at (-4,0) {$-\infty$}; + \node[anchor=south] at (-2,0) {$-1$}; + \node[anchor=south] at (0,0) {$0$}; + \node[anchor=south] at (2,0) {$1$}; + \node[anchor=south] at (4,0) {$\infty$}; + + \end{scope} + + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex new file mode 100644 index 0000000..3fc3cc6 --- /dev/null +++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex @@ -0,0 +1,45 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \begin{scope}[xscale=0.5] + + \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z_1$}; + \draw[gray, ->] (-10,0) -- (10,0) node[anchor=west]{$\mathrm{Re}~z_1$}; + + \begin{scope} + + \draw[>->, line width=0.05, thick, blue] (2, 1.5) -- (2,0.05) -- node[anchor=south, pos=0.5]{$N=1$} (0.1,0.05) -- (0.1,1.5); + \draw[>->, line width=0.05, thick, orange] (4, 1.5) -- (4,0) -- node[anchor=south, pos=0.25]{$N=2$} (0,0) -- (0,1.5); + \draw[>->, line width=0.05, thick, red] (6, 1.5) node[anchor=north west]{$-\infty$} -- (6,-0.05) node[anchor=west]{$-1$} -- node[anchor=north]{$0$} node[anchor=south, pos=0.1666]{$N=3$} (-0.1,-0.05) node[anchor=east]{$1$} -- (-0.1,1.5) node[anchor=north east]{$\infty$}; + + + \node[zero] at (-7,0) {}; + \node[zero] at (-5,0) {}; + \node[zero] at (-3,0) {}; + \node[zero] at (-1,0) {}; + \node[zero] at (1,0) {}; + \node[zero] at (3,0) {}; + \node[zero] at (5,0) {}; + \node[zero] at (7,0) {}; + + + \end{scope} + + \node[gray, anchor=north] at (-8,0) {$-4\pi$}; + \node[gray, anchor=north] at (-6,0) {$-3\pi$}; + \node[gray, anchor=north] at (-4,0) {$-2\pi$}; + \node[gray, anchor=north] at (-2,0) {$-\pi$}; + \node[gray, anchor=north] at (2,0) {$\pi$}; + \node[gray, anchor=north] at (4,0) {$2\pi$}; + \node[gray, anchor=north] at (6,0) {$3\pi$}; + \node[gray, anchor=north] at (8,0) {$4\pi$}; + + + \node[gray, anchor=east] at (0,-1.5) {$-\infty$}; + \node[gray, anchor=east] at (0, 1.5) {$\infty$}; + + \end{scope} + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tikz/cd.tikz.tex b/buch/papers/ellfilter/tikz/cd.tikz.tex new file mode 100644 index 0000000..7155a85 --- /dev/null +++ b/buch/papers/ellfilter/tikz/cd.tikz.tex @@ -0,0 +1,87 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \begin{scope}[xscale=1, yscale=2] + + \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$}; + \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$}; + + \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$}; + + \draw[gray] (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$}; + + + \begin{scope} + + \begin{scope}[xshift=0cm] + + \clip(-4.5,-1.25) rectangle (4.5,1.25); + + \fill[yellow!30] (0,0) rectangle (1, 0.5); + + + \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5); + \draw[thick, ->, orange] (1, 0) -- (0,0); + \draw[thick, ->, red] (2, 0) -- (1,0); + \draw[thick, ->, blue] (2,0.5) -- (2, 0); + \draw[thick, ->, purple] (1, 0.5) -- (2,0.5); + \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5); + + + + \foreach \i in {-2,...,1} { + \foreach \j in {-2,...,1} { + \begin{scope}[xshift=\i*4cm, yshift=\j*1cm] + \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5); + \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0); + \draw[opacity=0.5, ->, red] (2, 0) -- (1,0); + \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0); + \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5); + \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5); + \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5); + \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1); + \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5); + \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5); + \draw[opacity=0.5, ->, red] (2, 0) -- (3,0); + \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0); + + \node[zero] at ( 1, 0) {}; + \node[zero] at ( 3, 0) {}; + \node[pole] at ( 1,0.5) {}; + \node[pole] at ( 3,0.5) {}; + + \end{scope} + } + } + + \end{scope} + + \end{scope} + + \end{scope} + + \begin{scope}[yshift=-3.5cm, xscale=0.75] + + \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$}; + + \draw[thick, ->, purple] (-5, 0) -- (-3, 0); + \draw[thick, ->, blue] (-3, 0) -- (-2, 0); + \draw[thick, ->, red] (-2, 0) -- (0, 0); + \draw[thick, ->, orange] (0, 0) -- (2, 0); + \draw[thick, ->, darkgreen] (2, 0) -- (3, 0); + \draw[thick, ->, cyan] (3, 0) -- (5, 0); + + \node[anchor=south] at (-5,0) {$-\infty$}; + \node[anchor=south] at (-3,0) {$-1/k$}; + \node[anchor=south] at (-2,0) {$-1$}; + \node[anchor=south] at (0,0) {$0$}; + \node[anchor=south] at (2,0) {$1$}; + \node[anchor=south] at (3,0) {$1/k$}; + \node[anchor=south] at (5,0) {$\infty$}; + + \end{scope} + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex new file mode 100644 index 0000000..0743f7d --- /dev/null +++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex @@ -0,0 +1,84 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm] + + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \begin{scope}[xscale=1.25, yscale=2.5] + + \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z_1$}; + \draw[gray, ->] (-1.5,0) -- (6,0) node[anchor=west]{$\mathrm{Re}~z_1$}; + + \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$}; + \draw[gray] ( 5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $5K_1$}; + \draw[gray] (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime_1$}; + + \begin{scope} + + \clip(-1.5,-0.75) rectangle (6.8,1.25); + + % \draw[>->, line width=0.05, thick, blue] (1, 0.45) -- (2, 0.45) -- (2, 0.05) -- ( 0.1, 0.05) -- ( 0.1,0.45) -- (1, 0.45); + % \draw[>->, line width=0.05, thick, orange] (2, 0.5 ) -- (4, 0.5 ) -- (4, 0 ) -- ( 0 , 0 ) -- ( 0 ,0.5 ) -- (2, 0.5 ); + % \draw[>->, line width=0.05, thick, red] (3, 0.55) -- (6, 0.55) -- (6,-0.05) -- (-0.1,-0.05) -- (-0.1,0.55) -- (3, 0.55); + % \node[blue] at (1, 0.25) {$N=1$}; + % \node[orange] at (3, 0.25) {$N=2$}; + % \node[red] at (5, 0.25) {$N=3$}; + + + + % \draw[line width=0.1cm, fill, red!50] (0,0) rectangle (3, 0.5); + % \draw[line width=0.05cm, fill, orange!50] (0,0) rectangle (2, 0.5); + % \fill[yellow!50] (0,0) rectangle (1, 0.5); + % \node[] at (0.5, 0.25) {\small $N=1$}; + % \node[] at (1.5, 0.25) {\small $N=2$}; + % \node[] at (2.5, 0.25) {\small $N=3$}; + + \fill[orange!30] (0,0) rectangle (5, 0.5); + \fill[yellow!30] (0,0) rectangle (1, 0.5); + \node[] at (2.5, 0.25) {\small $N=5$}; + + + \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm] + (5,0.5) node(t_k_unten){} -- node[above, yshift=0.1cm]{$NK$} + (0,0.5) node(t_k_opt_unten){}; + + \draw[decorate,decoration={brace,amplitude=3pt,mirror}, xshift=0.1cm] + (5,0) node(t_k_unten){} -- node[right, xshift=0.1cm]{$K^\prime \frac{K_1N}{K} = K^\prime_1$} + (5,0.5) node(t_k_opt_unten){}; + + \foreach \i in {-2,...,1} { + \foreach \j in {-2,...,1} { + \begin{scope}[xshift=\i*4cm, yshift=\j*1cm] + + \node[zero] at ( 1, 0) {}; + \node[zero] at ( 3, 0) {}; + \node[pole] at ( 1,0.5) {}; + \node[pole] at ( 3,0.5) {}; + + \end{scope} + } + } + + + + + \draw[thick, ->, darkgreen] (5, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0); + \draw[thick, ->, orange] (-0, 0) -- node[align=center]{Übergangs-\\berech} (0,0.5); + \draw[thick, ->, red] (0,0.5) -- node[align=center, yshift=0.5cm]{Sperrbereich} (5, 0.5); + + \draw (4,0 ) node[dot]{} node[anchor=south] {\small $1$}; + \draw (2,0 ) node[dot]{} node[anchor=south] {\small $-1$}; + \draw (0,0 ) node[dot]{} node[anchor=south west] {\small $1$}; + \draw (0,0.5) node[dot]{} node[anchor=north west] {\small $1/k$}; + \draw (2,0.5) node[dot]{} node[anchor=north] {\small $-1/k$}; + \draw (4,0.5) node[dot]{} node[anchor=north] {\small $1/k$}; + + + + \end{scope} + + + \end{scope} + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex new file mode 100644 index 0000000..921dbfa --- /dev/null +++ b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex @@ -0,0 +1,26 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \begin{scope}[xscale=2, yscale=2] + + \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$}; + \draw[gray, ->] (-0.25,0) -- (1.5,0) node[anchor=west]{$\mathrm{Re}~z$}; + + \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$}; + + \draw[gray] (0, 1) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$}; + + \fill[yellow!50] (0,0) rectangle (1, 1); + + \node[anchor=south east] at ( 1,0) {$c$}; + \node[anchor=north east] at ( 1,1) {$d$}; + \node[anchor=north west] at ( 0,1) {$n$}; + \node[anchor=south west] at ( 0,0) {$s$}; + + \end{scope} + + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex new file mode 100644 index 0000000..87c63c0 --- /dev/null +++ b/buch/papers/ellfilter/tikz/sn.tikz.tex @@ -0,0 +1,86 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2] + + \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm] + + \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}} + + \begin{scope}[xscale=1, yscale=2] + + \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$}; + \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$}; + + \begin{scope} + + \clip(-4.5,-1.25) rectangle (4.5,1.25); + + \fill[yellow!30] (0,0) rectangle (1, 0.5); + + \begin{scope}[xshift=-1cm] + + \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5); + \draw[thick, ->, orange] (1, 0) -- (0,0); + \draw[thick, ->, red] (2, 0) -- (1,0); + \draw[thick, ->, blue] (2,0.5) -- (2, 0); + \draw[thick, ->, purple] (1, 0.5) -- (2,0.5); + \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5); + + + \foreach \i in {-2,...,2} { + \foreach \j in {-2,...,1} { + \begin{scope}[xshift=\i*4cm, yshift=\j*1cm] + \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5); + \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0); + \draw[opacity=0.5, ->, red] (2, 0) -- (1,0); + \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0); + \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5); + \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5); + \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5); + \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1); + \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5); + \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5); + \draw[opacity=0.5, ->, red] (2, 0) -- (3,0); + \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0); + + \node[zero] at ( 1, 0) {}; + \node[zero] at ( 3, 0) {}; + \node[pole] at ( 1,0.5) {}; + \node[pole] at ( 3,0.5) {}; + + \end{scope} + } + } + + \end{scope} + + \end{scope} + + \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$}; + \draw[gray] (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$}; + + + + \end{scope} + + \begin{scope}[yshift=-3.5cm, xscale=0.75] + + \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$}; + + \draw[thick, ->, purple] (-5, 0) -- (-3, 0); + \draw[thick, ->, blue] (-3, 0) -- (-2, 0); + \draw[thick, ->, red] (-2, 0) -- (0, 0); + \draw[thick, ->, orange] (0, 0) -- (2, 0); + \draw[thick, ->, darkgreen] (2, 0) -- (3, 0); + \draw[thick, ->, cyan] (3, 0) -- (5, 0); + + \node[anchor=south] at (-5,0) {$-\infty$}; + \node[anchor=south] at (-3,0) {$-1/k$}; + \node[anchor=south] at (-2,0) {$-1$}; + \node[anchor=south] at (0,0) {$0$}; + \node[anchor=south] at (2,0) {$1$}; + \node[anchor=south] at (3,0) {$1/k$}; + \node[anchor=south] at (5,0) {$\infty$}; + + \end{scope} + + +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex new file mode 100644 index 0000000..7d426b6 --- /dev/null +++ b/buch/papers/ellfilter/tschebyscheff.tex @@ -0,0 +1,133 @@ +\section{Tschebyscheff-Filter} + +Als Einstieg betrachent Wir das Tschebyscheff-Filter, welches sehr verwand ist mit dem elliptischen Filter. +Genauer ausgedrückt sind die Tschebyscheff-1 und -2 Filter Spezialfälle davon. + +Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ für das Filter relevant sind: +\begin{align} + T_{0}(x)&=1\\ + T_{1}(x)&=x\\ + T_{2}(x)&=2x^{2}-1\\ + T_{3}(x)&=4x^{3}-3x\\ + T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x). +\end{align} +Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometrischen Funktion +\begin{align} \label{ellfilter:eq:chebychef_polynomials} + T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\ + &= \cos \left(N~z \right), \quad w= \cos(z) +\end{align} +übereinstimmt. +Der Zusammenhang lässt sich mit den Doppel- und Mehrfachwinkelfunktionen der trigonometrischen Funktionen erklären. +Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-Polynome. +\begin{figure} + \centering + \input{papers/ellfilter/python/F_N_chebychev2.pgf} + \caption{Die Tschebyscheff-Polynome $C_N$.} + \label{ellfilter:fig:chebychef_polynomials} +\end{figure} +Da der Kosinus begrenzt zwischen $-1$ und $1$ ist, sind auch die Tschebyscheff-Polynome begrenzt. +Geht man aber über das Intervall $[-1, 1]$ hinaus, divergieren die Funktionen mit zunehmender Ordnung immer steiler gegen $\pm \infty$. +Diese Eigenschaft ist sehr nützlich für ein Filter. +Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Voraussetzungen für Filterfunktionen, wie es Abbildung \ref{ellfiter:fig:chebychef} demonstriert. +\begin{figure} + \centering + \input{papers/ellfilter/python/F_N_chebychev.pgf} + \caption{Die Tschebyscheff-Polynome füllen den erlaubten Bereich besser, und erhalten dadurch eine steilere Flanke im Sperrbereich.} + \label{ellfiter:fig:chebychef} +\end{figure} + + +Die analytische Fortsetzung von \eqref{ellfilter:eq:chebychef_polynomials} über das Intervall $[-1,1]$ hinaus stimmt mit den Polynomen überein, wie es zu erwarten ist. +Die genauere Betrachtung wird uns dann helfen die elliptischen Filter besser zu verstehen. + +Starten wir mit der Funktion, die als erstes auf $w$ angewendet wird, dem Arcuscosinus. +Die invertierte Funktion des Kosinus kann als definites Integral dargestellt werden: +\begin{align} + \cos^{-1}(x) + &= + \int_{x}^{1} + \frac{ + dz + }{ + \sqrt{ + 1-z^2 + } + }\\ + &= + \int_{0}^{x} + \frac{ + -1 + }{ + \sqrt{ + 1-z^2 + } + } + ~dz + + \frac{\pi}{2} +\end{align} +Der Integrand oder auch die Ableitung +\begin{equation} + \frac{ + -1 + }{ + \sqrt{ + 1-z^2 + } + } +\end{equation} +bestimmt dabei die Richtung, in der die Funktion verläuft. +Der reelle Arcuscosinus is bekanntlich nur für $|z| \leq 1$ definiert. +Hier bleibt der Wert unter der Wurzel positiv und das Integral liefert reelle Werte. +Doch wenn $|z|$ über 1 hinausgeht, wird der Term unter der Wurzel negativ. +Durch die Quadratwurzel entstehen für den Integranden zwei rein komplexe Lösungen. +Der Wert des Arcuscosinus verlässt also bei $z= \pm 1$ den reellen Zahlenstrahl und knickt in die komplexe Ebene ab. +Abbildung \ref{ellfilter:fig:arccos} zeigt den $\arccos$ in der komplexen Ebene. +\begin{figure} + \centering + \input{papers/ellfilter/tikz/arccos.tikz.tex} + \caption{Die Funktion $z = \cos^{-1}(w)$ dargestellt in der komplexen ebene.} + \label{ellfilter:fig:arccos} +\end{figure} +Wegen der Periodizität des Kosinus ist auch der Arcuscosinus $2\pi$-periodisch und es entstehen periodische Nullstellen. +% \begin{equation} +% \frac{ +% 1 +% }{ +% \sqrt{ +% 1-z^2 +% } +% } +% \in \mathbb{R} +% \quad +% \forall +% \quad +% -1 \leq z \leq 1 +% \end{equation} +% \begin{equation} +% \frac{ +% 1 +% }{ +% \sqrt{ +% 1-z^2 +% } +% } +% = i \xi \quad | \quad \xi \in \mathbb{R} +% \quad +% \forall +% \quad +% z \leq -1 \cup z \geq 1 +% \end{equation} + +Die Tschebyscheff-Polynome skalieren diese Nullstellen mit dem Ordnungsfaktor $N$, wie dargestellt in Abbildung \ref{ellfilter:fig:arccos2}. +\begin{figure} + \centering + \input{papers/ellfilter/tikz/arccos2.tikz.tex} + \caption{ + $z_1=N \cos^{-1}(w)$-Ebene der Tschebyscheff-Funktion. + Die eingefärbten Pfade sind Verläufe von $w~\forall~[-\infty, \infty]$ für verschiedene Ordnungen $N$. + Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert. + } + \label{ellfilter:fig:arccos2} +\end{figure} +Somit passert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen. +Durch die spezielle Anordnung der Nullstellen hat die Funktion Equirippel-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet. diff --git a/buch/papers/fm/.gitignore b/buch/papers/fm/.gitignore new file mode 100644 index 0000000..eae2913 --- /dev/null +++ b/buch/papers/fm/.gitignore @@ -0,0 +1 @@ +standalone
\ No newline at end of file diff --git a/buch/papers/fm/.vscode/settings.json b/buch/papers/fm/.vscode/settings.json new file mode 100644 index 0000000..5125289 --- /dev/null +++ b/buch/papers/fm/.vscode/settings.json @@ -0,0 +1,3 @@ +{ + "notebook.cellFocusIndicator": "border" +}
\ No newline at end of file diff --git a/buch/papers/fm/00_modulation.tex b/buch/papers/fm/00_modulation.tex new file mode 100644 index 0000000..e2ba39f --- /dev/null +++ b/buch/papers/fm/00_modulation.tex @@ -0,0 +1,32 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\subsection{Modulationsarten\label{fm:section:modulation}} + +Das sinusförmige Trägersignal hat die übliche Form: +\(x_c(t) = A_c \cdot \cos(\omega_c(t)+\varphi)\). +Wobei die konstanten Amplitude \(A_c\) und Phase \(\varphi\) vom Nachrichtensignal \(m(t)\) verändert wird. +Der Parameter \(\omega_c\), die Trägerkreisfrequenz bzw. die Trägerfrequenz \(f_c = \frac{\omega_c}{2\pi}\), +steht nicht für die modulation zur verfügung, statt dessen kann durch ihn die Frequenzachse frei gewählt werden. +\newblockpunct +Jedoch ist das für die Vielfalt der Modulationsarten keine Einschrenkung. +Ein Nachrichtensignal kann auch über die Momentanfrequenz (instantenous frequency) \(\omega_i\) eines trägers verändert werden. +Mathematisch wird dann daraus +\[ + \omega_i = \omega_c + \frac{d \varphi(t)}{dt} +\] +mit der Ableitung der Phase\cite{fm:NAT}. +Mit diesen drei Parameter ergeben sich auch drei Modulationsarten, die Amplitudenmodulation, welche \(A_c\) benutzt, +die Phasenmodulation \(\varphi\) und dann noch die Momentankreisfrequenz \(\omega_i\): +\begin{itemize} + \item AM + \item PM + \item FM +\end{itemize} + +To do: Bilder jeder Modulationsart + + + diff --git a/buch/papers/fm/01_AM.tex b/buch/papers/fm/01_AM.tex new file mode 100644 index 0000000..21927f5 --- /dev/null +++ b/buch/papers/fm/01_AM.tex @@ -0,0 +1,29 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Amplitudenmodulation\label{fm:section:teil0}} +\rhead{AM} + +Das Ziel ist FM zu verstehen doch dazu wird zuerst AM erklärt welches einwenig einfacher zu verstehen ist und erst dann übertragen wir die Ideen in FM. +Nun zur Amplitudenmodulation verwenden wir das bevorzugte Trägersignal +\[ + x_c(t) = A_c \cdot \cos(\omega_ct). +\] +Dies bringt den grossen Vorteil das, dass modulierend Signal sämtliche Anteile im Frequenzspektrum inanspruch nimmt +und das Trägersignal nur zwei komplexe Schwingungen besitzt. +Dies sieht man besonders in der Eulerischen Formel +\[ + x_c(t) = \frac{A_c}{2} \cdot e^{j\omega_ct}\;+\;\frac{A_c}{2} \cdot e^{-j\omega_ct}. +\] +Dabei ist die negative Frequenz der zweiten komplexen Schwingung zwingend erforderlich, damit in der Summe immer ein reellwertiges Trägersignal ergibt. +Nun wird der Parameter \(A_c\) durch das Modulierende Signal \(m(t)\) ersetzt, wobei so \(m(t) \leqslant |1|\) normiert wurde. +\newline +\newline +TODO: +Hier beschrieib ich was AmplitudenModulation ist und mache dan den link zu Frequenzmodulation inkl Formel \[\cos( \cos x)\] +so wird beschrieben das daraus eigentlich \(x_c(t) = A_c \cdot \cos(\omega_i)\) wird und somit \(x_c(t) = A_c \cdot \cos(\omega_c + \frac{d \varphi(t)}{dt})\). +Da \(\sin \) abgeleitet \(\cos \) ergibt, so wird aus dem \(m(t)\) ein \( \frac{d \varphi(t)}{dt}\) in der momentan frequenz. \[ \Rightarrow \cos( \cos x) \] + +\subsection{Frequenzspektrum}
\ No newline at end of file diff --git a/buch/papers/fm/02_FM.tex b/buch/papers/fm/02_FM.tex new file mode 100644 index 0000000..fedfaaa --- /dev/null +++ b/buch/papers/fm/02_FM.tex @@ -0,0 +1,56 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{FM +\label{fm:section:teil1}} +\rhead{FM} +\subsection{Frequenzspektrum} +TODO +Hier Beschreiben ich FM und FM im Frequenzspektrum. +%Sed ut perspiciatis unde omnis iste natus error sit voluptatem +%accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +%quae ab illo inventore veritatis et quasi architecto beatae vitae +%dicta sunt explicabo. +%Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit +%aut fugit, sed quia consequuntur magni dolores eos qui ratione +%voluptatem sequi nesciunt +%\begin{equation} +%\int_a^b x^2\, dx +%= +%\left[ \frac13 x^3 \right]_a^b +%= +%\frac{b^3-a^3}3. +%\label{fm:equation1} +%\end{equation} +%Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, +%consectetur, adipisci velit, sed quia non numquam eius modi tempora +%incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +% +%Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis +%suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? +%Quis autem vel eum iure reprehenderit qui in ea voluptate velit +%esse quam nihil molestiae consequatur, vel illum qui dolorem eum +%fugiat quo voluptas nulla pariatur? +% +%\subsection{De finibus bonorum et malorum +%\label{fm:subsection:finibus}} +%At vero eos et accusamus et iusto odio dignissimos ducimus qui +%blanditiis praesentium voluptatum deleniti atque corrupti quos +%dolores et quas molestias excepturi sint occaecati cupiditate non +%provident, similique sunt in culpa qui officia deserunt mollitia +%animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. +% +%Et harum quidem rerum facilis est et expedita distinctio +%\ref{fm:section:loesung}. +%Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil +%impedit quo minus id quod maxime placeat facere possimus, omnis +%voluptas assumenda est, omnis dolor repellendus +%\ref{fm:section:folgerung}. +%Temporibus autem quibusdam et aut officiis debitis aut rerum +%necessitatibus saepe eveniet ut et voluptates repudiandae sint et +%molestiae non recusandae. +%Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis +%voluptatibus maiores alias consequatur aut perferendis doloribus +%asperiores repellat. diff --git a/buch/papers/fm/03_bessel.tex b/buch/papers/fm/03_bessel.tex new file mode 100644 index 0000000..5f85dc6 --- /dev/null +++ b/buch/papers/fm/03_bessel.tex @@ -0,0 +1,212 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{FM und Bessel-Funktion +\label{fm:section:proof}} +\rhead{Herleitung} +Die momentane Trägerkreisfrequenz \(\omega_i\), wie schon in (ref) beschrieben ist, bringt die Ableitung \(\frac{d \varphi(t)}{dt}\) mit sich. +Diese wiederum kann durch \(\beta\sin(\omega_mt)\) ausgedrückt werden, wobei es das modulierende Signal \(m(t)\) ist. +Somit haben wir unser \(x_c\) welches +\[ +\cos(\omega_c t+\beta\sin(\omega_mt)) +\] +ist. + +\subsection{Herleitung} +Das Ziel ist, unser moduliertes Signal mit der Bessel-Funktion so auszudrücken: +\begin{align} + x_c(t) + = + \cos(\omega_ct+\beta\sin(\omega_mt)) + &= + \sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t) + \label{fm:eq:proof} +\end{align} + +\subsubsection{Hilfsmittel} +Doch dazu brauchen wir die Hilfe der Additionsthoerme +\begin{align} + \cos(A + B) + &= + \cos(A)\cos(B)-\sin(A)\sin(B) + \label{fm:eq:addth1} + \\ + 2\cos (A)\cos (B) + &= + \cos(A-B)+\cos(A+B) + \label{fm:eq:addth2} + \\ + 2\sin(A)\sin(B) + &= + \cos(A-B)-\cos(A+B) + \label{fm:eq:addth3} +\end{align} +und die drei Bessel-Funktionsindentitäten, +\begin{align} + \cos(\beta\sin\phi) + &= + J_0(\beta) + 2\sum_{k=1}^\infty J_{2k}(\beta) \cos(2k\phi) + \label{fm:eq:besselid1} + \\ + \sin(\beta\sin\phi) + &= + 2\sum_{k=0}^\infty J_{2k+1}(\beta) \cos((2k+1)\phi) + \label{fm:eq:besselid2} + \\ + J_{-n}(\beta) &= (-1)^n J_n(\beta) + \label{fm:eq:besselid3} +\end{align} +welche man im Kapitel \eqref{buch:fourier:eqn:expinphireal}, \eqref{buch:fourier:eqn:expinphiimaginary}, \eqref{buch:fourier:eqn:symetrie} findet. + +\subsubsection{Anwenden des Additionstheorem} +Mit dem \eqref{fm:eq:addth1} wird aus dem modulierten Signal +\[ + x_c(t) + = + \cos(\omega_c t + \beta\sin(\omega_mt)) + = + \cos(\omega_c t)\cos(\beta\sin(\omega_m t))-\sin(\omega_ct)\sin(\beta\sin(\omega_m t)). + \label{fm:eq:start} +\] +%----------------------------------------------------------------------------------------------------------- +\subsubsection{Cos-Teil} +Zu beginn wird der Cos-Teil +\begin{align*} + c(t) + &= + \cos(\omega_c t)\cdot\cos(\beta\sin(\omega_mt)) +\end{align*} +mit hilfe der Besselindentität \eqref{fm:eq:besselid1} zum +\begin{align*} + c(t) + &= + \cos(\omega_c t) \cdot \bigg[ J_0(\beta) + 2\sum_{k=1}^\infty J_{2k}(\beta) \cos( 2k \omega_m t)\, \bigg] + \\ + &= + J_0(\beta) \cdot \cos(\omega_c t) + \sum_{k=1}^\infty J_{2k}(\beta) \underbrace{2\cos(\omega_c t)\cos(2k\omega_m t)}_{\text{Additionstheorem \eqref{fm:eq:addth2}}} +\end{align*} +%intertext{} Funktioniert nicht. +wobei mit dem Additionstheorem \eqref{fm:eq:addth2} \(A = \omega_c t\) und \(B = 2k\omega_m t \) ersetzt wurden. +\begin{align*} + c(t) + &= + J_0(\beta) \cdot \cos(\omega_c t) + \sum_{k=1}^\infty J_{2k}(\beta) \{ \underbrace{\cos((\omega_c - 2k \omega_m) t)} \,+\, \cos((\omega_c + 2k \omega_m) t) \} + \\ + &= + \sum_{k=-\infty}^{-1} J_{2k}(\beta) \overbrace{\cos((\omega_c +2k \omega_m) t)} + \,+\,J_0(\beta)\cdot \cos(\omega_c t+ 2\cdot0 \omega_m) + \,+\, \sum_{k=1}^\infty J_{2k}(\beta)\cos((\omega_c + 2k \omega_m) t) +\end{align*} +wird. +Das Minus im Ersten Term wird zur negativen Summe \(\sum_{-\infty}^{-1}\) ersetzt. +Da \(2k\) immer gerade ist, wird es durch alle negativen und positiven Ganzzahlen \(n\) ersetzt: +\begin{align*} + \sum_{n\, \text{gerade}} J_{n}(\beta) \cos((\omega_c + n \omega_m) t), + \label{fm:eq:gerade} +\end{align*} +%---------------------------------------------------------------------------------------------------------------- +\subsubsection{Sin-Teil} +Nun zum zweiten Teil des Term \eqref{fm:eq:start}, den Sin-Teil +\begin{align*} + s(t) + &= + -\sin(\omega_c t)\cdot\sin(\beta\sin(\omega_m t)). +\end{align*} +Dieser wird mit der \eqref{fm:eq:besselid2} Besselindentität zu +\begin{align*} + s(t) + &= + -\sin(\omega_c t) \cdot \bigg[ 2 \sum_{k=0}^\infty J_{ 2k + 1}(\beta) \cos(( 2k + 1) \omega_m t) \bigg] + \\ + &= + \sum_{k=0}^\infty -1 \cdot J_{2k+1}(\beta) 2\sin(\omega_c t)\cos((2k+1)\omega_m t). +\end{align*} +Da \(2k + 1\) alle ungeraden positiven Ganzzahlen entspricht wird es durch \(n\) ersetzt. +Wird die Besselindentität \eqref{fm:eq:besselid3} gebraucht, so ersetzten wird \(J_{-n}(\beta) = -1\cdot J_n(\beta)\) ersetzt: +\begin{align*} + s(t) + &= + \sum_{n=0}^\infty J_{-n}(\beta) \underbrace{2\sin(\omega_c t)\cos(n \omega_m t)}_{\text{Additionstheorem \eqref{fm:eq:addth3}}}. +\end{align*} +Auch hier wird ein Additionstheorem \eqref{fm:eq:addth3} gebraucht, dabei ist \(A = \omega_c t\) und \(B = n \omega_m t \), +somit wird daraus: +\begin{align*} + s(t) + &= + \sum_{n=0}^\infty J_{-n}(\beta) \{ \underbrace{\cos((\omega_c - n\omega_m) t)} \,-\, \cos((\omega_c + n\omega_m) t) \} + \\ + &= + \sum_{n=- \infty}^{0} J_{n}(\beta) \overbrace{\cos((\omega_c + n \omega_m) t)} + \,-\, \sum_{n=0}^\infty J_{-n}(\beta) \cos((\omega_c + n\omega_m) t) +\end{align*} +Auch hier wurde wieder eine zweite Summe \(\sum_{-\infty}^{-1}\) gebraucht um das Minus zu einem Plus zu wandeln. +Wenn \(n = 0 \) ist der Minuend gleich dem Subtrahend und somit dieser Teil \(=0\), das bedeutet \(n\) ended bei \(-1\) und started bei \(1\). +\begin{align*} + s(t) + &= + \sum_{n=- \infty}^{-1} J_{n}(\beta) \cos((\omega_c + n \omega_m) t) + \underbrace{\,-\, \sum_{n=1}^\infty J_{-n}(\beta)} \cos((\omega_c + n\omega_m) t) +\end{align*} +Um aus diesem Subtrahend eine Addition zu kreiernen, wird die Besselindentität \eqref{fm:eq:besselid3} gebraucht, +jedoch so \(-1 \cdot J_{-n}(\beta) = J_n(\beta)\) und daraus wird dann: +\begin{align*} + s(t) + &= + \sum_{n=- \infty}^{-1} J_{n}(\beta) \cos((\omega_c + n \omega_m) t) + \,+\, \sum_{n=1}^\infty J_{n}(\beta) \cos((\omega_c + n\omega_m) t) +\end{align*} +Da \(n\) immer ungerade ist und \(0\) nicht zu den ungeraden zahlen zählt, kann man dies so vereinfacht +\[ + s(t) + = + \sum_{n\, \text{ungerade}} -1 \cdot J_{n}(\beta) \cos((\omega_c + n\omega_m) t). + \label{fm:eq:ungerade} +\] +schreiben. +%------------------------------------------------------------------------------------------ +\subsubsection{Summe Zusammenführen} +Beide Teile \eqref{fm:eq:gerade} Gerade +\[ + \sum_{n\, \text{gerade}} J_{n}(\beta) \cos((\omega_c + n\omega_m) t) +\] +und \eqref{fm:eq:ungerade} Ungerade +\[ + \sum_{n\, \text{ungerade}} J_{n}(\beta) \cos((\omega_c + n\omega_m) t) +\] +ergeben zusammen +\[ + \cos(\omega_ct+\beta\sin(\omega_mt)) + = + \sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t). +\] +Somit ist \eqref{fm:eq:proof} bewiesen. +\newpage +%----------------------------------------------------------------------------------------- +\subsection{Bessel und Frequenzspektrum} +Um sich das ganze noch einwenig Bildlicher vorzustellenhier einmal die Bessel-Funktion \(J_{k}(\beta)\) in geplottet. +\begin{figure} + \centering + \input{papers/fm/Python animation/bessel.pgf} + \caption{Bessle Funktion \(J_{k}(\beta)\)} + \label{fig:bessel} +\end{figure} +TODO Grafik einfügen, +\newline +Nun einmal das Modulierte FM signal im Frequenzspektrum mit den einzelen Summen dargestellt + +TODO +Hier wird beschrieben wie die Bessel Funktion der FM im Frequenzspektrum hilft, wieso diese gebrauch wird und ihre Vorteile. +\begin{itemize} + \item Zuerest einmal die Herleitung von FM zu der Bessel-Funktion + \item Im Frequenzspektrum darstellen mit Farben, ersichtlich machen. + \item Parameter tuing der Trägerfrequenz, Modulierende frequenz und Beta. +\end{itemize} + + +%\subsection{De finibus bonorum et malorum +%\label{fm:subsection:bonorum}} + + + diff --git a/buch/papers/fm/04_fazit.tex b/buch/papers/fm/04_fazit.tex new file mode 100644 index 0000000..8d5eca4 --- /dev/null +++ b/buch/papers/fm/04_fazit.tex @@ -0,0 +1,12 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Fazit +\label{fm:section:fazit}} +\rhead{Zusamenfassend} + +TODO Anwendungen erklären und Sinn des Ganzen. + + diff --git a/buch/papers/fm/FM presentation/FM_presentation.pdf b/buch/papers/fm/FM presentation/FM_presentation.pdf Binary files differnew file mode 100644 index 0000000..496e35e --- /dev/null +++ b/buch/papers/fm/FM presentation/FM_presentation.pdf diff --git a/buch/papers/fm/FM presentation/FM_presentation.tex b/buch/papers/fm/FM presentation/FM_presentation.tex new file mode 100644 index 0000000..2801e69 --- /dev/null +++ b/buch/papers/fm/FM presentation/FM_presentation.tex @@ -0,0 +1,125 @@ +%% !TeX root = .tex + +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär} + \title{FM - Bessel} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{16.5.2022} + \subject{Mathematisches Seminar - Spezielle Funktionen} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Frequenzmodulation} + + \visible<1->{ + \begin{equation} \cos(\omega_c t+\beta\sin(\omega_mt)) + \end{equation}} + + \only<2>{\includegraphics[scale= 0.7]{images/fm_in_time.png}} + \only<3>{\includegraphics[scale= 0.7]{images/fm_frequenz.png}} + \only<4>{\includegraphics[scale= 0.7]{images/bessel_frequenz.png}} + + + \end{frame} +%------------------------------------------------------------------------------- +\section{Proof} +\begin{frame} + \frametitle{Bessel} + + \visible<1->{\begin{align} + \cos(\beta\sin\varphi) + &= + J_0(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi) + \\ + \sin(\beta\sin\varphi) + &= + J_0(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi) + \\ + J_{-n}(\beta) &= (-1)^n J_n(\beta) + \end{align}} + \visible<2->{\begin{align} + \cos(A + B) + &= + \cos(A)\cos(B)-\sin(A)\sin(B) + \\ + 2\cos (A)\cos (B) + &= + \cos(A-B)+\cos(A+B) + \\ + 2\sin(A)\sin(B) + &= + \cos(A-B)-\cos(A+B) + \end{align}} +\end{frame} + +%------------------------------------------------------------------------------- +\begin{frame} + \frametitle{Prof->Done} + \begin{align} + \cos(\omega_ct+\beta\sin(\omega_mt)) + &= + \sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t) + \end{align} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{\includegraphics[scale = 0.75]{images/fm_frequenz.png}} + \only<2>{\includegraphics[scale = 0.75]{images/bessel_frequenz.png}} + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- +\section{Input Parameter} + \begin{frame} + \frametitle{Träger-Frequenz Parameter} + \onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}} + \only<1>{\includegraphics[scale=0.75]{images/100HZ.png}} + \only<2>{\includegraphics[scale=0.75]{images/200HZ.png}} + \only<3>{\includegraphics[scale=0.75]{images/300HZ.png}} + \only<4>{\includegraphics[scale=0.75]{images/400HZ.png}} + \end{frame} +%------------------------------------------------------------------------------- +\begin{frame} +\frametitle{Modulations-Frequenz Parameter} +\onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}} +\only<1>{\includegraphics[scale=0.75]{images/fm_3Hz.png}} +\only<2>{\includegraphics[scale=0.75]{images/fm_5Hz.png}} +\only<3>{\includegraphics[scale=0.75]{images/fm_7Hz.png}} +\only<4>{\includegraphics[scale=0.75]{images/fm_10Hz.png}} +\only<5>{\includegraphics[scale=0.75]{images/fm_20Hz.png}} +\only<6>{\includegraphics[scale=0.75]{images/fm_30Hz.png}} +\end{frame} +%------------------------------------------------------------------------------- +\begin{frame} +\frametitle{Beta Parameter} + \onslide<1->{\begin{equation}\sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t)\end{equation}} + \only<1>{\includegraphics[scale=0.7]{images/beta_0.001.png}} + \only<2>{\includegraphics[scale=0.7]{images/beta_0.1.png}} + \only<3>{\includegraphics[scale=0.7]{images/beta_0.5.png}} + \only<4>{\includegraphics[scale=0.7]{images/beta_1.png}} + \only<5>{\includegraphics[scale=0.7]{images/beta_2.png}} + \only<6>{\includegraphics[scale=0.7]{images/beta_3.png}} + \only<7>{\includegraphics[scale=0.7]{images/bessel.png}} +\end{frame} +%------------------------------------------------------------------------------- +\begin{frame} + \includegraphics[scale=0.5]{images/beta_1.png} + \includegraphics[scale=0.5]{images/bessel.png} +\end{frame} +\end{document} diff --git a/buch/papers/fm/FM presentation/README.txt b/buch/papers/fm/FM presentation/README.txt new file mode 100644 index 0000000..65f390d --- /dev/null +++ b/buch/papers/fm/FM presentation/README.txt @@ -0,0 +1 @@ +Dies ist die Presentation des FM - Bessel
\ No newline at end of file diff --git a/buch/papers/fm/FM presentation/images/100HZ.png b/buch/papers/fm/FM presentation/images/100HZ.png Binary files differnew file mode 100644 index 0000000..371b9bf --- /dev/null +++ b/buch/papers/fm/FM presentation/images/100HZ.png diff --git a/buch/papers/fm/FM presentation/images/200HZ.png b/buch/papers/fm/FM presentation/images/200HZ.png Binary files differnew file mode 100644 index 0000000..f6836bd --- /dev/null +++ b/buch/papers/fm/FM presentation/images/200HZ.png diff --git a/buch/papers/fm/FM presentation/images/300HZ.png b/buch/papers/fm/FM presentation/images/300HZ.png Binary files differnew file mode 100644 index 0000000..6762c1a --- /dev/null +++ b/buch/papers/fm/FM presentation/images/300HZ.png diff --git a/buch/papers/fm/FM presentation/images/400HZ.png b/buch/papers/fm/FM presentation/images/400HZ.png Binary files differnew file mode 100644 index 0000000..236c428 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/400HZ.png diff --git a/buch/papers/fm/FM presentation/images/bessel.png b/buch/papers/fm/FM presentation/images/bessel.png Binary files differnew file mode 100644 index 0000000..f4c83ea --- /dev/null +++ b/buch/papers/fm/FM presentation/images/bessel.png diff --git a/buch/papers/fm/FM presentation/images/bessel2.png b/buch/papers/fm/FM presentation/images/bessel2.png Binary files differnew file mode 100644 index 0000000..ccda3f9 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/bessel2.png diff --git a/buch/papers/fm/FM presentation/images/bessel_beta1.png b/buch/papers/fm/FM presentation/images/bessel_beta1.png Binary files differnew file mode 100644 index 0000000..1f5c47e --- /dev/null +++ b/buch/papers/fm/FM presentation/images/bessel_beta1.png diff --git a/buch/papers/fm/FM presentation/images/bessel_frequenz.png b/buch/papers/fm/FM presentation/images/bessel_frequenz.png Binary files differnew file mode 100644 index 0000000..4f228b9 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/bessel_frequenz.png diff --git a/buch/papers/fm/FM presentation/images/beta_0.001.png b/buch/papers/fm/FM presentation/images/beta_0.001.png Binary files differnew file mode 100644 index 0000000..7e4e276 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_0.001.png diff --git a/buch/papers/fm/FM presentation/images/beta_0.1.png b/buch/papers/fm/FM presentation/images/beta_0.1.png Binary files differnew file mode 100644 index 0000000..e7722b3 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_0.1.png diff --git a/buch/papers/fm/FM presentation/images/beta_0.5.png b/buch/papers/fm/FM presentation/images/beta_0.5.png Binary files differnew file mode 100644 index 0000000..5261b43 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_0.5.png diff --git a/buch/papers/fm/FM presentation/images/beta_1.png b/buch/papers/fm/FM presentation/images/beta_1.png Binary files differnew file mode 100644 index 0000000..6d3535c --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_1.png diff --git a/buch/papers/fm/FM presentation/images/beta_2.png b/buch/papers/fm/FM presentation/images/beta_2.png Binary files differnew file mode 100644 index 0000000..6930eae --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_2.png diff --git a/buch/papers/fm/FM presentation/images/beta_3.png b/buch/papers/fm/FM presentation/images/beta_3.png Binary files differnew file mode 100644 index 0000000..c6df82c --- /dev/null +++ b/buch/papers/fm/FM presentation/images/beta_3.png diff --git a/buch/papers/fm/FM presentation/images/fm_10Hz.png b/buch/papers/fm/FM presentation/images/fm_10Hz.png Binary files differnew file mode 100644 index 0000000..51bddc7 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_10Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_20hz.png b/buch/papers/fm/FM presentation/images/fm_20hz.png Binary files differnew file mode 100644 index 0000000..126ecf3 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_20hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_30Hz.png b/buch/papers/fm/FM presentation/images/fm_30Hz.png Binary files differnew file mode 100644 index 0000000..371b9bf --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_30Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_3Hz.png b/buch/papers/fm/FM presentation/images/fm_3Hz.png Binary files differnew file mode 100644 index 0000000..d4098af --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_3Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_40Hz.png b/buch/papers/fm/FM presentation/images/fm_40Hz.png Binary files differnew file mode 100644 index 0000000..4cf11d4 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_40Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_5Hz.png b/buch/papers/fm/FM presentation/images/fm_5Hz.png Binary files differnew file mode 100644 index 0000000..e495b5c --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_5Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_7Hz.png b/buch/papers/fm/FM presentation/images/fm_7Hz.png Binary files differnew file mode 100644 index 0000000..b3dd7e3 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_7Hz.png diff --git a/buch/papers/fm/FM presentation/images/fm_frequenz.png b/buch/papers/fm/FM presentation/images/fm_frequenz.png Binary files differnew file mode 100644 index 0000000..26bfd86 --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_frequenz.png diff --git a/buch/papers/fm/FM presentation/images/fm_in_time.png b/buch/papers/fm/FM presentation/images/fm_in_time.png Binary files differnew file mode 100644 index 0000000..068eafc --- /dev/null +++ b/buch/papers/fm/FM presentation/images/fm_in_time.png diff --git a/buch/papers/fm/Makefile b/buch/papers/fm/Makefile index f43d497..f30c4a9 100644 --- a/buch/papers/fm/Makefile +++ b/buch/papers/fm/Makefile @@ -4,6 +4,37 @@ # (c) 2020 Prof Dr Andreas Mueller # -images: - @echo "no images to be created in fm" +SOURCES := \ + 00_modulation.tex \ + 01_AM.tex \ + 02_FM.tex \ + 03_bessel.tex \ + 04_fazit.tex \ + main.tex +#TIKZFIGURES := \ + tikz/atoms-grid-still.tex \ + +#FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) + +all: images standalone + +.PHONY: images +images: $(FIGURES) + +#figures/%.pdf: tikz/%.tex +# mkdir -p figures +# pdflatex --output-directory=figures $< + +.PHONY: standalone +standalone: standalone.tex $(SOURCES) $(FIGURES) + mkdir -p standalone + cd ../..; \ + pdflatex \ + --halt-on-error \ + --shell-escape \ + --output-directory=papers/fm/standalone \ + papers/fm/standalone.tex; + cd standalone; \ + bibtex standalone; \ + makeindex standalone;
\ No newline at end of file diff --git a/buch/papers/fm/Makefile.inc b/buch/papers/fm/Makefile.inc index 0f144b6..40f23b1 100644 --- a/buch/papers/fm/Makefile.inc +++ b/buch/papers/fm/Makefile.inc @@ -6,9 +6,10 @@ dependencies-fm = \ papers/fm/packages.tex \ papers/fm/main.tex \ - papers/fm/references.bib \ - papers/fm/teil0.tex \ - papers/fm/teil1.tex \ - papers/fm/teil2.tex \ - papers/fm/teil3.tex + papers/fm/00_modulation.tex \ + papers/fm/01_AM.tex \ + papers/fm/02_FM.tex \ + papers/fm/03_bessel.tex \ + papers/fm/04_fazit.tex \ + papers/fm/references.bib diff --git a/buch/papers/fm/Python animation/Bessel-FM.ipynb b/buch/papers/fm/Python animation/Bessel-FM.ipynb new file mode 100644 index 0000000..74f1011 --- /dev/null +++ b/buch/papers/fm/Python animation/Bessel-FM.ipynb @@ -0,0 +1,217 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "from scipy import signal\n", + "from scipy.fft import fft, ifft, fftfreq\n", + "import scipy.special as sc\n", + "import scipy.fftpack\n", + "import matplotlib as mpl\n", + "# Use the pgf backend (must be set before pyplot imported)\n", + "mpl.use('pgf')\n", + "import matplotlib.pyplot as plt\n", + "from matplotlib.widgets import Slider\n", + "def fm(beta):\n", + " # Number of samplepoints\n", + " N = 600\n", + " # sample spacing\n", + " T = 1.0 / 1000.0\n", + " fc = 100.0\n", + " fm = 30.0\n", + " x = np.linspace(0.01, N*T, N)\n", + " #beta = 1.0\n", + " y_old = np.sin(fc * 2.0*np.pi*x+beta*np.sin(fm * 2.0*np.pi*x))\n", + " y = 0*x;\n", + " xf = fftfreq(N, 1 / 400)\n", + " for k in range (-4, 4):\n", + " y = sc.jv(k,beta)*np.sin((fc+k*fm) * 2.0*np.pi*x)\n", + " yf = fft(y)/(fc*np.pi)\n", + " plt.plot(xf, np.abs(yf))\n", + " plt.xlim(-150, 150)\n", + " plt.show()\n", + " #yf_old = fft(y_old)\n", + " #plt.plot(xf, np.abs(yf_old))\n", + " #plt.show()\n", + " \n" + ] + }, + { + "cell_type": "code", + "execution_count": 114, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Number of samplepoints\n", + "N = 800\n", + "# sample spacing\n", + "T = 1.0 / 1000.0\n", + "x = np.linspace(0.01, N*T, N)\n", + "\n", + "y_old = np.sin(100* 2.0*np.pi*x+1*np.sin(15* 2.0*np.pi*x))\n", + "yf_old = fft(y_old)/(100*np.pi)\n", + "xf = fftfreq(N, 1 / 1000)\n", + "plt.plot(xf, np.abs(yf_old))\n", + "#plt.xlim(-150, 150)\n", + "plt.show()\n", + "\n", + "fm(1)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.7651976865579666\n" + ] + }, + { + "data": { + "image/png": "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", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "for n in range (-2,4):\n", + " x = np.linspace(-11,11,1000)\n", + " y = sc.jv(n,x)\n", + " plt.plot(x, y, '-',label='n='+str(n))\n", + "#plt.plot([1,1],[sc.jv(0,1),sc.jv(-1,1)],)\n", + "plt.xlim(-10,10)\n", + "plt.grid(True)\n", + "plt.ylabel('Bessel $J_n(\\\\beta)$')\n", + "plt.xlabel(' $ \\\\beta $ ')\n", + "plt.plot(x, y)\n", + "plt.legend()\n", + "#plt.show()\n", + "plt.savefig('bessel.pgf', format='pgf')\n", + "print(sc.jv(0,1))" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "from scipy import special\n", + "\n", + "def drumhead_height(n, k, distance, angle, t):\n", + " kth_zero = special.jn_zeros(n, k)[-1]\n", + " return np.cos(t) * np.cos(n*angle) * special.jn(n, distance*kth_zero)\n", + "\n", + "theta = np.r_[0:2*np.pi:50j]\n", + "radius = np.r_[0:1:50j]\n", + "x = np.array([r * np.cos(theta) for r in radius])\n", + "y = np.array([r * np.sin(theta) for r in radius])\n", + "z = np.array([drumhead_height(1, 1, r, theta, 0.5) for r in radius])\n", + "\n", + "import matplotlib.pyplot as plt\n", + "fig = plt.figure()\n", + "ax = fig.add_axes(rect=(0, 0.05, 0.95, 0.95), projection='3d')\n", + "ax.plot_surface(x, y, z, rstride=1, cstride=1, cmap='RdBu_r', vmin=-0.5, vmax=0.5)\n", + "ax.set_xlabel('X')\n", + "ax.set_ylabel('Y')\n", + "ax.set_xticks(np.arange(-1, 1.1, 0.5))\n", + "ax.set_yticks(np.arange(-1, 1.1, 0.5))\n", + "ax.set_zlabel('Z')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "x = np.linspace(0,0.1,1000)\n", + "y = np.sin(100 * 2.0*np.pi*x+1.5*np.sin(30 * 2.0*np.pi*x))\n", + "plt.plot(x, y, '-')\n", + "plt.show()" + ] + } + ], + "metadata": { + "interpreter": { + "hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1" + }, + "kernelspec": { + "display_name": "Python 3.8.10 64-bit", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/buch/papers/fm/Python animation/Bessel-FM.py b/buch/papers/fm/Python animation/Bessel-FM.py new file mode 100644 index 0000000..cf30e16 --- /dev/null +++ b/buch/papers/fm/Python animation/Bessel-FM.py @@ -0,0 +1,42 @@ +import numpy as np +from scipy import signal +from scipy.fft import fft, ifft, fftfreq +import scipy.special as sc +import scipy.fftpack +import matplotlib.pyplot as plt +from matplotlib.widgets import Slider + +# Number of samplepoints +N = 600 +# sample spacing +T = 1.0 / 800.0 +x = np.linspace(0.01, N*T, N) +beta = 1.0 +y_old = np.sin(100.0 * 2.0*np.pi*x+beta*np.sin(50.0 * 2.0*np.pi*x)) +y = 0*x; +xf = fftfreq(N, 1 / 400) +for k in range (-5, 5): + y = sc.jv(k,beta)*np.sin((100.0+k*50) * 2.0*np.pi*x) + yf = fft(y) + plt.plot(xf, np.abs(yf)) + +axbeta =plt.axes([0.25, 0.1, 0.65, 0.03]) +beta_slider = Slider( +ax=axbeta, +label="Beta", +valmin=0.1, +valmax=3, +valinit=beta, +) + +def update(val): + line.set_ydata(fm(beta_slider.val)) + fig.canvas.draw_idle() + + +beta_slider.on_changed(update) +plt.show() + +yf_old = fft(y_old) +plt.plot(xf, np.abs(yf_old)) +plt.show()
\ No newline at end of file diff --git a/buch/papers/fm/Python animation/bessel.pgf b/buch/papers/fm/Python animation/bessel.pgf new file mode 100644 index 0000000..cc7af1e --- /dev/null +++ b/buch/papers/fm/Python animation/bessel.pgf @@ -0,0 +1,2057 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/joshua/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/joshua/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/joshua/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.000000in}{4.000000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{6.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{6.000000in}{4.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{4.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathclose% +\pgfusepath{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.000000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.520000in}}% +\pgfpathlineto{\pgfqpoint{0.750000in}{3.520000in}}% +\pgfpathlineto{\pgfqpoint{0.750000in}{0.500000in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{0.750000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.750000in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}10.0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.331250in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{1.331250in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.331250in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.331250in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}7.5}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.912500in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{1.912500in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{1.912500in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=1.912500in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}5.0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{2.493750in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{2.493750in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{2.493750in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=2.493750in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}2.5}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.075000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{3.075000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.075000in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.075000in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.656250in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{3.656250in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{3.656250in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.656250in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 2.5}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.237500in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{4.237500in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{4.237500in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.237500in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 5.0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.818750in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{4.818750in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{4.818750in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.818750in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 7.5}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{5.400000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{5.400000in}{0.500000in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=5.400000in,y=0.402778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 10.0}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=3.075000in,y=0.212809in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \beta \) }% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.605796in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{0.605796in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{0.605796in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.323873in, y=0.553034in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}0.6}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.952919in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{0.952919in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{0.952919in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.323873in, y=0.900157in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}0.4}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{1.300042in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{1.300042in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{1.300042in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.323873in, y=1.247280in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont \ensuremath{-}0.2}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{1.647165in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{1.647165in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{1.647165in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=1.594403in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{1.994288in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{1.994288in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{1.994288in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=1.941526in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.2}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{2.341411in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{2.341411in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{2.341411in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=2.288649in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.4}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{2.688534in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{2.688534in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{2.688534in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=2.635772in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.6}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{3.035657in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.035657in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{3.035657in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=2.982895in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0.8}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{3.382780in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.382780in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.750000in}{3.382780in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.431898in, y=3.330018in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 1.0}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=0.268318in,y=2.010000in,,bottom,rotate=90.000000]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont Bessel \(\displaystyle J_n(\beta)\)}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{2.087558in}}% +\pgfpathlineto{\pgfqpoint{0.753026in}{2.089238in}}% +\pgfpathlineto{\pgfqpoint{0.773506in}{2.088244in}}% +\pgfpathlineto{\pgfqpoint{0.793986in}{2.083946in}}% +\pgfpathlineto{\pgfqpoint{0.814467in}{2.076349in}}% +\pgfpathlineto{\pgfqpoint{0.834947in}{2.065480in}}% +\pgfpathlineto{\pgfqpoint{0.855428in}{2.051393in}}% +\pgfpathlineto{\pgfqpoint{0.881029in}{2.029378in}}% +\pgfpathlineto{\pgfqpoint{0.906629in}{2.002657in}}% +\pgfpathlineto{\pgfqpoint{0.932230in}{1.971486in}}% +\pgfpathlineto{\pgfqpoint{0.962950in}{1.928646in}}% +\pgfpathlineto{\pgfqpoint{0.993671in}{1.880474in}}% +\pgfpathlineto{\pgfqpoint{1.029512in}{1.818502in}}% +\pgfpathlineto{\pgfqpoint{1.075593in}{1.731696in}}% +\pgfpathlineto{\pgfqpoint{1.137035in}{1.608436in}}% +\pgfpathlineto{\pgfqpoint{1.229197in}{1.423447in}}% +\pgfpathlineto{\pgfqpoint{1.275278in}{1.338266in}}% +\pgfpathlineto{\pgfqpoint{1.311119in}{1.278235in}}% +\pgfpathlineto{\pgfqpoint{1.341839in}{1.232297in}}% +\pgfpathlineto{\pgfqpoint{1.372560in}{1.192309in}}% +\pgfpathlineto{\pgfqpoint{1.398161in}{1.164059in}}% +\pgfpathlineto{\pgfqpoint{1.423761in}{1.140817in}}% +\pgfpathlineto{\pgfqpoint{1.444242in}{1.126046in}}% +\pgfpathlineto{\pgfqpoint{1.464722in}{1.114825in}}% +\pgfpathlineto{\pgfqpoint{1.485203in}{1.107268in}}% +\pgfpathlineto{\pgfqpoint{1.505683in}{1.103467in}}% +\pgfpathlineto{\pgfqpoint{1.526164in}{1.103488in}}% +\pgfpathlineto{\pgfqpoint{1.546644in}{1.107370in}}% +\pgfpathlineto{\pgfqpoint{1.567125in}{1.115129in}}% +\pgfpathlineto{\pgfqpoint{1.587605in}{1.126753in}}% +\pgfpathlineto{\pgfqpoint{1.608086in}{1.142205in}}% +\pgfpathlineto{\pgfqpoint{1.628566in}{1.161422in}}% +\pgfpathlineto{\pgfqpoint{1.649047in}{1.184315in}}% +\pgfpathlineto{\pgfqpoint{1.674647in}{1.217924in}}% +\pgfpathlineto{\pgfqpoint{1.700248in}{1.256823in}}% +\pgfpathlineto{\pgfqpoint{1.730968in}{1.310014in}}% +\pgfpathlineto{\pgfqpoint{1.761689in}{1.369671in}}% +\pgfpathlineto{\pgfqpoint{1.797530in}{1.446409in}}% +\pgfpathlineto{\pgfqpoint{1.838491in}{1.541833in}}% +\pgfpathlineto{\pgfqpoint{1.889692in}{1.669363in}}% +\pgfpathlineto{\pgfqpoint{2.053536in}{2.084842in}}% +\pgfpathlineto{\pgfqpoint{2.094497in}{2.177528in}}% +\pgfpathlineto{\pgfqpoint{2.130338in}{2.251280in}}% +\pgfpathlineto{\pgfqpoint{2.161059in}{2.308003in}}% +\pgfpathlineto{\pgfqpoint{2.191779in}{2.357982in}}% +\pgfpathlineto{\pgfqpoint{2.217380in}{2.394040in}}% +\pgfpathlineto{\pgfqpoint{2.242980in}{2.424708in}}% +\pgfpathlineto{\pgfqpoint{2.263461in}{2.445205in}}% +\pgfpathlineto{\pgfqpoint{2.283941in}{2.462016in}}% +\pgfpathlineto{\pgfqpoint{2.304422in}{2.475078in}}% +\pgfpathlineto{\pgfqpoint{2.324902in}{2.484355in}}% +\pgfpathlineto{\pgfqpoint{2.345383in}{2.489835in}}% +\pgfpathlineto{\pgfqpoint{2.365863in}{2.491535in}}% +\pgfpathlineto{\pgfqpoint{2.386344in}{2.489498in}}% +\pgfpathlineto{\pgfqpoint{2.406824in}{2.483791in}}% +\pgfpathlineto{\pgfqpoint{2.427305in}{2.474508in}}% +\pgfpathlineto{\pgfqpoint{2.447785in}{2.461767in}}% +\pgfpathlineto{\pgfqpoint{2.468266in}{2.445710in}}% +\pgfpathlineto{\pgfqpoint{2.493866in}{2.421230in}}% +\pgfpathlineto{\pgfqpoint{2.519467in}{2.392195in}}% +\pgfpathlineto{\pgfqpoint{2.550188in}{2.351932in}}% +\pgfpathlineto{\pgfqpoint{2.580908in}{2.306515in}}% +\pgfpathlineto{\pgfqpoint{2.616749in}{2.248188in}}% +\pgfpathlineto{\pgfqpoint{2.662830in}{2.167068in}}% +\pgfpathlineto{\pgfqpoint{2.826674in}{1.871952in}}% +\pgfpathlineto{\pgfqpoint{2.862515in}{1.816084in}}% +\pgfpathlineto{\pgfqpoint{2.893236in}{1.773136in}}% +\pgfpathlineto{\pgfqpoint{2.923956in}{1.735550in}}% +\pgfpathlineto{\pgfqpoint{2.949557in}{1.708802in}}% +\pgfpathlineto{\pgfqpoint{2.975158in}{1.686562in}}% +\pgfpathlineto{\pgfqpoint{2.995638in}{1.672198in}}% +\pgfpathlineto{\pgfqpoint{3.016119in}{1.661005in}}% +\pgfpathlineto{\pgfqpoint{3.036599in}{1.653070in}}% +\pgfpathlineto{\pgfqpoint{3.057080in}{1.648453in}}% +\pgfpathlineto{\pgfqpoint{3.077560in}{1.647191in}}% +\pgfpathlineto{\pgfqpoint{3.098041in}{1.649294in}}% +\pgfpathlineto{\pgfqpoint{3.118521in}{1.654744in}}% +\pgfpathlineto{\pgfqpoint{3.139002in}{1.663501in}}% +\pgfpathlineto{\pgfqpoint{3.159482in}{1.675496in}}% +\pgfpathlineto{\pgfqpoint{3.179962in}{1.690635in}}% +\pgfpathlineto{\pgfqpoint{3.205563in}{1.713801in}}% +\pgfpathlineto{\pgfqpoint{3.231164in}{1.741413in}}% +\pgfpathlineto{\pgfqpoint{3.261884in}{1.779941in}}% +\pgfpathlineto{\pgfqpoint{3.292605in}{1.823711in}}% +\pgfpathlineto{\pgfqpoint{3.328446in}{1.880369in}}% +\pgfpathlineto{\pgfqpoint{3.374527in}{1.959949in}}% +\pgfpathlineto{\pgfqpoint{3.446209in}{2.091693in}}% +\pgfpathlineto{\pgfqpoint{3.517890in}{2.221778in}}% +\pgfpathlineto{\pgfqpoint{3.558851in}{2.290381in}}% +\pgfpathlineto{\pgfqpoint{3.594692in}{2.344698in}}% +\pgfpathlineto{\pgfqpoint{3.625413in}{2.385878in}}% +\pgfpathlineto{\pgfqpoint{3.651014in}{2.415775in}}% +\pgfpathlineto{\pgfqpoint{3.676614in}{2.441197in}}% +\pgfpathlineto{\pgfqpoint{3.702215in}{2.461767in}}% +\pgfpathlineto{\pgfqpoint{3.722695in}{2.474508in}}% +\pgfpathlineto{\pgfqpoint{3.743176in}{2.483791in}}% +\pgfpathlineto{\pgfqpoint{3.763656in}{2.489498in}}% +\pgfpathlineto{\pgfqpoint{3.784137in}{2.491535in}}% +\pgfpathlineto{\pgfqpoint{3.804617in}{2.489835in}}% +\pgfpathlineto{\pgfqpoint{3.825098in}{2.484355in}}% +\pgfpathlineto{\pgfqpoint{3.845578in}{2.475078in}}% +\pgfpathlineto{\pgfqpoint{3.866059in}{2.462016in}}% +\pgfpathlineto{\pgfqpoint{3.886539in}{2.445205in}}% +\pgfpathlineto{\pgfqpoint{3.907020in}{2.424708in}}% +\pgfpathlineto{\pgfqpoint{3.932620in}{2.394040in}}% +\pgfpathlineto{\pgfqpoint{3.958221in}{2.357982in}}% +\pgfpathlineto{\pgfqpoint{3.983821in}{2.316819in}}% +\pgfpathlineto{\pgfqpoint{4.014542in}{2.261173in}}% +\pgfpathlineto{\pgfqpoint{4.045263in}{2.199366in}}% +\pgfpathlineto{\pgfqpoint{4.081104in}{2.120526in}}% +\pgfpathlineto{\pgfqpoint{4.127185in}{2.010684in}}% +\pgfpathlineto{\pgfqpoint{4.188626in}{1.854728in}}% +\pgfpathlineto{\pgfqpoint{4.301269in}{1.566729in}}% +\pgfpathlineto{\pgfqpoint{4.347350in}{1.457926in}}% +\pgfpathlineto{\pgfqpoint{4.383191in}{1.380189in}}% +\pgfpathlineto{\pgfqpoint{4.419032in}{1.310014in}}% +\pgfpathlineto{\pgfqpoint{4.449752in}{1.256823in}}% +\pgfpathlineto{\pgfqpoint{4.475353in}{1.217924in}}% +\pgfpathlineto{\pgfqpoint{4.500953in}{1.184315in}}% +\pgfpathlineto{\pgfqpoint{4.526554in}{1.156269in}}% +\pgfpathlineto{\pgfqpoint{4.547035in}{1.137987in}}% +\pgfpathlineto{\pgfqpoint{4.567515in}{1.123487in}}% +\pgfpathlineto{\pgfqpoint{4.587995in}{1.112826in}}% +\pgfpathlineto{\pgfqpoint{4.608476in}{1.106036in}}% +\pgfpathlineto{\pgfqpoint{4.628956in}{1.103122in}}% +\pgfpathlineto{\pgfqpoint{4.649437in}{1.104062in}}% +\pgfpathlineto{\pgfqpoint{4.669917in}{1.108809in}}% +\pgfpathlineto{\pgfqpoint{4.690398in}{1.117291in}}% +\pgfpathlineto{\pgfqpoint{4.710878in}{1.129411in}}% +\pgfpathlineto{\pgfqpoint{4.731359in}{1.145049in}}% +\pgfpathlineto{\pgfqpoint{4.751839in}{1.164059in}}% +\pgfpathlineto{\pgfqpoint{4.777440in}{1.192309in}}% +\pgfpathlineto{\pgfqpoint{4.803041in}{1.225198in}}% +\pgfpathlineto{\pgfqpoint{4.833761in}{1.270199in}}% +\pgfpathlineto{\pgfqpoint{4.869602in}{1.329314in}}% +\pgfpathlineto{\pgfqpoint{4.910563in}{1.403883in}}% +\pgfpathlineto{\pgfqpoint{4.961764in}{1.504307in}}% +\pgfpathlineto{\pgfqpoint{5.105128in}{1.790337in}}% +\pgfpathlineto{\pgfqpoint{5.146089in}{1.863354in}}% +\pgfpathlineto{\pgfqpoint{5.181929in}{1.920969in}}% +\pgfpathlineto{\pgfqpoint{5.212650in}{1.964744in}}% +\pgfpathlineto{\pgfqpoint{5.243371in}{2.002657in}}% +\pgfpathlineto{\pgfqpoint{5.268971in}{2.029378in}}% +\pgfpathlineto{\pgfqpoint{5.294572in}{2.051393in}}% +\pgfpathlineto{\pgfqpoint{5.315053in}{2.065480in}}% +\pgfpathlineto{\pgfqpoint{5.335533in}{2.076349in}}% +\pgfpathlineto{\pgfqpoint{5.356014in}{2.083946in}}% +\pgfpathlineto{\pgfqpoint{5.376494in}{2.088244in}}% +\pgfpathlineto{\pgfqpoint{5.396974in}{2.089238in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{2.087558in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{2.087558in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{1.696626in}}% +\pgfpathlineto{\pgfqpoint{0.814467in}{1.840343in}}% +\pgfpathlineto{\pgfqpoint{0.855428in}{1.909768in}}% +\pgfpathlineto{\pgfqpoint{0.891269in}{1.964990in}}% +\pgfpathlineto{\pgfqpoint{0.921989in}{2.007118in}}% +\pgfpathlineto{\pgfqpoint{0.952710in}{2.043605in}}% +\pgfpathlineto{\pgfqpoint{0.978311in}{2.069190in}}% +\pgfpathlineto{\pgfqpoint{1.003911in}{2.090009in}}% +\pgfpathlineto{\pgfqpoint{1.024392in}{2.103027in}}% +\pgfpathlineto{\pgfqpoint{1.044872in}{2.112669in}}% +\pgfpathlineto{\pgfqpoint{1.065353in}{2.118827in}}% +\pgfpathlineto{\pgfqpoint{1.085833in}{2.121422in}}% +\pgfpathlineto{\pgfqpoint{1.106314in}{2.120397in}}% +\pgfpathlineto{\pgfqpoint{1.126794in}{2.115724in}}% +\pgfpathlineto{\pgfqpoint{1.147275in}{2.107401in}}% +\pgfpathlineto{\pgfqpoint{1.167755in}{2.095453in}}% +\pgfpathlineto{\pgfqpoint{1.188236in}{2.079933in}}% +\pgfpathlineto{\pgfqpoint{1.208716in}{2.060921in}}% +\pgfpathlineto{\pgfqpoint{1.234317in}{2.032409in}}% +\pgfpathlineto{\pgfqpoint{1.259917in}{1.998874in}}% +\pgfpathlineto{\pgfqpoint{1.285518in}{1.960642in}}% +\pgfpathlineto{\pgfqpoint{1.316239in}{1.909112in}}% +\pgfpathlineto{\pgfqpoint{1.352080in}{1.842184in}}% +\pgfpathlineto{\pgfqpoint{1.393041in}{1.758442in}}% +\pgfpathlineto{\pgfqpoint{1.444242in}{1.646346in}}% +\pgfpathlineto{\pgfqpoint{1.572245in}{1.362289in}}% +\pgfpathlineto{\pgfqpoint{1.613206in}{1.280362in}}% +\pgfpathlineto{\pgfqpoint{1.649047in}{1.215890in}}% +\pgfpathlineto{\pgfqpoint{1.679767in}{1.167290in}}% +\pgfpathlineto{\pgfqpoint{1.705368in}{1.132210in}}% +\pgfpathlineto{\pgfqpoint{1.730968in}{1.102582in}}% +\pgfpathlineto{\pgfqpoint{1.751449in}{1.083102in}}% +\pgfpathlineto{\pgfqpoint{1.771929in}{1.067590in}}% +\pgfpathlineto{\pgfqpoint{1.792410in}{1.056219in}}% +\pgfpathlineto{\pgfqpoint{1.812890in}{1.049132in}}% +\pgfpathlineto{\pgfqpoint{1.828251in}{1.046700in}}% +\pgfpathlineto{\pgfqpoint{1.843611in}{1.046782in}}% +\pgfpathlineto{\pgfqpoint{1.858971in}{1.049407in}}% +\pgfpathlineto{\pgfqpoint{1.874332in}{1.054594in}}% +\pgfpathlineto{\pgfqpoint{1.889692in}{1.062353in}}% +\pgfpathlineto{\pgfqpoint{1.910173in}{1.076696in}}% +\pgfpathlineto{\pgfqpoint{1.930653in}{1.095578in}}% +\pgfpathlineto{\pgfqpoint{1.951134in}{1.118939in}}% +\pgfpathlineto{\pgfqpoint{1.971614in}{1.146687in}}% +\pgfpathlineto{\pgfqpoint{1.997215in}{1.187350in}}% +\pgfpathlineto{\pgfqpoint{2.022815in}{1.234366in}}% +\pgfpathlineto{\pgfqpoint{2.048416in}{1.287354in}}% +\pgfpathlineto{\pgfqpoint{2.079137in}{1.358190in}}% +\pgfpathlineto{\pgfqpoint{2.114977in}{1.449684in}}% +\pgfpathlineto{\pgfqpoint{2.155938in}{1.563913in}}% +\pgfpathlineto{\pgfqpoint{2.207140in}{1.717153in}}% +\pgfpathlineto{\pgfqpoint{2.370983in}{2.217374in}}% +\pgfpathlineto{\pgfqpoint{2.411944in}{2.327896in}}% +\pgfpathlineto{\pgfqpoint{2.442665in}{2.403052in}}% +\pgfpathlineto{\pgfqpoint{2.473386in}{2.470271in}}% +\pgfpathlineto{\pgfqpoint{2.498986in}{2.519452in}}% +\pgfpathlineto{\pgfqpoint{2.524587in}{2.561823in}}% +\pgfpathlineto{\pgfqpoint{2.545068in}{2.590485in}}% +\pgfpathlineto{\pgfqpoint{2.565548in}{2.614254in}}% +\pgfpathlineto{\pgfqpoint{2.586029in}{2.632938in}}% +\pgfpathlineto{\pgfqpoint{2.601389in}{2.643518in}}% +\pgfpathlineto{\pgfqpoint{2.616749in}{2.651092in}}% +\pgfpathlineto{\pgfqpoint{2.632110in}{2.655617in}}% +\pgfpathlineto{\pgfqpoint{2.647470in}{2.657057in}}% +\pgfpathlineto{\pgfqpoint{2.662830in}{2.655390in}}% +\pgfpathlineto{\pgfqpoint{2.678191in}{2.650603in}}% +\pgfpathlineto{\pgfqpoint{2.693551in}{2.642696in}}% +\pgfpathlineto{\pgfqpoint{2.708911in}{2.631677in}}% +\pgfpathlineto{\pgfqpoint{2.724272in}{2.617568in}}% +\pgfpathlineto{\pgfqpoint{2.744752in}{2.594005in}}% +\pgfpathlineto{\pgfqpoint{2.765233in}{2.565112in}}% +\pgfpathlineto{\pgfqpoint{2.785713in}{2.531028in}}% +\pgfpathlineto{\pgfqpoint{2.811314in}{2.481388in}}% +\pgfpathlineto{\pgfqpoint{2.836914in}{2.424317in}}% +\pgfpathlineto{\pgfqpoint{2.862515in}{2.360295in}}% +\pgfpathlineto{\pgfqpoint{2.893236in}{2.275073in}}% +\pgfpathlineto{\pgfqpoint{2.929077in}{2.165443in}}% +\pgfpathlineto{\pgfqpoint{2.970038in}{2.029041in}}% +\pgfpathlineto{\pgfqpoint{3.021239in}{1.846491in}}% +\pgfpathlineto{\pgfqpoint{3.195323in}{1.212928in}}% +\pgfpathlineto{\pgfqpoint{3.236284in}{1.080664in}}% +\pgfpathlineto{\pgfqpoint{3.272125in}{0.975558in}}% +\pgfpathlineto{\pgfqpoint{3.302845in}{0.894817in}}% +\pgfpathlineto{\pgfqpoint{3.328446in}{0.834903in}}% +\pgfpathlineto{\pgfqpoint{3.354047in}{0.782238in}}% +\pgfpathlineto{\pgfqpoint{3.379647in}{0.737258in}}% +\pgfpathlineto{\pgfqpoint{3.400128in}{0.707053in}}% +\pgfpathlineto{\pgfqpoint{3.420608in}{0.682146in}}% +\pgfpathlineto{\pgfqpoint{3.441089in}{0.662653in}}% +\pgfpathlineto{\pgfqpoint{3.456449in}{0.651634in}}% +\pgfpathlineto{\pgfqpoint{3.471809in}{0.643726in}}% +\pgfpathlineto{\pgfqpoint{3.487170in}{0.638940in}}% +\pgfpathlineto{\pgfqpoint{3.502530in}{0.637273in}}% +\pgfpathlineto{\pgfqpoint{3.517890in}{0.638713in}}% +\pgfpathlineto{\pgfqpoint{3.533251in}{0.643238in}}% +\pgfpathlineto{\pgfqpoint{3.548611in}{0.650812in}}% +\pgfpathlineto{\pgfqpoint{3.563971in}{0.661392in}}% +\pgfpathlineto{\pgfqpoint{3.584452in}{0.680076in}}% +\pgfpathlineto{\pgfqpoint{3.604932in}{0.703844in}}% +\pgfpathlineto{\pgfqpoint{3.625413in}{0.732506in}}% +\pgfpathlineto{\pgfqpoint{3.645893in}{0.765842in}}% +\pgfpathlineto{\pgfqpoint{3.671494in}{0.813703in}}% +\pgfpathlineto{\pgfqpoint{3.697095in}{0.867935in}}% +\pgfpathlineto{\pgfqpoint{3.727815in}{0.940570in}}% +\pgfpathlineto{\pgfqpoint{3.763656in}{1.034286in}}% +\pgfpathlineto{\pgfqpoint{3.804617in}{1.150816in}}% +\pgfpathlineto{\pgfqpoint{3.860938in}{1.321912in}}% +\pgfpathlineto{\pgfqpoint{3.988941in}{1.715539in}}% +\pgfpathlineto{\pgfqpoint{4.029902in}{1.830883in}}% +\pgfpathlineto{\pgfqpoint{4.065743in}{1.923621in}}% +\pgfpathlineto{\pgfqpoint{4.096464in}{1.995699in}}% +\pgfpathlineto{\pgfqpoint{4.127185in}{2.059964in}}% +\pgfpathlineto{\pgfqpoint{4.152785in}{2.106979in}}% +\pgfpathlineto{\pgfqpoint{4.178386in}{2.147643in}}% +\pgfpathlineto{\pgfqpoint{4.198866in}{2.175391in}}% +\pgfpathlineto{\pgfqpoint{4.219347in}{2.198752in}}% +\pgfpathlineto{\pgfqpoint{4.239827in}{2.217634in}}% +\pgfpathlineto{\pgfqpoint{4.260308in}{2.231976in}}% +\pgfpathlineto{\pgfqpoint{4.280788in}{2.241750in}}% +\pgfpathlineto{\pgfqpoint{4.296149in}{2.246082in}}% +\pgfpathlineto{\pgfqpoint{4.311509in}{2.247857in}}% +\pgfpathlineto{\pgfqpoint{4.326869in}{2.247097in}}% +\pgfpathlineto{\pgfqpoint{4.342230in}{2.243835in}}% +\pgfpathlineto{\pgfqpoint{4.357590in}{2.238111in}}% +\pgfpathlineto{\pgfqpoint{4.378071in}{2.226740in}}% +\pgfpathlineto{\pgfqpoint{4.398551in}{2.211228in}}% +\pgfpathlineto{\pgfqpoint{4.419032in}{2.191748in}}% +\pgfpathlineto{\pgfqpoint{4.439512in}{2.168497in}}% +\pgfpathlineto{\pgfqpoint{4.465113in}{2.134469in}}% +\pgfpathlineto{\pgfqpoint{4.490713in}{2.095377in}}% +\pgfpathlineto{\pgfqpoint{4.521434in}{2.042531in}}% +\pgfpathlineto{\pgfqpoint{4.557275in}{1.973957in}}% +\pgfpathlineto{\pgfqpoint{4.598236in}{1.888550in}}% +\pgfpathlineto{\pgfqpoint{4.659677in}{1.752047in}}% +\pgfpathlineto{\pgfqpoint{4.751839in}{1.546791in}}% +\pgfpathlineto{\pgfqpoint{4.797920in}{1.452145in}}% +\pgfpathlineto{\pgfqpoint{4.833761in}{1.385218in}}% +\pgfpathlineto{\pgfqpoint{4.864482in}{1.333687in}}% +\pgfpathlineto{\pgfqpoint{4.895203in}{1.288362in}}% +\pgfpathlineto{\pgfqpoint{4.920803in}{1.255807in}}% +\pgfpathlineto{\pgfqpoint{4.946404in}{1.228334in}}% +\pgfpathlineto{\pgfqpoint{4.966884in}{1.210185in}}% +\pgfpathlineto{\pgfqpoint{4.987365in}{1.195552in}}% +\pgfpathlineto{\pgfqpoint{5.007845in}{1.184507in}}% +\pgfpathlineto{\pgfqpoint{5.028326in}{1.177095in}}% +\pgfpathlineto{\pgfqpoint{5.048806in}{1.173335in}}% +\pgfpathlineto{\pgfqpoint{5.069287in}{1.173219in}}% +\pgfpathlineto{\pgfqpoint{5.089767in}{1.176711in}}% +\pgfpathlineto{\pgfqpoint{5.110248in}{1.183749in}}% +\pgfpathlineto{\pgfqpoint{5.130728in}{1.194245in}}% +\pgfpathlineto{\pgfqpoint{5.151209in}{1.208087in}}% +\pgfpathlineto{\pgfqpoint{5.171689in}{1.225140in}}% +\pgfpathlineto{\pgfqpoint{5.197290in}{1.250725in}}% +\pgfpathlineto{\pgfqpoint{5.222890in}{1.280719in}}% +\pgfpathlineto{\pgfqpoint{5.253611in}{1.321959in}}% +\pgfpathlineto{\pgfqpoint{5.289452in}{1.376320in}}% +\pgfpathlineto{\pgfqpoint{5.330413in}{1.445009in}}% +\pgfpathlineto{\pgfqpoint{5.381614in}{1.537494in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.597703in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.597703in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{1.216606in}}% +\pgfpathlineto{\pgfqpoint{0.753026in}{1.221334in}}% +\pgfpathlineto{\pgfqpoint{0.773506in}{1.230156in}}% +\pgfpathlineto{\pgfqpoint{0.793986in}{1.242306in}}% +\pgfpathlineto{\pgfqpoint{0.814467in}{1.257718in}}% +\pgfpathlineto{\pgfqpoint{0.840068in}{1.281435in}}% +\pgfpathlineto{\pgfqpoint{0.865668in}{1.309880in}}% +\pgfpathlineto{\pgfqpoint{0.891269in}{1.342765in}}% +\pgfpathlineto{\pgfqpoint{0.921989in}{1.387602in}}% +\pgfpathlineto{\pgfqpoint{0.957830in}{1.446431in}}% +\pgfpathlineto{\pgfqpoint{0.998791in}{1.520692in}}% +\pgfpathlineto{\pgfqpoint{1.049992in}{1.620888in}}% +\pgfpathlineto{\pgfqpoint{1.193356in}{1.906162in}}% +\pgfpathlineto{\pgfqpoint{1.234317in}{1.978026in}}% +\pgfpathlineto{\pgfqpoint{1.265038in}{2.026309in}}% +\pgfpathlineto{\pgfqpoint{1.295758in}{2.068752in}}% +\pgfpathlineto{\pgfqpoint{1.321359in}{2.099055in}}% +\pgfpathlineto{\pgfqpoint{1.346959in}{2.124290in}}% +\pgfpathlineto{\pgfqpoint{1.367440in}{2.140574in}}% +\pgfpathlineto{\pgfqpoint{1.387920in}{2.153205in}}% +\pgfpathlineto{\pgfqpoint{1.408401in}{2.162042in}}% +\pgfpathlineto{\pgfqpoint{1.428881in}{2.166971in}}% +\pgfpathlineto{\pgfqpoint{1.449362in}{2.167905in}}% +\pgfpathlineto{\pgfqpoint{1.469842in}{2.164788in}}% +\pgfpathlineto{\pgfqpoint{1.490323in}{2.157590in}}% +\pgfpathlineto{\pgfqpoint{1.510803in}{2.146316in}}% +\pgfpathlineto{\pgfqpoint{1.531284in}{2.130996in}}% +\pgfpathlineto{\pgfqpoint{1.551764in}{2.111695in}}% +\pgfpathlineto{\pgfqpoint{1.572245in}{2.088506in}}% +\pgfpathlineto{\pgfqpoint{1.597845in}{2.054244in}}% +\pgfpathlineto{\pgfqpoint{1.623446in}{2.014409in}}% +\pgfpathlineto{\pgfqpoint{1.654167in}{1.959785in}}% +\pgfpathlineto{\pgfqpoint{1.684887in}{1.898458in}}% +\pgfpathlineto{\pgfqpoint{1.720728in}{1.819647in}}% +\pgfpathlineto{\pgfqpoint{1.761689in}{1.722017in}}% +\pgfpathlineto{\pgfqpoint{1.823131in}{1.566276in}}% +\pgfpathlineto{\pgfqpoint{1.915293in}{1.332110in}}% +\pgfpathlineto{\pgfqpoint{1.956254in}{1.236020in}}% +\pgfpathlineto{\pgfqpoint{1.992095in}{1.159534in}}% +\pgfpathlineto{\pgfqpoint{2.022815in}{1.101214in}}% +\pgfpathlineto{\pgfqpoint{2.048416in}{1.058625in}}% +\pgfpathlineto{\pgfqpoint{2.074017in}{1.022179in}}% +\pgfpathlineto{\pgfqpoint{2.094497in}{0.997838in}}% +\pgfpathlineto{\pgfqpoint{2.114977in}{0.978071in}}% +\pgfpathlineto{\pgfqpoint{2.135458in}{0.963123in}}% +\pgfpathlineto{\pgfqpoint{2.150818in}{0.955205in}}% +\pgfpathlineto{\pgfqpoint{2.166179in}{0.950198in}}% +\pgfpathlineto{\pgfqpoint{2.181539in}{0.948173in}}% +\pgfpathlineto{\pgfqpoint{2.196899in}{0.949189in}}% +\pgfpathlineto{\pgfqpoint{2.212260in}{0.953295in}}% +\pgfpathlineto{\pgfqpoint{2.227620in}{0.960530in}}% +\pgfpathlineto{\pgfqpoint{2.242980in}{0.970921in}}% +\pgfpathlineto{\pgfqpoint{2.258341in}{0.984483in}}% +\pgfpathlineto{\pgfqpoint{2.278821in}{1.007506in}}% +\pgfpathlineto{\pgfqpoint{2.299302in}{1.036155in}}% +\pgfpathlineto{\pgfqpoint{2.319782in}{1.070372in}}% +\pgfpathlineto{\pgfqpoint{2.340263in}{1.110064in}}% +\pgfpathlineto{\pgfqpoint{2.365863in}{1.167179in}}% +\pgfpathlineto{\pgfqpoint{2.391464in}{1.232315in}}% +\pgfpathlineto{\pgfqpoint{2.422185in}{1.320479in}}% +\pgfpathlineto{\pgfqpoint{2.452905in}{1.418727in}}% +\pgfpathlineto{\pgfqpoint{2.488746in}{1.544721in}}% +\pgfpathlineto{\pgfqpoint{2.529707in}{1.701401in}}% +\pgfpathlineto{\pgfqpoint{2.580908in}{1.911607in}}% +\pgfpathlineto{\pgfqpoint{2.662830in}{2.264717in}}% +\pgfpathlineto{\pgfqpoint{2.739632in}{2.590813in}}% +\pgfpathlineto{\pgfqpoint{2.785713in}{2.773303in}}% +\pgfpathlineto{\pgfqpoint{2.821554in}{2.904226in}}% +\pgfpathlineto{\pgfqpoint{2.857395in}{3.023000in}}% +\pgfpathlineto{\pgfqpoint{2.888116in}{3.113554in}}% +\pgfpathlineto{\pgfqpoint{2.913716in}{3.180177in}}% +\pgfpathlineto{\pgfqpoint{2.939317in}{3.238121in}}% +\pgfpathlineto{\pgfqpoint{2.959797in}{3.277873in}}% +\pgfpathlineto{\pgfqpoint{2.980278in}{3.311504in}}% +\pgfpathlineto{\pgfqpoint{3.000758in}{3.338818in}}% +\pgfpathlineto{\pgfqpoint{3.016119in}{3.355062in}}% +\pgfpathlineto{\pgfqpoint{3.031479in}{3.367610in}}% +\pgfpathlineto{\pgfqpoint{3.046839in}{3.376420in}}% +\pgfpathlineto{\pgfqpoint{3.062200in}{3.381465in}}% +\pgfpathlineto{\pgfqpoint{3.077560in}{3.382727in}}% +\pgfpathlineto{\pgfqpoint{3.092920in}{3.380203in}}% +\pgfpathlineto{\pgfqpoint{3.108281in}{3.373901in}}% +\pgfpathlineto{\pgfqpoint{3.123641in}{3.363840in}}% +\pgfpathlineto{\pgfqpoint{3.139002in}{3.350056in}}% +\pgfpathlineto{\pgfqpoint{3.154362in}{3.332591in}}% +\pgfpathlineto{\pgfqpoint{3.174842in}{3.303681in}}% +\pgfpathlineto{\pgfqpoint{3.195323in}{3.268501in}}% +\pgfpathlineto{\pgfqpoint{3.215803in}{3.227253in}}% +\pgfpathlineto{\pgfqpoint{3.241404in}{3.167529in}}% +\pgfpathlineto{\pgfqpoint{3.267005in}{3.099239in}}% +\pgfpathlineto{\pgfqpoint{3.297725in}{3.006858in}}% +\pgfpathlineto{\pgfqpoint{3.328446in}{2.904226in}}% +\pgfpathlineto{\pgfqpoint{3.364287in}{2.773303in}}% +\pgfpathlineto{\pgfqpoint{3.405248in}{2.611735in}}% +\pgfpathlineto{\pgfqpoint{3.461569in}{2.375278in}}% +\pgfpathlineto{\pgfqpoint{3.610053in}{1.742341in}}% +\pgfpathlineto{\pgfqpoint{3.656134in}{1.563621in}}% +\pgfpathlineto{\pgfqpoint{3.691974in}{1.436012in}}% +\pgfpathlineto{\pgfqpoint{3.722695in}{1.336179in}}% +\pgfpathlineto{\pgfqpoint{3.753416in}{1.246270in}}% +\pgfpathlineto{\pgfqpoint{3.779017in}{1.179576in}}% +\pgfpathlineto{\pgfqpoint{3.804617in}{1.120828in}}% +\pgfpathlineto{\pgfqpoint{3.830218in}{1.070372in}}% +\pgfpathlineto{\pgfqpoint{3.850698in}{1.036155in}}% +\pgfpathlineto{\pgfqpoint{3.871179in}{1.007506in}}% +\pgfpathlineto{\pgfqpoint{3.891659in}{0.984483in}}% +\pgfpathlineto{\pgfqpoint{3.912140in}{0.967105in}}% +\pgfpathlineto{\pgfqpoint{3.927500in}{0.957769in}}% +\pgfpathlineto{\pgfqpoint{3.942860in}{0.951581in}}% +\pgfpathlineto{\pgfqpoint{3.958221in}{0.948509in}}% +\pgfpathlineto{\pgfqpoint{3.973581in}{0.948513in}}% +\pgfpathlineto{\pgfqpoint{3.988941in}{0.951539in}}% +\pgfpathlineto{\pgfqpoint{4.004302in}{0.957524in}}% +\pgfpathlineto{\pgfqpoint{4.019662in}{0.966396in}}% +\pgfpathlineto{\pgfqpoint{4.040143in}{0.982570in}}% +\pgfpathlineto{\pgfqpoint{4.060623in}{1.003504in}}% +\pgfpathlineto{\pgfqpoint{4.081104in}{1.028948in}}% +\pgfpathlineto{\pgfqpoint{4.106704in}{1.066671in}}% +\pgfpathlineto{\pgfqpoint{4.132305in}{1.110414in}}% +\pgfpathlineto{\pgfqpoint{4.163026in}{1.169944in}}% +\pgfpathlineto{\pgfqpoint{4.198866in}{1.247581in}}% +\pgfpathlineto{\pgfqpoint{4.239827in}{1.344626in}}% +\pgfpathlineto{\pgfqpoint{4.296149in}{1.486961in}}% +\pgfpathlineto{\pgfqpoint{4.398551in}{1.747046in}}% +\pgfpathlineto{\pgfqpoint{4.439512in}{1.842878in}}% +\pgfpathlineto{\pgfqpoint{4.475353in}{1.919592in}}% +\pgfpathlineto{\pgfqpoint{4.506074in}{1.978779in}}% +\pgfpathlineto{\pgfqpoint{4.536794in}{2.030989in}}% +\pgfpathlineto{\pgfqpoint{4.562395in}{2.068636in}}% +\pgfpathlineto{\pgfqpoint{4.587995in}{2.100580in}}% +\pgfpathlineto{\pgfqpoint{4.608476in}{2.121838in}}% +\pgfpathlineto{\pgfqpoint{4.628956in}{2.139158in}}% +\pgfpathlineto{\pgfqpoint{4.649437in}{2.152462in}}% +\pgfpathlineto{\pgfqpoint{4.669917in}{2.161700in}}% +\pgfpathlineto{\pgfqpoint{4.690398in}{2.166855in}}% +\pgfpathlineto{\pgfqpoint{4.710878in}{2.167941in}}% +\pgfpathlineto{\pgfqpoint{4.731359in}{2.165001in}}% +\pgfpathlineto{\pgfqpoint{4.751839in}{2.158105in}}% +\pgfpathlineto{\pgfqpoint{4.772320in}{2.147355in}}% +\pgfpathlineto{\pgfqpoint{4.792800in}{2.132879in}}% +\pgfpathlineto{\pgfqpoint{4.813281in}{2.114830in}}% +\pgfpathlineto{\pgfqpoint{4.838881in}{2.087519in}}% +\pgfpathlineto{\pgfqpoint{4.864482in}{2.055307in}}% +\pgfpathlineto{\pgfqpoint{4.895203in}{2.010820in}}% +\pgfpathlineto{\pgfqpoint{4.925923in}{1.960790in}}% +\pgfpathlineto{\pgfqpoint{4.961764in}{1.896680in}}% +\pgfpathlineto{\pgfqpoint{5.007845in}{1.807644in}}% +\pgfpathlineto{\pgfqpoint{5.176809in}{1.473501in}}% +\pgfpathlineto{\pgfqpoint{5.212650in}{1.412018in}}% +\pgfpathlineto{\pgfqpoint{5.243371in}{1.364487in}}% +\pgfpathlineto{\pgfqpoint{5.274092in}{1.322519in}}% +\pgfpathlineto{\pgfqpoint{5.299692in}{1.292260in}}% +\pgfpathlineto{\pgfqpoint{5.325293in}{1.266622in}}% +\pgfpathlineto{\pgfqpoint{5.350893in}{1.245856in}}% +\pgfpathlineto{\pgfqpoint{5.371374in}{1.232884in}}% +\pgfpathlineto{\pgfqpoint{5.391854in}{1.223225in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.216606in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.216606in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{1.597703in}}% +\pgfpathlineto{\pgfqpoint{0.814467in}{1.453986in}}% +\pgfpathlineto{\pgfqpoint{0.855428in}{1.384562in}}% +\pgfpathlineto{\pgfqpoint{0.891269in}{1.329340in}}% +\pgfpathlineto{\pgfqpoint{0.921989in}{1.287212in}}% +\pgfpathlineto{\pgfqpoint{0.952710in}{1.250725in}}% +\pgfpathlineto{\pgfqpoint{0.978311in}{1.225140in}}% +\pgfpathlineto{\pgfqpoint{1.003911in}{1.204320in}}% +\pgfpathlineto{\pgfqpoint{1.024392in}{1.191302in}}% +\pgfpathlineto{\pgfqpoint{1.044872in}{1.181661in}}% +\pgfpathlineto{\pgfqpoint{1.065353in}{1.175502in}}% +\pgfpathlineto{\pgfqpoint{1.085833in}{1.172908in}}% +\pgfpathlineto{\pgfqpoint{1.106314in}{1.173933in}}% +\pgfpathlineto{\pgfqpoint{1.126794in}{1.178606in}}% +\pgfpathlineto{\pgfqpoint{1.147275in}{1.186929in}}% +\pgfpathlineto{\pgfqpoint{1.167755in}{1.198876in}}% +\pgfpathlineto{\pgfqpoint{1.188236in}{1.214396in}}% +\pgfpathlineto{\pgfqpoint{1.208716in}{1.233409in}}% +\pgfpathlineto{\pgfqpoint{1.234317in}{1.261921in}}% +\pgfpathlineto{\pgfqpoint{1.259917in}{1.295456in}}% +\pgfpathlineto{\pgfqpoint{1.285518in}{1.333687in}}% +\pgfpathlineto{\pgfqpoint{1.316239in}{1.385218in}}% +\pgfpathlineto{\pgfqpoint{1.352080in}{1.452145in}}% +\pgfpathlineto{\pgfqpoint{1.393041in}{1.535887in}}% +\pgfpathlineto{\pgfqpoint{1.444242in}{1.647984in}}% +\pgfpathlineto{\pgfqpoint{1.572245in}{1.932041in}}% +\pgfpathlineto{\pgfqpoint{1.613206in}{2.013968in}}% +\pgfpathlineto{\pgfqpoint{1.649047in}{2.078440in}}% +\pgfpathlineto{\pgfqpoint{1.679767in}{2.127039in}}% +\pgfpathlineto{\pgfqpoint{1.705368in}{2.162120in}}% +\pgfpathlineto{\pgfqpoint{1.730968in}{2.191748in}}% +\pgfpathlineto{\pgfqpoint{1.751449in}{2.211228in}}% +\pgfpathlineto{\pgfqpoint{1.771929in}{2.226740in}}% +\pgfpathlineto{\pgfqpoint{1.792410in}{2.238111in}}% +\pgfpathlineto{\pgfqpoint{1.812890in}{2.245198in}}% +\pgfpathlineto{\pgfqpoint{1.828251in}{2.247630in}}% +\pgfpathlineto{\pgfqpoint{1.843611in}{2.247548in}}% +\pgfpathlineto{\pgfqpoint{1.858971in}{2.244923in}}% +\pgfpathlineto{\pgfqpoint{1.874332in}{2.239735in}}% +\pgfpathlineto{\pgfqpoint{1.889692in}{2.231976in}}% +\pgfpathlineto{\pgfqpoint{1.910173in}{2.217634in}}% +\pgfpathlineto{\pgfqpoint{1.930653in}{2.198752in}}% +\pgfpathlineto{\pgfqpoint{1.951134in}{2.175391in}}% +\pgfpathlineto{\pgfqpoint{1.971614in}{2.147643in}}% +\pgfpathlineto{\pgfqpoint{1.997215in}{2.106979in}}% +\pgfpathlineto{\pgfqpoint{2.022815in}{2.059964in}}% +\pgfpathlineto{\pgfqpoint{2.048416in}{2.006975in}}% +\pgfpathlineto{\pgfqpoint{2.079137in}{1.936139in}}% +\pgfpathlineto{\pgfqpoint{2.114977in}{1.844645in}}% +\pgfpathlineto{\pgfqpoint{2.155938in}{1.730417in}}% +\pgfpathlineto{\pgfqpoint{2.207140in}{1.577176in}}% +\pgfpathlineto{\pgfqpoint{2.370983in}{1.076956in}}% +\pgfpathlineto{\pgfqpoint{2.411944in}{0.966434in}}% +\pgfpathlineto{\pgfqpoint{2.442665in}{0.891278in}}% +\pgfpathlineto{\pgfqpoint{2.473386in}{0.824058in}}% +\pgfpathlineto{\pgfqpoint{2.498986in}{0.774878in}}% +\pgfpathlineto{\pgfqpoint{2.524587in}{0.732506in}}% +\pgfpathlineto{\pgfqpoint{2.545068in}{0.703844in}}% +\pgfpathlineto{\pgfqpoint{2.565548in}{0.680076in}}% +\pgfpathlineto{\pgfqpoint{2.586029in}{0.661392in}}% +\pgfpathlineto{\pgfqpoint{2.601389in}{0.650812in}}% +\pgfpathlineto{\pgfqpoint{2.616749in}{0.643238in}}% +\pgfpathlineto{\pgfqpoint{2.632110in}{0.638713in}}% +\pgfpathlineto{\pgfqpoint{2.647470in}{0.637273in}}% +\pgfpathlineto{\pgfqpoint{2.662830in}{0.638940in}}% +\pgfpathlineto{\pgfqpoint{2.678191in}{0.643726in}}% +\pgfpathlineto{\pgfqpoint{2.693551in}{0.651634in}}% +\pgfpathlineto{\pgfqpoint{2.708911in}{0.662653in}}% +\pgfpathlineto{\pgfqpoint{2.724272in}{0.676762in}}% +\pgfpathlineto{\pgfqpoint{2.744752in}{0.700324in}}% +\pgfpathlineto{\pgfqpoint{2.765233in}{0.729218in}}% +\pgfpathlineto{\pgfqpoint{2.785713in}{0.763302in}}% +\pgfpathlineto{\pgfqpoint{2.811314in}{0.812941in}}% +\pgfpathlineto{\pgfqpoint{2.836914in}{0.870013in}}% +\pgfpathlineto{\pgfqpoint{2.862515in}{0.934035in}}% +\pgfpathlineto{\pgfqpoint{2.893236in}{1.019257in}}% +\pgfpathlineto{\pgfqpoint{2.929077in}{1.128887in}}% +\pgfpathlineto{\pgfqpoint{2.970038in}{1.265288in}}% +\pgfpathlineto{\pgfqpoint{3.021239in}{1.447839in}}% +\pgfpathlineto{\pgfqpoint{3.195323in}{2.081402in}}% +\pgfpathlineto{\pgfqpoint{3.236284in}{2.213666in}}% +\pgfpathlineto{\pgfqpoint{3.272125in}{2.318771in}}% +\pgfpathlineto{\pgfqpoint{3.302845in}{2.399513in}}% +\pgfpathlineto{\pgfqpoint{3.328446in}{2.459426in}}% +\pgfpathlineto{\pgfqpoint{3.354047in}{2.512092in}}% +\pgfpathlineto{\pgfqpoint{3.379647in}{2.557072in}}% +\pgfpathlineto{\pgfqpoint{3.400128in}{2.587277in}}% +\pgfpathlineto{\pgfqpoint{3.420608in}{2.612183in}}% +\pgfpathlineto{\pgfqpoint{3.441089in}{2.631677in}}% +\pgfpathlineto{\pgfqpoint{3.456449in}{2.642696in}}% +\pgfpathlineto{\pgfqpoint{3.471809in}{2.650603in}}% +\pgfpathlineto{\pgfqpoint{3.487170in}{2.655390in}}% +\pgfpathlineto{\pgfqpoint{3.502530in}{2.657057in}}% +\pgfpathlineto{\pgfqpoint{3.517890in}{2.655617in}}% +\pgfpathlineto{\pgfqpoint{3.533251in}{2.651092in}}% +\pgfpathlineto{\pgfqpoint{3.548611in}{2.643518in}}% +\pgfpathlineto{\pgfqpoint{3.563971in}{2.632938in}}% +\pgfpathlineto{\pgfqpoint{3.584452in}{2.614254in}}% +\pgfpathlineto{\pgfqpoint{3.604932in}{2.590485in}}% +\pgfpathlineto{\pgfqpoint{3.625413in}{2.561823in}}% +\pgfpathlineto{\pgfqpoint{3.645893in}{2.528487in}}% +\pgfpathlineto{\pgfqpoint{3.671494in}{2.480626in}}% +\pgfpathlineto{\pgfqpoint{3.697095in}{2.426395in}}% +\pgfpathlineto{\pgfqpoint{3.727815in}{2.353759in}}% +\pgfpathlineto{\pgfqpoint{3.763656in}{2.260043in}}% +\pgfpathlineto{\pgfqpoint{3.804617in}{2.143513in}}% +\pgfpathlineto{\pgfqpoint{3.860938in}{1.972417in}}% +\pgfpathlineto{\pgfqpoint{3.988941in}{1.578791in}}% +\pgfpathlineto{\pgfqpoint{4.029902in}{1.463447in}}% +\pgfpathlineto{\pgfqpoint{4.065743in}{1.370709in}}% +\pgfpathlineto{\pgfqpoint{4.096464in}{1.298630in}}% +\pgfpathlineto{\pgfqpoint{4.127185in}{1.234366in}}% +\pgfpathlineto{\pgfqpoint{4.152785in}{1.187350in}}% +\pgfpathlineto{\pgfqpoint{4.178386in}{1.146687in}}% +\pgfpathlineto{\pgfqpoint{4.198866in}{1.118939in}}% +\pgfpathlineto{\pgfqpoint{4.219347in}{1.095578in}}% +\pgfpathlineto{\pgfqpoint{4.239827in}{1.076696in}}% +\pgfpathlineto{\pgfqpoint{4.260308in}{1.062353in}}% +\pgfpathlineto{\pgfqpoint{4.280788in}{1.052580in}}% +\pgfpathlineto{\pgfqpoint{4.296149in}{1.048248in}}% +\pgfpathlineto{\pgfqpoint{4.311509in}{1.046473in}}% +\pgfpathlineto{\pgfqpoint{4.326869in}{1.047233in}}% +\pgfpathlineto{\pgfqpoint{4.342230in}{1.050495in}}% +\pgfpathlineto{\pgfqpoint{4.357590in}{1.056219in}}% +\pgfpathlineto{\pgfqpoint{4.378071in}{1.067590in}}% +\pgfpathlineto{\pgfqpoint{4.398551in}{1.083102in}}% +\pgfpathlineto{\pgfqpoint{4.419032in}{1.102582in}}% +\pgfpathlineto{\pgfqpoint{4.439512in}{1.125833in}}% +\pgfpathlineto{\pgfqpoint{4.465113in}{1.159861in}}% +\pgfpathlineto{\pgfqpoint{4.490713in}{1.198952in}}% +\pgfpathlineto{\pgfqpoint{4.521434in}{1.251799in}}% +\pgfpathlineto{\pgfqpoint{4.557275in}{1.320372in}}% +\pgfpathlineto{\pgfqpoint{4.598236in}{1.405779in}}% +\pgfpathlineto{\pgfqpoint{4.659677in}{1.542283in}}% +\pgfpathlineto{\pgfqpoint{4.751839in}{1.747538in}}% +\pgfpathlineto{\pgfqpoint{4.797920in}{1.842184in}}% +\pgfpathlineto{\pgfqpoint{4.833761in}{1.909112in}}% +\pgfpathlineto{\pgfqpoint{4.864482in}{1.960642in}}% +\pgfpathlineto{\pgfqpoint{4.895203in}{2.005968in}}% +\pgfpathlineto{\pgfqpoint{4.920803in}{2.038523in}}% +\pgfpathlineto{\pgfqpoint{4.946404in}{2.065996in}}% +\pgfpathlineto{\pgfqpoint{4.966884in}{2.084144in}}% +\pgfpathlineto{\pgfqpoint{4.987365in}{2.098778in}}% +\pgfpathlineto{\pgfqpoint{5.007845in}{2.109823in}}% +\pgfpathlineto{\pgfqpoint{5.028326in}{2.117235in}}% +\pgfpathlineto{\pgfqpoint{5.048806in}{2.120995in}}% +\pgfpathlineto{\pgfqpoint{5.069287in}{2.121111in}}% +\pgfpathlineto{\pgfqpoint{5.089767in}{2.117619in}}% +\pgfpathlineto{\pgfqpoint{5.110248in}{2.110581in}}% +\pgfpathlineto{\pgfqpoint{5.130728in}{2.100085in}}% +\pgfpathlineto{\pgfqpoint{5.151209in}{2.086242in}}% +\pgfpathlineto{\pgfqpoint{5.171689in}{2.069190in}}% +\pgfpathlineto{\pgfqpoint{5.197290in}{2.043605in}}% +\pgfpathlineto{\pgfqpoint{5.222890in}{2.013611in}}% +\pgfpathlineto{\pgfqpoint{5.253611in}{1.972370in}}% +\pgfpathlineto{\pgfqpoint{5.289452in}{1.918010in}}% +\pgfpathlineto{\pgfqpoint{5.330413in}{1.849320in}}% +\pgfpathlineto{\pgfqpoint{5.381614in}{1.756835in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.696626in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.696626in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.580392,0.403922,0.741176}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{2.087558in}}% +\pgfpathlineto{\pgfqpoint{0.753026in}{2.089238in}}% +\pgfpathlineto{\pgfqpoint{0.773506in}{2.088244in}}% +\pgfpathlineto{\pgfqpoint{0.793986in}{2.083946in}}% +\pgfpathlineto{\pgfqpoint{0.814467in}{2.076349in}}% +\pgfpathlineto{\pgfqpoint{0.834947in}{2.065480in}}% +\pgfpathlineto{\pgfqpoint{0.855428in}{2.051393in}}% +\pgfpathlineto{\pgfqpoint{0.881029in}{2.029378in}}% +\pgfpathlineto{\pgfqpoint{0.906629in}{2.002657in}}% +\pgfpathlineto{\pgfqpoint{0.932230in}{1.971486in}}% +\pgfpathlineto{\pgfqpoint{0.962950in}{1.928646in}}% +\pgfpathlineto{\pgfqpoint{0.993671in}{1.880474in}}% +\pgfpathlineto{\pgfqpoint{1.029512in}{1.818502in}}% +\pgfpathlineto{\pgfqpoint{1.075593in}{1.731696in}}% +\pgfpathlineto{\pgfqpoint{1.137035in}{1.608436in}}% +\pgfpathlineto{\pgfqpoint{1.229197in}{1.423447in}}% +\pgfpathlineto{\pgfqpoint{1.275278in}{1.338266in}}% +\pgfpathlineto{\pgfqpoint{1.311119in}{1.278235in}}% +\pgfpathlineto{\pgfqpoint{1.341839in}{1.232297in}}% +\pgfpathlineto{\pgfqpoint{1.372560in}{1.192309in}}% +\pgfpathlineto{\pgfqpoint{1.398161in}{1.164059in}}% +\pgfpathlineto{\pgfqpoint{1.423761in}{1.140817in}}% +\pgfpathlineto{\pgfqpoint{1.444242in}{1.126046in}}% +\pgfpathlineto{\pgfqpoint{1.464722in}{1.114825in}}% +\pgfpathlineto{\pgfqpoint{1.485203in}{1.107268in}}% +\pgfpathlineto{\pgfqpoint{1.505683in}{1.103467in}}% +\pgfpathlineto{\pgfqpoint{1.526164in}{1.103488in}}% +\pgfpathlineto{\pgfqpoint{1.546644in}{1.107370in}}% +\pgfpathlineto{\pgfqpoint{1.567125in}{1.115129in}}% +\pgfpathlineto{\pgfqpoint{1.587605in}{1.126753in}}% +\pgfpathlineto{\pgfqpoint{1.608086in}{1.142205in}}% +\pgfpathlineto{\pgfqpoint{1.628566in}{1.161422in}}% +\pgfpathlineto{\pgfqpoint{1.649047in}{1.184315in}}% +\pgfpathlineto{\pgfqpoint{1.674647in}{1.217924in}}% +\pgfpathlineto{\pgfqpoint{1.700248in}{1.256823in}}% +\pgfpathlineto{\pgfqpoint{1.730968in}{1.310014in}}% +\pgfpathlineto{\pgfqpoint{1.761689in}{1.369671in}}% +\pgfpathlineto{\pgfqpoint{1.797530in}{1.446409in}}% +\pgfpathlineto{\pgfqpoint{1.838491in}{1.541833in}}% +\pgfpathlineto{\pgfqpoint{1.889692in}{1.669363in}}% +\pgfpathlineto{\pgfqpoint{2.053536in}{2.084842in}}% +\pgfpathlineto{\pgfqpoint{2.094497in}{2.177528in}}% +\pgfpathlineto{\pgfqpoint{2.130338in}{2.251280in}}% +\pgfpathlineto{\pgfqpoint{2.161059in}{2.308003in}}% +\pgfpathlineto{\pgfqpoint{2.191779in}{2.357982in}}% +\pgfpathlineto{\pgfqpoint{2.217380in}{2.394040in}}% +\pgfpathlineto{\pgfqpoint{2.242980in}{2.424708in}}% +\pgfpathlineto{\pgfqpoint{2.263461in}{2.445205in}}% +\pgfpathlineto{\pgfqpoint{2.283941in}{2.462016in}}% +\pgfpathlineto{\pgfqpoint{2.304422in}{2.475078in}}% +\pgfpathlineto{\pgfqpoint{2.324902in}{2.484355in}}% +\pgfpathlineto{\pgfqpoint{2.345383in}{2.489835in}}% +\pgfpathlineto{\pgfqpoint{2.365863in}{2.491535in}}% +\pgfpathlineto{\pgfqpoint{2.386344in}{2.489498in}}% +\pgfpathlineto{\pgfqpoint{2.406824in}{2.483791in}}% +\pgfpathlineto{\pgfqpoint{2.427305in}{2.474508in}}% +\pgfpathlineto{\pgfqpoint{2.447785in}{2.461767in}}% +\pgfpathlineto{\pgfqpoint{2.468266in}{2.445710in}}% +\pgfpathlineto{\pgfqpoint{2.493866in}{2.421230in}}% +\pgfpathlineto{\pgfqpoint{2.519467in}{2.392195in}}% +\pgfpathlineto{\pgfqpoint{2.550188in}{2.351932in}}% +\pgfpathlineto{\pgfqpoint{2.580908in}{2.306515in}}% +\pgfpathlineto{\pgfqpoint{2.616749in}{2.248188in}}% +\pgfpathlineto{\pgfqpoint{2.662830in}{2.167068in}}% +\pgfpathlineto{\pgfqpoint{2.826674in}{1.871952in}}% +\pgfpathlineto{\pgfqpoint{2.862515in}{1.816084in}}% +\pgfpathlineto{\pgfqpoint{2.893236in}{1.773136in}}% +\pgfpathlineto{\pgfqpoint{2.923956in}{1.735550in}}% +\pgfpathlineto{\pgfqpoint{2.949557in}{1.708802in}}% +\pgfpathlineto{\pgfqpoint{2.975158in}{1.686562in}}% +\pgfpathlineto{\pgfqpoint{2.995638in}{1.672198in}}% +\pgfpathlineto{\pgfqpoint{3.016119in}{1.661005in}}% +\pgfpathlineto{\pgfqpoint{3.036599in}{1.653070in}}% +\pgfpathlineto{\pgfqpoint{3.057080in}{1.648453in}}% +\pgfpathlineto{\pgfqpoint{3.077560in}{1.647191in}}% +\pgfpathlineto{\pgfqpoint{3.098041in}{1.649294in}}% +\pgfpathlineto{\pgfqpoint{3.118521in}{1.654744in}}% +\pgfpathlineto{\pgfqpoint{3.139002in}{1.663501in}}% +\pgfpathlineto{\pgfqpoint{3.159482in}{1.675496in}}% +\pgfpathlineto{\pgfqpoint{3.179962in}{1.690635in}}% +\pgfpathlineto{\pgfqpoint{3.205563in}{1.713801in}}% +\pgfpathlineto{\pgfqpoint{3.231164in}{1.741413in}}% +\pgfpathlineto{\pgfqpoint{3.261884in}{1.779941in}}% +\pgfpathlineto{\pgfqpoint{3.292605in}{1.823711in}}% +\pgfpathlineto{\pgfqpoint{3.328446in}{1.880369in}}% +\pgfpathlineto{\pgfqpoint{3.374527in}{1.959949in}}% +\pgfpathlineto{\pgfqpoint{3.446209in}{2.091693in}}% +\pgfpathlineto{\pgfqpoint{3.517890in}{2.221778in}}% +\pgfpathlineto{\pgfqpoint{3.558851in}{2.290381in}}% +\pgfpathlineto{\pgfqpoint{3.594692in}{2.344698in}}% +\pgfpathlineto{\pgfqpoint{3.625413in}{2.385878in}}% +\pgfpathlineto{\pgfqpoint{3.651014in}{2.415775in}}% +\pgfpathlineto{\pgfqpoint{3.676614in}{2.441197in}}% +\pgfpathlineto{\pgfqpoint{3.702215in}{2.461767in}}% +\pgfpathlineto{\pgfqpoint{3.722695in}{2.474508in}}% +\pgfpathlineto{\pgfqpoint{3.743176in}{2.483791in}}% +\pgfpathlineto{\pgfqpoint{3.763656in}{2.489498in}}% +\pgfpathlineto{\pgfqpoint{3.784137in}{2.491535in}}% +\pgfpathlineto{\pgfqpoint{3.804617in}{2.489835in}}% +\pgfpathlineto{\pgfqpoint{3.825098in}{2.484355in}}% +\pgfpathlineto{\pgfqpoint{3.845578in}{2.475078in}}% +\pgfpathlineto{\pgfqpoint{3.866059in}{2.462016in}}% +\pgfpathlineto{\pgfqpoint{3.886539in}{2.445205in}}% +\pgfpathlineto{\pgfqpoint{3.907020in}{2.424708in}}% +\pgfpathlineto{\pgfqpoint{3.932620in}{2.394040in}}% +\pgfpathlineto{\pgfqpoint{3.958221in}{2.357982in}}% +\pgfpathlineto{\pgfqpoint{3.983821in}{2.316819in}}% +\pgfpathlineto{\pgfqpoint{4.014542in}{2.261173in}}% +\pgfpathlineto{\pgfqpoint{4.045263in}{2.199366in}}% +\pgfpathlineto{\pgfqpoint{4.081104in}{2.120526in}}% +\pgfpathlineto{\pgfqpoint{4.127185in}{2.010684in}}% +\pgfpathlineto{\pgfqpoint{4.188626in}{1.854728in}}% +\pgfpathlineto{\pgfqpoint{4.301269in}{1.566729in}}% +\pgfpathlineto{\pgfqpoint{4.347350in}{1.457926in}}% +\pgfpathlineto{\pgfqpoint{4.383191in}{1.380189in}}% +\pgfpathlineto{\pgfqpoint{4.419032in}{1.310014in}}% +\pgfpathlineto{\pgfqpoint{4.449752in}{1.256823in}}% +\pgfpathlineto{\pgfqpoint{4.475353in}{1.217924in}}% +\pgfpathlineto{\pgfqpoint{4.500953in}{1.184315in}}% +\pgfpathlineto{\pgfqpoint{4.526554in}{1.156269in}}% +\pgfpathlineto{\pgfqpoint{4.547035in}{1.137987in}}% +\pgfpathlineto{\pgfqpoint{4.567515in}{1.123487in}}% +\pgfpathlineto{\pgfqpoint{4.587995in}{1.112826in}}% +\pgfpathlineto{\pgfqpoint{4.608476in}{1.106036in}}% +\pgfpathlineto{\pgfqpoint{4.628956in}{1.103122in}}% +\pgfpathlineto{\pgfqpoint{4.649437in}{1.104062in}}% +\pgfpathlineto{\pgfqpoint{4.669917in}{1.108809in}}% +\pgfpathlineto{\pgfqpoint{4.690398in}{1.117291in}}% +\pgfpathlineto{\pgfqpoint{4.710878in}{1.129411in}}% +\pgfpathlineto{\pgfqpoint{4.731359in}{1.145049in}}% +\pgfpathlineto{\pgfqpoint{4.751839in}{1.164059in}}% +\pgfpathlineto{\pgfqpoint{4.777440in}{1.192309in}}% +\pgfpathlineto{\pgfqpoint{4.803041in}{1.225198in}}% +\pgfpathlineto{\pgfqpoint{4.833761in}{1.270199in}}% +\pgfpathlineto{\pgfqpoint{4.869602in}{1.329314in}}% +\pgfpathlineto{\pgfqpoint{4.910563in}{1.403883in}}% +\pgfpathlineto{\pgfqpoint{4.961764in}{1.504307in}}% +\pgfpathlineto{\pgfqpoint{5.105128in}{1.790337in}}% +\pgfpathlineto{\pgfqpoint{5.146089in}{1.863354in}}% +\pgfpathlineto{\pgfqpoint{5.181929in}{1.920969in}}% +\pgfpathlineto{\pgfqpoint{5.212650in}{1.964744in}}% +\pgfpathlineto{\pgfqpoint{5.243371in}{2.002657in}}% +\pgfpathlineto{\pgfqpoint{5.268971in}{2.029378in}}% +\pgfpathlineto{\pgfqpoint{5.294572in}{2.051393in}}% +\pgfpathlineto{\pgfqpoint{5.315053in}{2.065480in}}% +\pgfpathlineto{\pgfqpoint{5.335533in}{2.076349in}}% +\pgfpathlineto{\pgfqpoint{5.356014in}{2.083946in}}% +\pgfpathlineto{\pgfqpoint{5.376494in}{2.088244in}}% +\pgfpathlineto{\pgfqpoint{5.396974in}{2.089238in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{2.087558in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{2.087558in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.549020,0.337255,0.294118}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{1.521515in}}% +\pgfpathlineto{\pgfqpoint{0.783746in}{1.606794in}}% +\pgfpathlineto{\pgfqpoint{0.947590in}{1.905224in}}% +\pgfpathlineto{\pgfqpoint{0.988551in}{1.970384in}}% +\pgfpathlineto{\pgfqpoint{1.024392in}{2.021154in}}% +\pgfpathlineto{\pgfqpoint{1.055113in}{2.059124in}}% +\pgfpathlineto{\pgfqpoint{1.080713in}{2.086366in}}% +\pgfpathlineto{\pgfqpoint{1.106314in}{2.109260in}}% +\pgfpathlineto{\pgfqpoint{1.131914in}{2.127526in}}% +\pgfpathlineto{\pgfqpoint{1.152395in}{2.138650in}}% +\pgfpathlineto{\pgfqpoint{1.172875in}{2.146565in}}% +\pgfpathlineto{\pgfqpoint{1.193356in}{2.151195in}}% +\pgfpathlineto{\pgfqpoint{1.213836in}{2.152486in}}% +\pgfpathlineto{\pgfqpoint{1.234317in}{2.150404in}}% +\pgfpathlineto{\pgfqpoint{1.254797in}{2.144938in}}% +\pgfpathlineto{\pgfqpoint{1.275278in}{2.136097in}}% +\pgfpathlineto{\pgfqpoint{1.295758in}{2.123915in}}% +\pgfpathlineto{\pgfqpoint{1.316239in}{2.108444in}}% +\pgfpathlineto{\pgfqpoint{1.341839in}{2.084598in}}% +\pgfpathlineto{\pgfqpoint{1.367440in}{2.055922in}}% +\pgfpathlineto{\pgfqpoint{1.393041in}{2.022656in}}% +\pgfpathlineto{\pgfqpoint{1.423761in}{1.977081in}}% +\pgfpathlineto{\pgfqpoint{1.454482in}{1.925878in}}% +\pgfpathlineto{\pgfqpoint{1.490323in}{1.859898in}}% +\pgfpathlineto{\pgfqpoint{1.531284in}{1.777673in}}% +\pgfpathlineto{\pgfqpoint{1.582485in}{1.667422in}}% +\pgfpathlineto{\pgfqpoint{1.771929in}{1.250513in}}% +\pgfpathlineto{\pgfqpoint{1.812890in}{1.171326in}}% +\pgfpathlineto{\pgfqpoint{1.848731in}{1.108376in}}% +\pgfpathlineto{\pgfqpoint{1.879452in}{1.059883in}}% +\pgfpathlineto{\pgfqpoint{1.910173in}{1.016970in}}% +\pgfpathlineto{\pgfqpoint{1.935773in}{0.985773in}}% +\pgfpathlineto{\pgfqpoint{1.961374in}{0.958934in}}% +\pgfpathlineto{\pgfqpoint{1.986974in}{0.936595in}}% +\pgfpathlineto{\pgfqpoint{2.012575in}{0.918855in}}% +\pgfpathlineto{\pgfqpoint{2.033056in}{0.908010in}}% +\pgfpathlineto{\pgfqpoint{2.053536in}{0.900144in}}% +\pgfpathlineto{\pgfqpoint{2.074017in}{0.895242in}}% +\pgfpathlineto{\pgfqpoint{2.094497in}{0.893271in}}% +\pgfpathlineto{\pgfqpoint{2.114977in}{0.894176in}}% +\pgfpathlineto{\pgfqpoint{2.135458in}{0.897889in}}% +\pgfpathlineto{\pgfqpoint{2.155938in}{0.904320in}}% +\pgfpathlineto{\pgfqpoint{2.181539in}{0.916024in}}% +\pgfpathlineto{\pgfqpoint{2.207140in}{0.931580in}}% +\pgfpathlineto{\pgfqpoint{2.232740in}{0.950717in}}% +\pgfpathlineto{\pgfqpoint{2.263461in}{0.977990in}}% +\pgfpathlineto{\pgfqpoint{2.294182in}{1.009426in}}% +\pgfpathlineto{\pgfqpoint{2.330023in}{1.050554in}}% +\pgfpathlineto{\pgfqpoint{2.370983in}{1.102180in}}% +\pgfpathlineto{\pgfqpoint{2.427305in}{1.178446in}}% +\pgfpathlineto{\pgfqpoint{2.565548in}{1.368082in}}% +\pgfpathlineto{\pgfqpoint{2.611629in}{1.425304in}}% +\pgfpathlineto{\pgfqpoint{2.652590in}{1.471530in}}% +\pgfpathlineto{\pgfqpoint{2.688431in}{1.507772in}}% +\pgfpathlineto{\pgfqpoint{2.724272in}{1.539721in}}% +\pgfpathlineto{\pgfqpoint{2.760113in}{1.567175in}}% +\pgfpathlineto{\pgfqpoint{2.790833in}{1.587085in}}% +\pgfpathlineto{\pgfqpoint{2.821554in}{1.603704in}}% +\pgfpathlineto{\pgfqpoint{2.852275in}{1.617160in}}% +\pgfpathlineto{\pgfqpoint{2.882995in}{1.627654in}}% +\pgfpathlineto{\pgfqpoint{2.918836in}{1.636514in}}% +\pgfpathlineto{\pgfqpoint{2.954677in}{1.642236in}}% +\pgfpathlineto{\pgfqpoint{2.995638in}{1.645737in}}% +\pgfpathlineto{\pgfqpoint{3.051959in}{1.647130in}}% +\pgfpathlineto{\pgfqpoint{3.159482in}{1.648885in}}% +\pgfpathlineto{\pgfqpoint{3.200443in}{1.652741in}}% +\pgfpathlineto{\pgfqpoint{3.236284in}{1.658876in}}% +\pgfpathlineto{\pgfqpoint{3.272125in}{1.668230in}}% +\pgfpathlineto{\pgfqpoint{3.302845in}{1.679201in}}% +\pgfpathlineto{\pgfqpoint{3.333566in}{1.693172in}}% +\pgfpathlineto{\pgfqpoint{3.364287in}{1.710332in}}% +\pgfpathlineto{\pgfqpoint{3.395008in}{1.730798in}}% +\pgfpathlineto{\pgfqpoint{3.425728in}{1.754608in}}% +\pgfpathlineto{\pgfqpoint{3.461569in}{1.786557in}}% +\pgfpathlineto{\pgfqpoint{3.497410in}{1.822800in}}% +\pgfpathlineto{\pgfqpoint{3.538371in}{1.869025in}}% +\pgfpathlineto{\pgfqpoint{3.584452in}{1.926248in}}% +\pgfpathlineto{\pgfqpoint{3.640773in}{2.001687in}}% +\pgfpathlineto{\pgfqpoint{3.799497in}{2.218490in}}% +\pgfpathlineto{\pgfqpoint{3.840458in}{2.267812in}}% +\pgfpathlineto{\pgfqpoint{3.876299in}{2.306295in}}% +\pgfpathlineto{\pgfqpoint{3.907020in}{2.335020in}}% +\pgfpathlineto{\pgfqpoint{3.937740in}{2.359200in}}% +\pgfpathlineto{\pgfqpoint{3.963341in}{2.375494in}}% +\pgfpathlineto{\pgfqpoint{3.988941in}{2.387988in}}% +\pgfpathlineto{\pgfqpoint{4.014542in}{2.396441in}}% +\pgfpathlineto{\pgfqpoint{4.035023in}{2.400153in}}% +\pgfpathlineto{\pgfqpoint{4.055503in}{2.401059in}}% +\pgfpathlineto{\pgfqpoint{4.075983in}{2.399087in}}% +\pgfpathlineto{\pgfqpoint{4.096464in}{2.394186in}}% +\pgfpathlineto{\pgfqpoint{4.116944in}{2.386320in}}% +\pgfpathlineto{\pgfqpoint{4.137425in}{2.375474in}}% +\pgfpathlineto{\pgfqpoint{4.157905in}{2.361653in}}% +\pgfpathlineto{\pgfqpoint{4.183506in}{2.340228in}}% +\pgfpathlineto{\pgfqpoint{4.209107in}{2.314278in}}% +\pgfpathlineto{\pgfqpoint{4.234707in}{2.283939in}}% +\pgfpathlineto{\pgfqpoint{4.265428in}{2.241999in}}% +\pgfpathlineto{\pgfqpoint{4.296149in}{2.194403in}}% +\pgfpathlineto{\pgfqpoint{4.331989in}{2.132385in}}% +\pgfpathlineto{\pgfqpoint{4.367830in}{2.064259in}}% +\pgfpathlineto{\pgfqpoint{4.413911in}{1.969404in}}% +\pgfpathlineto{\pgfqpoint{4.470233in}{1.845837in}}% +\pgfpathlineto{\pgfqpoint{4.613596in}{1.527360in}}% +\pgfpathlineto{\pgfqpoint{4.659677in}{1.434431in}}% +\pgfpathlineto{\pgfqpoint{4.695518in}{1.368451in}}% +\pgfpathlineto{\pgfqpoint{4.731359in}{1.309246in}}% +\pgfpathlineto{\pgfqpoint{4.762080in}{1.264666in}}% +\pgfpathlineto{\pgfqpoint{4.787680in}{1.232297in}}% +\pgfpathlineto{\pgfqpoint{4.813281in}{1.204569in}}% +\pgfpathlineto{\pgfqpoint{4.838881in}{1.181713in}}% +\pgfpathlineto{\pgfqpoint{4.859362in}{1.167058in}}% +\pgfpathlineto{\pgfqpoint{4.879842in}{1.155707in}}% +\pgfpathlineto{\pgfqpoint{4.900323in}{1.147708in}}% +\pgfpathlineto{\pgfqpoint{4.920803in}{1.143088in}}% +\pgfpathlineto{\pgfqpoint{4.941284in}{1.141851in}}% +\pgfpathlineto{\pgfqpoint{4.961764in}{1.143981in}}% +\pgfpathlineto{\pgfqpoint{4.982245in}{1.149438in}}% +\pgfpathlineto{\pgfqpoint{5.002725in}{1.158163in}}% +\pgfpathlineto{\pgfqpoint{5.023206in}{1.170075in}}% +\pgfpathlineto{\pgfqpoint{5.043686in}{1.185070in}}% +\pgfpathlineto{\pgfqpoint{5.069287in}{1.207964in}}% +\pgfpathlineto{\pgfqpoint{5.094887in}{1.235205in}}% +\pgfpathlineto{\pgfqpoint{5.125608in}{1.273176in}}% +\pgfpathlineto{\pgfqpoint{5.156329in}{1.316305in}}% +\pgfpathlineto{\pgfqpoint{5.192170in}{1.372195in}}% +\pgfpathlineto{\pgfqpoint{5.233131in}{1.441903in}}% +\pgfpathlineto{\pgfqpoint{5.289452in}{1.544396in}}% +\pgfpathlineto{\pgfqpoint{5.412335in}{1.770126in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.772815in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.772815in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.750000in}{0.500000in}}{\pgfqpoint{4.650000in}{3.020000in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.890196,0.466667,0.760784}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.736111in}{1.521515in}}% +\pgfpathlineto{\pgfqpoint{0.783746in}{1.606794in}}% +\pgfpathlineto{\pgfqpoint{0.947590in}{1.905224in}}% +\pgfpathlineto{\pgfqpoint{0.988551in}{1.970384in}}% +\pgfpathlineto{\pgfqpoint{1.024392in}{2.021154in}}% +\pgfpathlineto{\pgfqpoint{1.055113in}{2.059124in}}% +\pgfpathlineto{\pgfqpoint{1.080713in}{2.086366in}}% +\pgfpathlineto{\pgfqpoint{1.106314in}{2.109260in}}% +\pgfpathlineto{\pgfqpoint{1.131914in}{2.127526in}}% +\pgfpathlineto{\pgfqpoint{1.152395in}{2.138650in}}% +\pgfpathlineto{\pgfqpoint{1.172875in}{2.146565in}}% +\pgfpathlineto{\pgfqpoint{1.193356in}{2.151195in}}% +\pgfpathlineto{\pgfqpoint{1.213836in}{2.152486in}}% +\pgfpathlineto{\pgfqpoint{1.234317in}{2.150404in}}% +\pgfpathlineto{\pgfqpoint{1.254797in}{2.144938in}}% +\pgfpathlineto{\pgfqpoint{1.275278in}{2.136097in}}% +\pgfpathlineto{\pgfqpoint{1.295758in}{2.123915in}}% +\pgfpathlineto{\pgfqpoint{1.316239in}{2.108444in}}% +\pgfpathlineto{\pgfqpoint{1.341839in}{2.084598in}}% +\pgfpathlineto{\pgfqpoint{1.367440in}{2.055922in}}% +\pgfpathlineto{\pgfqpoint{1.393041in}{2.022656in}}% +\pgfpathlineto{\pgfqpoint{1.423761in}{1.977081in}}% +\pgfpathlineto{\pgfqpoint{1.454482in}{1.925878in}}% +\pgfpathlineto{\pgfqpoint{1.490323in}{1.859898in}}% +\pgfpathlineto{\pgfqpoint{1.531284in}{1.777673in}}% +\pgfpathlineto{\pgfqpoint{1.582485in}{1.667422in}}% +\pgfpathlineto{\pgfqpoint{1.771929in}{1.250513in}}% +\pgfpathlineto{\pgfqpoint{1.812890in}{1.171326in}}% +\pgfpathlineto{\pgfqpoint{1.848731in}{1.108376in}}% +\pgfpathlineto{\pgfqpoint{1.879452in}{1.059883in}}% +\pgfpathlineto{\pgfqpoint{1.910173in}{1.016970in}}% +\pgfpathlineto{\pgfqpoint{1.935773in}{0.985773in}}% +\pgfpathlineto{\pgfqpoint{1.961374in}{0.958934in}}% +\pgfpathlineto{\pgfqpoint{1.986974in}{0.936595in}}% +\pgfpathlineto{\pgfqpoint{2.012575in}{0.918855in}}% +\pgfpathlineto{\pgfqpoint{2.033056in}{0.908010in}}% +\pgfpathlineto{\pgfqpoint{2.053536in}{0.900144in}}% +\pgfpathlineto{\pgfqpoint{2.074017in}{0.895242in}}% +\pgfpathlineto{\pgfqpoint{2.094497in}{0.893271in}}% +\pgfpathlineto{\pgfqpoint{2.114977in}{0.894176in}}% +\pgfpathlineto{\pgfqpoint{2.135458in}{0.897889in}}% +\pgfpathlineto{\pgfqpoint{2.155938in}{0.904320in}}% +\pgfpathlineto{\pgfqpoint{2.181539in}{0.916024in}}% +\pgfpathlineto{\pgfqpoint{2.207140in}{0.931580in}}% +\pgfpathlineto{\pgfqpoint{2.232740in}{0.950717in}}% +\pgfpathlineto{\pgfqpoint{2.263461in}{0.977990in}}% +\pgfpathlineto{\pgfqpoint{2.294182in}{1.009426in}}% +\pgfpathlineto{\pgfqpoint{2.330023in}{1.050554in}}% +\pgfpathlineto{\pgfqpoint{2.370983in}{1.102180in}}% +\pgfpathlineto{\pgfqpoint{2.427305in}{1.178446in}}% +\pgfpathlineto{\pgfqpoint{2.565548in}{1.368082in}}% +\pgfpathlineto{\pgfqpoint{2.611629in}{1.425304in}}% +\pgfpathlineto{\pgfqpoint{2.652590in}{1.471530in}}% +\pgfpathlineto{\pgfqpoint{2.688431in}{1.507772in}}% +\pgfpathlineto{\pgfqpoint{2.724272in}{1.539721in}}% +\pgfpathlineto{\pgfqpoint{2.760113in}{1.567175in}}% +\pgfpathlineto{\pgfqpoint{2.790833in}{1.587085in}}% +\pgfpathlineto{\pgfqpoint{2.821554in}{1.603704in}}% +\pgfpathlineto{\pgfqpoint{2.852275in}{1.617160in}}% +\pgfpathlineto{\pgfqpoint{2.882995in}{1.627654in}}% +\pgfpathlineto{\pgfqpoint{2.918836in}{1.636514in}}% +\pgfpathlineto{\pgfqpoint{2.954677in}{1.642236in}}% +\pgfpathlineto{\pgfqpoint{2.995638in}{1.645737in}}% +\pgfpathlineto{\pgfqpoint{3.051959in}{1.647130in}}% +\pgfpathlineto{\pgfqpoint{3.159482in}{1.648885in}}% +\pgfpathlineto{\pgfqpoint{3.200443in}{1.652741in}}% +\pgfpathlineto{\pgfqpoint{3.236284in}{1.658876in}}% +\pgfpathlineto{\pgfqpoint{3.272125in}{1.668230in}}% +\pgfpathlineto{\pgfqpoint{3.302845in}{1.679201in}}% +\pgfpathlineto{\pgfqpoint{3.333566in}{1.693172in}}% +\pgfpathlineto{\pgfqpoint{3.364287in}{1.710332in}}% +\pgfpathlineto{\pgfqpoint{3.395008in}{1.730798in}}% +\pgfpathlineto{\pgfqpoint{3.425728in}{1.754608in}}% +\pgfpathlineto{\pgfqpoint{3.461569in}{1.786557in}}% +\pgfpathlineto{\pgfqpoint{3.497410in}{1.822800in}}% +\pgfpathlineto{\pgfqpoint{3.538371in}{1.869025in}}% +\pgfpathlineto{\pgfqpoint{3.584452in}{1.926248in}}% +\pgfpathlineto{\pgfqpoint{3.640773in}{2.001687in}}% +\pgfpathlineto{\pgfqpoint{3.799497in}{2.218490in}}% +\pgfpathlineto{\pgfqpoint{3.840458in}{2.267812in}}% +\pgfpathlineto{\pgfqpoint{3.876299in}{2.306295in}}% +\pgfpathlineto{\pgfqpoint{3.907020in}{2.335020in}}% +\pgfpathlineto{\pgfqpoint{3.937740in}{2.359200in}}% +\pgfpathlineto{\pgfqpoint{3.963341in}{2.375494in}}% +\pgfpathlineto{\pgfqpoint{3.988941in}{2.387988in}}% +\pgfpathlineto{\pgfqpoint{4.014542in}{2.396441in}}% +\pgfpathlineto{\pgfqpoint{4.035023in}{2.400153in}}% +\pgfpathlineto{\pgfqpoint{4.055503in}{2.401059in}}% +\pgfpathlineto{\pgfqpoint{4.075983in}{2.399087in}}% +\pgfpathlineto{\pgfqpoint{4.096464in}{2.394186in}}% +\pgfpathlineto{\pgfqpoint{4.116944in}{2.386320in}}% +\pgfpathlineto{\pgfqpoint{4.137425in}{2.375474in}}% +\pgfpathlineto{\pgfqpoint{4.157905in}{2.361653in}}% +\pgfpathlineto{\pgfqpoint{4.183506in}{2.340228in}}% +\pgfpathlineto{\pgfqpoint{4.209107in}{2.314278in}}% +\pgfpathlineto{\pgfqpoint{4.234707in}{2.283939in}}% +\pgfpathlineto{\pgfqpoint{4.265428in}{2.241999in}}% +\pgfpathlineto{\pgfqpoint{4.296149in}{2.194403in}}% +\pgfpathlineto{\pgfqpoint{4.331989in}{2.132385in}}% +\pgfpathlineto{\pgfqpoint{4.367830in}{2.064259in}}% +\pgfpathlineto{\pgfqpoint{4.413911in}{1.969404in}}% +\pgfpathlineto{\pgfqpoint{4.470233in}{1.845837in}}% +\pgfpathlineto{\pgfqpoint{4.613596in}{1.527360in}}% +\pgfpathlineto{\pgfqpoint{4.659677in}{1.434431in}}% +\pgfpathlineto{\pgfqpoint{4.695518in}{1.368451in}}% +\pgfpathlineto{\pgfqpoint{4.731359in}{1.309246in}}% +\pgfpathlineto{\pgfqpoint{4.762080in}{1.264666in}}% +\pgfpathlineto{\pgfqpoint{4.787680in}{1.232297in}}% +\pgfpathlineto{\pgfqpoint{4.813281in}{1.204569in}}% +\pgfpathlineto{\pgfqpoint{4.838881in}{1.181713in}}% +\pgfpathlineto{\pgfqpoint{4.859362in}{1.167058in}}% +\pgfpathlineto{\pgfqpoint{4.879842in}{1.155707in}}% +\pgfpathlineto{\pgfqpoint{4.900323in}{1.147708in}}% +\pgfpathlineto{\pgfqpoint{4.920803in}{1.143088in}}% +\pgfpathlineto{\pgfqpoint{4.941284in}{1.141851in}}% +\pgfpathlineto{\pgfqpoint{4.961764in}{1.143981in}}% +\pgfpathlineto{\pgfqpoint{4.982245in}{1.149438in}}% +\pgfpathlineto{\pgfqpoint{5.002725in}{1.158163in}}% +\pgfpathlineto{\pgfqpoint{5.023206in}{1.170075in}}% +\pgfpathlineto{\pgfqpoint{5.043686in}{1.185070in}}% +\pgfpathlineto{\pgfqpoint{5.069287in}{1.207964in}}% +\pgfpathlineto{\pgfqpoint{5.094887in}{1.235205in}}% +\pgfpathlineto{\pgfqpoint{5.125608in}{1.273176in}}% +\pgfpathlineto{\pgfqpoint{5.156329in}{1.316305in}}% +\pgfpathlineto{\pgfqpoint{5.192170in}{1.372195in}}% +\pgfpathlineto{\pgfqpoint{5.233131in}{1.441903in}}% +\pgfpathlineto{\pgfqpoint{5.289452in}{1.544396in}}% +\pgfpathlineto{\pgfqpoint{5.412335in}{1.770126in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.772815in}}% +\pgfpathlineto{\pgfqpoint{5.413889in}{1.772815in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{0.750000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{5.400000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{0.500000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{0.500000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetmiterjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.750000in}{3.520000in}}% +\pgfpathlineto{\pgfqpoint{5.400000in}{3.520000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetfillopacity{0.800000}% +\pgfsetlinewidth{1.003750pt}% +\definecolor{currentstroke}{rgb}{0.800000,0.800000,0.800000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetstrokeopacity{0.800000}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.515451in}{2.185746in}}% +\pgfpathlineto{\pgfqpoint{5.302778in}{2.185746in}}% +\pgfpathquadraticcurveto{\pgfqpoint{5.330556in}{2.185746in}}{\pgfqpoint{5.330556in}{2.213523in}}% +\pgfpathlineto{\pgfqpoint{5.330556in}{3.422778in}}% +\pgfpathquadraticcurveto{\pgfqpoint{5.330556in}{3.450556in}}{\pgfqpoint{5.302778in}{3.450556in}}% +\pgfpathlineto{\pgfqpoint{4.515451in}{3.450556in}}% +\pgfpathquadraticcurveto{\pgfqpoint{4.487674in}{3.450556in}}{\pgfqpoint{4.487674in}{3.422778in}}% +\pgfpathlineto{\pgfqpoint{4.487674in}{2.213523in}}% +\pgfpathquadraticcurveto{\pgfqpoint{4.487674in}{2.185746in}}{\pgfqpoint{4.515451in}{2.185746in}}% +\pgfpathlineto{\pgfqpoint{4.515451in}{2.185746in}}% +\pgfpathclose% +\pgfusepath{stroke,fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{3.338088in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{3.338088in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{3.338088in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=3.289477in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=-2}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{3.134231in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{3.134231in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{3.134231in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=3.085620in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=-1}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.172549,0.627451,0.172549}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{2.930374in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{2.930374in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{2.930374in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=2.881762in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=0}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.839216,0.152941,0.156863}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{2.726516in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{2.726516in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{2.726516in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=2.677905in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=1}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.580392,0.403922,0.741176}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{2.522659in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{2.522659in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{2.522659in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=2.474048in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=2}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{1.505625pt}% +\definecolor{currentstroke}{rgb}{0.549020,0.337255,0.294118}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{4.543229in}{2.318802in}}% +\pgfpathlineto{\pgfqpoint{4.682118in}{2.318802in}}% +\pgfpathlineto{\pgfqpoint{4.821007in}{2.318802in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{textcolor}% +\pgfsetfillcolor{textcolor}% +\pgftext[x=4.932118in,y=2.270191in,left,base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont n=3}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% diff --git a/buch/papers/fm/Quellen/A2-14.pdf b/buch/papers/fm/Quellen/A2-14.pdf Binary files differnew file mode 100644 index 0000000..7348cca --- /dev/null +++ b/buch/papers/fm/Quellen/A2-14.pdf diff --git a/buch/papers/fm/Quellen/FM_presentation.pdf b/buch/papers/fm/Quellen/FM_presentation.pdf Binary files differnew file mode 100644 index 0000000..496e35e --- /dev/null +++ b/buch/papers/fm/Quellen/FM_presentation.pdf diff --git a/buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf b/buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf Binary files differnew file mode 100644 index 0000000..a6e701c --- /dev/null +++ b/buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf diff --git a/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf b/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf Binary files differnew file mode 100644 index 0000000..2a0bddd --- /dev/null +++ b/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf diff --git a/buch/papers/fm/anim/Makefile b/buch/papers/fm/anim/Makefile new file mode 100644 index 0000000..f4c7850 --- /dev/null +++ b/buch/papers/fm/anim/Makefile @@ -0,0 +1,12 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller +# +all: animation.pdf + +parts.tex: fm.m + octave fm.m + +animation.pdf: animation.tex parts.tex + pdflatex animation.tex diff --git a/buch/papers/fm/anim/animation.tex b/buch/papers/fm/anim/animation.tex new file mode 100644 index 0000000..4a6f428 --- /dev/null +++ b/buch/papers/fm/anim/animation.tex @@ -0,0 +1,85 @@ +% +% animation.tex +% +% (c) 2022 Prof Dr Andreas Müller, +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{nccmath} +\usepackage{mathtools} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\begin{document} + +\def\spektrum#1#2{ +\only<#1>{ + \begin{scope} + \color{red} + \input{#2} + \end{scope} +} +} + +\begin{frame} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\df{0.37} +\def\da{1} + +\draw[->,color=gray] (0,-0.1) -- (0,6.3) [right] coordinate[label={right:$a$}]; + +\foreach \a in {1,...,5}{ + \draw[color=gray!50] (-6,{(6-\a)*\da}) -- (6,{(6-\a)*\da}); +} +\draw[color=gray!50] (-6,{6*\da}) -- (6,{6*\da}); +\foreach \f in {-15,-10,-5,5,10,15}{ + \draw[color=gray!50] ({\f*\df},0) -- ({\f*\df},{6*\da}); +} + +\input{parts.tex} + +\draw[->] (-6.1,0) -- (6.9,0) coordinate[label={$f$}]; +\foreach \f in {-16,...,16}{ + \draw ({\f*\df},-0.05) -- ({\f*\df},0.05); +} +\foreach \f in {-15,-10,-5,5,10,15}{ + \node at ({\f*\df},-0.1) [below] {$\f f_m$}; + \draw ({\f*\df},-0.1) -- ({\f*\df},0.1); +} +\node at (0,-0.1) [below] {$0$}; + +\foreach \a in {1,...,5}{ + \node at (6,{(6-\a)*\da}) [right] {$-\a$}; +} +\node at (6,{6*\da}) [right] {$\phantom{-}0$}; + +\end{tikzpicture} +\end{center} +\end{frame} + +\end{document} diff --git a/buch/papers/fm/anim/fm.m b/buch/papers/fm/anim/fm.m new file mode 100644 index 0000000..9062818 --- /dev/null +++ b/buch/papers/fm/anim/fm.m @@ -0,0 +1,98 @@ +# +# fm.m -- animation frequenzspektrum +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global fc; +fc = 1e6; +global width; +width = 16; +global fm; +fm = 1000; +global gamma; +gamma = 2; +global resolution; +resolution = 300; + +function retval = spektrum(beta, fm) + global width; + global fc; + retval = zeros(2 * width + 1, 2); + center = width + 1; + for k = (0:width) + retval(center - k, 1) = fc - k * fm; + retval(center + k, 1) = fc + k * fm; + a = besselj(k, beta); + retval(center - k, 2) = a; + retval(center + k, 2) = a; + endfor +endfunction + +function drawspectrum(fn, spectrum, foffset, fscale, beta) + n = size(spectrum)(1,1); + for i = (1:n) + f = (spectrum(i, 1) - foffset)/fscale; + a = log10(spectrum(i, 2)) + 6; + if (a < 0) + a = 0; + end + fprintf(fn, "\\draw[line width=3.5pt] "); + fprintf(fn, "({%.2f*\\df},0) -- ({%.2f*\\df},{%.5f*\\da});\n", + f, f, abs(a)); + fprintf(fn, "\\node at ({-15*\\df},5.5) [right] {$\\beta = %.3f$};", beta); + endfor +endfunction + +function drawhull(fn, beta) + global resolution; + fprintf(fn, "\\begin{scope}\n"); + fprintf(fn, "\\clip ({-16.5*\\df},0) rectangle ({16.5*\\df},{6*\\da});\n"); + p = zeros(resolution, 2); + for k = (1:resolution) + nu = 16.5 * (k - 1) / resolution; + p(k,1) = nu; + y = log10(abs(besselj(nu, beta))) + 6; + p(k,2) = y; + end + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + -p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\end{scope}\n"); +endfunction + +function animation(betamin, betamax, steps) + global fm; + global fc; + global gamma; + fa = fopen("parts.tex", "w"); + for k = (1:steps) + % add entry to parts.tex + fprintf(fa, "\\spektrum{%d}{texfiles/a%04d.tex}\n", k, k); + % compute beta + x = (k - 1) / (steps - 1); + beta = betamin + (betamax - betamin) * (x ^ gamma); + % create a new file + name = sprintf("texfiles/a%04d.tex", k); + fn = fopen(name, "w"); + % write the hull + drawhull(fn, beta); + % compute and write the spectrum + spectrum = spektrum(beta, fm); + drawspectrum(fn, spectrum, fc, fm, beta); + fclose(fn); + endfor + fclose(fa); +endfunction + +animation(0.001,10.1,200) diff --git a/buch/papers/fm/main.tex b/buch/papers/fm/main.tex index 1e75235..731f56f 100644 --- a/buch/papers/fm/main.tex +++ b/buch/papers/fm/main.tex @@ -1,36 +1,42 @@ +% !TeX root = ../../buch.tex % % main.tex -- Paper zum Thema <fm> % % (c) 2020 Hochschule Rapperswil -% -\chapter{Thema\label{chapter:fm}} -\lhead{Thema} +% + +\chapter{FM Bessel\label{chapter:fm}} +\lhead{FM} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/fm/teil0.tex} -\input{papers/fm/teil1.tex} -\input{papers/fm/teil2.tex} -\input{papers/fm/teil3.tex} + +\chapterauthor{Joshua Bär} + +Die Frequenzmodulation ist eine Modulation die man auch schon im alten Radio findet. +Falls du dich an die Zeit erinnerst, konnte man zwischen \textit{FM-AM} Umschalten, +dies bedeutete so viel wie: \textit{F}requenz-\textit{M}odulation und \textit{A}mplituden-\textit{M}odulation. +Durch die Modulation wird ein Nachrichtensignal \(m(t)\) auf ein Trägersignal (z.B. ein Sinus- oder Rechtecksignal) abgebildet (kombiniert). +Durch dieses Auftragen vom Nachrichtensignal \(m(t)\) kann das modulierte Signal in einem gewünschten Frequenzbereich übertragen werden. +Der ursprünglich Frequenzbereich des Nachrichtensignal \(m(t)\) erstreckt sich typischerweise von 0 Hz bis zur Bandbreite \(B_m\). +\newline +Beim Empfänger wird dann durch Demodulation das ursprüngliche Nachrichtensignal \(m(t)\) so originalgetreu wie möglich zurückgewonnen. +\newline +Beim Trägersignal \(x_c(t)\) handelt es sich um ein informationsloses Hilfssignal. +Durch die Modulation mit dem Nachrichtensignal \(m(t)\) wird es zum modulierten zu übertragenden Signal. +Für alle Erklärungen wird ein sinusförmiges Trägersignal benutzt, jedoch kann auch ein Rechtecksignal, +welches Digital einfach umzusetzten ist, +genauso als Trägersignal genutzt werden kann. +Zuerst wird erklärt was \textit{FM-AM} ist, danach wie sich diese im Frequenzspektrum verhalten. +Erst dann erklär ich dir wie die Besselfunktion mit der Frequenzmodulation( acro?) zusammenhängt. +Nun zur Modulation im nächsten Abschnitt.\cite{fm:NAT} + + +\input{papers/fm/00_modulation.tex} +\input{papers/fm/01_AM.tex} +\input{papers/fm/02_FM.tex} +\input{papers/fm/03_bessel.tex} +\input{papers/fm/04_fazit.tex} \printbibliography[heading=subbibliography] \end{refsection} + + diff --git a/buch/papers/fm/packages.tex b/buch/papers/fm/packages.tex index 4cba2b6..7bbbe35 100644 --- a/buch/papers/fm/packages.tex +++ b/buch/papers/fm/packages.tex @@ -7,4 +7,5 @@ % if your paper needs special packages, add package commands as in the % following example %\usepackage{packagename} - +\usepackage{xcolor} +\usepackage{pgf} diff --git a/buch/papers/fm/references.bib b/buch/papers/fm/references.bib index 76eb265..21b910b 100644 --- a/buch/papers/fm/references.bib +++ b/buch/papers/fm/references.bib @@ -23,6 +23,17 @@ volume = {2} } +@book{fm:NAT, + title = {Nachrichtentechnik 1 + 2}, + author = {Thomas Kneubühler}, + publisher = {None}, + year = {2021}, + isbn = {}, + inseries = {Script for students}, + volume = {} +} + + @article{fm:mendezmueller, author = { Tabea Méndez and Andreas Müller }, title = { Noncommutative harmonic analysis and image registration }, diff --git a/buch/papers/fm/standalone.tex b/buch/papers/fm/standalone.tex new file mode 100644 index 0000000..c161ed5 --- /dev/null +++ b/buch/papers/fm/standalone.tex @@ -0,0 +1,31 @@ +\documentclass{book} + +\def\IncludeBookCover{0} +\input{common/packages.tex} + +% additional packages used by the individual papers, add a line for +% each paper +\input{papers/common/addpackages.tex} + +% workaround for biblatex bug +\makeatletter +\def\blx@maxline{77} +\makeatother +\addbibresource{chapters/references.bib} + +% Bibresources for each article +\input{papers/common/addbibresources.tex} + +% make sure the last index starts on an odd page +\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} +\makeindex + +%\pgfplotsset{compat=1.12} +\setlength{\headheight}{15pt} % fix headheight warning +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + \input{common/macros.tex} + \def\chapterauthor#1{{\large #1}\bigskip\bigskip} + \input{papers/fm/main.tex} +\end{document} diff --git a/buch/papers/fm/teil0.tex b/buch/papers/fm/teil0.tex deleted file mode 100644 index 55697df..0000000 --- a/buch/papers/fm/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{fm:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{fm:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/fm/teil1.tex b/buch/papers/fm/teil1.tex deleted file mode 100644 index 6f9edf1..0000000 --- a/buch/papers/fm/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{fm:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{fm:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{fm:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{fm:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{fm:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/fm/teil2.tex b/buch/papers/fm/teil2.tex deleted file mode 100644 index 6ab6fa0..0000000 --- a/buch/papers/fm/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{fm:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{fm:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/fm/teil3.tex b/buch/papers/fm/teil3.tex deleted file mode 100644 index 3bcfc4d..0000000 --- a/buch/papers/fm/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{fm:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{fm:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/fresnel/Makefile b/buch/papers/fresnel/Makefile index c8aa073..ed74861 100644 --- a/buch/papers/fresnel/Makefile +++ b/buch/papers/fresnel/Makefile @@ -1,9 +1,8 @@ # # Makefile -- make file for the paper fresnel # -# (c) 2020 Prof Dr Andreas Mueller +# (c) 2022 Prof Dr Andreas Mueller # - images: @echo "no images to be created in fresnel" diff --git a/buch/papers/fresnel/images/Makefile b/buch/papers/fresnel/images/Makefile new file mode 100644 index 0000000..eb7dc57 --- /dev/null +++ b/buch/papers/fresnel/images/Makefile @@ -0,0 +1,38 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: schale.pdf \ + fresnelgraph.pdf \ + eulerspirale.pdf \ + pfad.pdf \ + apfel.pdf \ + kruemmung.pdf + +schale.png: schale.pov + povray +A0.1 -W1920 -H1080 -Oschale.png schale.pov + +schale.jpg: schale.png Makefile + convert -extract 1240x1080+340 schale.png -density 300 -units PixelsPerInch schale.jpg + +schale.pdf: schale.tex schale.jpg + pdflatex schale.tex + +eulerpath.tex: eulerspirale.m + octave eulerspirale.m + +fresnelgraph.pdf: fresnelgraph.tex eulerpath.tex + pdflatex fresnelgraph.tex + +eulerspirale.pdf: eulerspirale.tex eulerpath.tex + pdflatex eulerspirale.tex + +pfad.pdf: pfad.tex + pdflatex pfad.tex + +apfel.pdf: apfel.tex apfel.jpg eulerpath.tex + pdflatex apfel.tex + +kruemmung.pdf: kruemmung.tex + pdflatex kruemmung.tex diff --git a/buch/papers/fresnel/images/apfel.jpg b/buch/papers/fresnel/images/apfel.jpg Binary files differnew file mode 100644 index 0000000..76e48e7 --- /dev/null +++ b/buch/papers/fresnel/images/apfel.jpg diff --git a/buch/papers/fresnel/images/apfel.pdf b/buch/papers/fresnel/images/apfel.pdf Binary files differnew file mode 100644 index 0000000..69e5092 --- /dev/null +++ b/buch/papers/fresnel/images/apfel.pdf diff --git a/buch/papers/fresnel/images/apfel.tex b/buch/papers/fresnel/images/apfel.tex new file mode 100644 index 0000000..754886b --- /dev/null +++ b/buch/papers/fresnel/images/apfel.tex @@ -0,0 +1,49 @@ +% +% apfel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{7} +\def\hoehe{4} + +\input{eulerpath.tex} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\begin{scope} +\clip(-0.6,-0.6) rectangle (7,6); +\node at (3.1,2.2) [rotate=-3] {\includegraphics[width=9.4cm]{apfel.jpg}}; +\end{scope} + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\draw[color=gray!50] (0,0) rectangle (4,4); +\draw[->] (-0.5,0) -- (7.5,0) coordinate[label={$C(t)$}]; +\draw[->] (0,-0.5) -- (0,6.0) coordinate[label={left:$S(t)$}]; +\begin{scope}[scale=8] +\draw[color=red,opacity=0.5,line width=1.4pt] \fresnela; +\end{scope} + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/fresnel/images/eulerspirale.m b/buch/papers/fresnel/images/eulerspirale.m new file mode 100644 index 0000000..84e3696 --- /dev/null +++ b/buch/papers/fresnel/images/eulerspirale.m @@ -0,0 +1,61 @@ +# +# eulerspirale.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +# +global n; +n = 1000; +global tmax; +tmax = 10; +global N; +N = round(n*5/tmax); + +function retval = f(x, t) + x = pi * t^2 / 2; + retval = [ cos(x); sin(x) ]; +endfunction + +x0 = [ 0; 0 ]; +t = tmax * (0:n) / n; + +c = lsode(@f, x0, t); + +fn = fopen("eulerpath.tex", "w"); + +fprintf(fn, "\\def\\fresnela{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\fresnelb{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fclose(fn); diff --git a/buch/papers/fresnel/images/eulerspirale.pdf b/buch/papers/fresnel/images/eulerspirale.pdf Binary files differnew file mode 100644 index 0000000..db74e4b --- /dev/null +++ b/buch/papers/fresnel/images/eulerspirale.pdf diff --git a/buch/papers/fresnel/images/eulerspirale.tex b/buch/papers/fresnel/images/eulerspirale.tex new file mode 100644 index 0000000..38ef756 --- /dev/null +++ b/buch/papers/fresnel/images/eulerspirale.tex @@ -0,0 +1,41 @@ +% +% eulerspirale.tex -- Darstellung der Eulerspirale +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} + +\def\s{8} + +\begin{scope}[scale=\s] +\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5); +\draw[color=darkgreen,line width=1.4pt] \fresnela; +\draw[color=darkgreen,line width=1.4pt] \fresnelb; +\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}]; +\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}]; +\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s}); +\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s}); +\node at (-0.5,0) [above left] {$\frac12$}; +\node at (0.5,0) [below right] {$\frac12$}; +\node at (0,-0.5) [below right] {$\frac12$}; +\node at (0,0.5) [above left] {$\frac12$}; +\end{scope} + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];; +\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/fresnelgraph.pdf b/buch/papers/fresnel/images/fresnelgraph.pdf Binary files differnew file mode 100644 index 0000000..c658901 --- /dev/null +++ b/buch/papers/fresnel/images/fresnelgraph.pdf diff --git a/buch/papers/fresnel/images/fresnelgraph.tex b/buch/papers/fresnel/images/fresnelgraph.tex new file mode 100644 index 0000000..20df951 --- /dev/null +++ b/buch/papers/fresnel/images/fresnelgraph.tex @@ -0,0 +1,46 @@ +% +% fresnelgraph.tex -- Graphs of the fresnel functions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} +\def\dx{1.3} +\def\dy{2.6} + +\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy}); +\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy}); + +\draw[color=blue,line width=1.4pt] \Splotright; +\draw[color=blue,line width=1.4pt] \Splotleft; + +\draw[color=red,line width=1.4pt] \Cplotright; +\draw[color=red,line width=1.4pt] \Cplotleft; + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}]; +\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}]; + +\foreach \x in {1,2,3,4,5}{ + \draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05); + \draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05); + \node at ({\x*\dx},-0.05) [below] {$\x$}; + \node at ({-\x*\dx},0.05) [above] {$-\x$}; +} +\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy}); +\node at (-0.05,{0.5*\dy}) [left] {$\frac12$}; +\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy}); +\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/kruemmung.pdf b/buch/papers/fresnel/images/kruemmung.pdf Binary files differnew file mode 100644 index 0000000..1180116 --- /dev/null +++ b/buch/papers/fresnel/images/kruemmung.pdf diff --git a/buch/papers/fresnel/images/kruemmung.tex b/buch/papers/fresnel/images/kruemmung.tex new file mode 100644 index 0000000..af0a1a9 --- /dev/null +++ b/buch/papers/fresnel/images/kruemmung.tex @@ -0,0 +1,51 @@ +% +% kruemmung.tex -- Krümmung einer ebenen Kurve +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope} +\clip (-1,-1) rectangle (4,4); + +\def\r{3} +\def\winkel{30} + +\fill[color=blue!20] (0,0) -- (0:{0.6*\r}) arc (0:\winkel:{0.6*\r}) -- cycle; +\fill[color=blue!20] (\winkel:\r) + -- ($(\winkel:\r)+(0,{0.6*\r})$) arc (90:{90+\winkel}:{0.6*\r}) -- cycle; +\node[color=blue] at ({0.5*\winkel}:{0.45*\r}) {$\Delta\varphi$}; + +\node[color=blue] at ($(\winkel:\r)+({90+0.5*\winkel}:{0.45*\r})$) + {$\Delta\varphi$}; + +\draw[line width=0.3pt] (0,0) circle[radius=\r]; + +\draw[->] (0,0) -- (0:\r); +\draw[->] (0,0) -- (\winkel:\r); + +\draw[->] (0:\r) -- ($(0:\r)+(90:0.7*\r)$); +\draw[->] (\winkel:\r) -- ($(\winkel:\r)+({90+\winkel}:0.7*\r)$); +\draw[->,color=gray] (\winkel:\r) -- ($(\winkel:\r)+(0,0.7*\r)$); + +\draw[color=red,line width=1.4pt] (0:\r) arc (0:\winkel:\r); +\node[color=red] at ({0.5*\winkel}:\r) [left] {$\Delta s$}; +\fill[color=red] (0:\r) circle[radius=0.05]; +\fill[color=red] (\winkel:\r) circle[radius=0.05]; + +\node at (\winkel:{0.5*\r}) [above] {$r$}; +\node at (0:{0.5*\r}) [below] {$r$}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/pfad.pdf b/buch/papers/fresnel/images/pfad.pdf Binary files differnew file mode 100644 index 0000000..df3c7af --- /dev/null +++ b/buch/papers/fresnel/images/pfad.pdf diff --git a/buch/papers/fresnel/images/pfad.tex b/buch/papers/fresnel/images/pfad.tex new file mode 100644 index 0000000..680cd78 --- /dev/null +++ b/buch/papers/fresnel/images/pfad.tex @@ -0,0 +1,37 @@ +% +% pfad.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\fill[color=gray!40] (0,0) -- (2,0) arc (0:45:2) -- cycle; +\node at (22.5:1.4) {$\displaystyle\frac{\pi}4$}; + +\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}]; +\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}]; + +\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0); +\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7); +\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0); + +\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$}; +\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$}; +\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$}; + +\node at (7,0) [below] {$R$}; +\node at (45:7) [above] {$Re^{i\pi/4}$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/schale.pdf b/buch/papers/fresnel/images/schale.pdf Binary files differnew file mode 100644 index 0000000..9c21951 --- /dev/null +++ b/buch/papers/fresnel/images/schale.pdf diff --git a/buch/papers/fresnel/images/schale.pov b/buch/papers/fresnel/images/schale.pov new file mode 100644 index 0000000..085a6a4 --- /dev/null +++ b/buch/papers/fresnel/images/schale.pov @@ -0,0 +1,191 @@ +// +// schale.pov -- +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +#declare O = <0,0,0>; + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.036; + +camera { + location <40, 20, -20> + look_at <0, 0.5, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +sphere { + <0, 0, 0>, 1 + pigment { + color rgb<0.8,0.8,0.8> + } + finish { + specular 0.95 + metallic + } +} + +#declare stripcolor = rgb<0.2,0.2,0.8>; + +#declare R = 1.002; + +#macro punkt(phi,theta) +R * < cos(phi) * cos(theta), sin(theta), sin(phi) * cos(theta) > +#end + +#declare N = 24; +#declare thetaphi = 0.01; +#declare thetawidth = pi * 0.008; +#declare theta = function(phi) { phi * thetaphi } + +#declare axisdiameter = 0.007; + +cylinder { + < 0, -2, 0>, < 0, 2, 0>, axisdiameter + pigment { + color White + } + finish { + specular 0.95 + metallic + } +} + +#declare curvaturecircle = 0.008; +#declare curvaturecirclecolor = rgb<0.4,0.8,0.4>; + +#declare phit = 12.8 * 2 * pi; +#declare P = punkt(phit, theta(phit)); +#declare Q = <0, R / sin(theta(phit)), 0>; + +#declare e1 = vnormalize(P - Q) / tan(theta(phit)); +#declare e2 = vnormalize(vcross(e1, <0,1,0>)) / tan(theta(phit)); +#declare psimin = -0.1 * pi; +#declare psimax = 0.1 * pi; +#declare psistep = (psimax - psimin) / 30; + +union { + #declare psi = psimin; + #declare K = Q + cos(psi) * e1 + sin(psi) * e2; + #while (psi < psimax - psistep/2) + sphere { K, curvaturecircle } + #declare psi = psi + psistep; + #declare K2 = Q + cos(psi) * e1 + sin(psi) * e2; + cylinder { K, K2, curvaturecircle } + #declare K = K2; + #end + sphere { K, curvaturecircle } + pigment { + color curvaturecirclecolor + } + finish { + specular 0.95 + metallic + } +} + +object { + mesh { + #declare psi = psimin; + #declare K = Q + cos(psi) * e1 + sin(psi) * e2; + #while (psi < psimax - psistep/2) + #declare psi = psi + psistep; + #declare K2 = Q + cos(psi) * e1 + sin(psi) * e2; + triangle { K, K2, Q } + #declare K = K2; + #end + } + pigment { + color rgbt<0.4,0.8,0.4,0.5> + } + finish { + specular 0.95 + metallic + } +} + +union { + sphere { P, 0.02 } + sphere { Q, 0.02 } + cylinder { P, Q, 0.01 } + pigment { + color Red + } + finish { + specular 0.95 + metallic + } +} + +#declare phisteps = 300; +#declare phistep = 2 * pi / phisteps; +#declare phimin = 0; +#declare phimax = N * 2 * pi; + +object { + mesh { + #declare phi = phimin; + #declare Poben = punkt(phi, theta(phi) + thetawidth); + #declare Punten = punkt(phi, theta(phi) - thetawidth); + triangle { O, Punten, Poben } + #while (phi < phimax - phistep/2) + #declare phi = phi + phistep; + #declare Poben2 = punkt(phi, theta(phi) + thetawidth); + #declare Punten2 = punkt(phi, theta(phi) - thetawidth); + triangle { O, Punten, Punten2 } + triangle { O, Poben, Poben2 } + triangle { Punten, Punten2, Poben } + triangle { Punten2, Poben2, Poben } + #declare Poben = Poben2; + #declare Punten = Punten2; + #end + triangle { O, Punten, Poben } + } + pigment { + color stripcolor + } + finish { + specular 0.8 + metallic + } +} + +union { + #declare phi = phimin; + #declare P = punkt(phi, theta(phi)); + #while (phi < phimax - phistep/2) + sphere { P, 0.003 } + #declare phi = phi + phistep; + #declare P2 = punkt(phi, theta(phi)); + cylinder { P, P2, 0.003 } + #declare P = P2; + #end + sphere { P, 0.003 } + pigment { + color stripcolor + } + finish { + specular 0.8 + metallic + } +} diff --git a/buch/papers/fresnel/images/schale.tex b/buch/papers/fresnel/images/schale.tex new file mode 100644 index 0000000..577ede4 --- /dev/null +++ b/buch/papers/fresnel/images/schale.tex @@ -0,0 +1,77 @@ +% +% schlange.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math,calc} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} +\def\a{47} +\def\r{3.3} +\def\skala{0.95} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-7.4cm,yshift=-1.2cm] + \clip (-3.6,-2.2) rectangle (3.6,5.1); + + \fill[color=blue!20] (0,0) + -- ({180-\a}:{0.4*\r}) arc ({180-\a}:180:{0.4*\r}) + -- cycle; + \node[color=blue] at ({180-\a/2}:{0.3*\r}) {$\vartheta$}; + + \fill[color=blue!20] (0,{\r/sin(\a)}) + -- ($(0,{\r/sin(\a)})+({270-\a}:{0.3*\r})$) + arc ({270-\a}:270:{0.3*\r}) + -- cycle; + \node[color=blue] at ($(0,{\r/sin(\a)})+({270-\a/2}:{0.2*\r})$) + {$\vartheta$}; + + + \draw (0,0) circle[radius=\r]; + \draw[->] (0,-3.0) -- (0,5); + \draw ({-\r-0.2},0) -- ({\r+0.2},0); + \fill (0,0) circle[radius=0.06]; + + \draw (0,0) -- ({180-\a}:\r); + \node at ({180-\a+3}:{0.65*\r}) [above right] {$1$}; + + \draw[color=red,line width=1.4pt] + ({180-\a}:\r) -- (0,{\r/cos(90-\a)}); + \fill[color=red] ({180-\a}:\r) circle[radius=0.08]; + \fill[color=red] (0,{\r/cos(90-\a)}) circle[radius=0.08]; + \node[color=red] at (-1.0,3.7) [left] {$r=\cot\vartheta$}; + \node[color=red] at ({180-\a}:\r) [above left] {$P$}; + \node[color=red] at (0,{\r/sin(\a)}) [right] {$Q$}; +\end{scope} + +% Povray Bild +\node at (0,0) {\includegraphics[width=7.6cm]{schale.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node[color=red] at (-1.4,1.4) {$r$}; +\node[color=red] at (-2.2,-0.2) {$P$}; +\node[color=red] at (0,3.3) [right] {$Q$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/fresnel/main.tex b/buch/papers/fresnel/main.tex index bbaf7e6..2050fd4 100644 --- a/buch/papers/fresnel/main.tex +++ b/buch/papers/fresnel/main.tex @@ -3,29 +3,16 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:fresnel}} -\lhead{Thema} +\chapter{Fresnel-Integrale\label{chapter:fresnel}} +\lhead{Fresnel-Integrale} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Andreas Müller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +{\parindent0pt Die} Fresnel-Integrale tauchen in der Untersuchung der Beugung +in paraxialer Näherung auf, auch bekannt als die Fresnel-Approximation. +In diesem Kapitel betrachen wir jedoch nur die geometrische +Anwendung der Fresnel-Integrale als Parametrisierung der Euler-Spirale +und zeigen, dass letztere eine Klothoide ist. \input{papers/fresnel/teil0.tex} \input{papers/fresnel/teil1.tex} diff --git a/buch/papers/fresnel/references.bib b/buch/papers/fresnel/references.bib index 84cd3bc..cf8fb21 100644 --- a/buch/papers/fresnel/references.bib +++ b/buch/papers/fresnel/references.bib @@ -33,3 +33,20 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@online{fresnel:fresnelC, + url = { https://functions.wolfram.com/GammaBetaErf/FresnelC/introductions/FresnelIntegrals/ShowAll.html }, + title = { FresnelC }, + date = { 2022-05-13 } +} + +@online{fresnel:wikipedia, + url = { https://en.wikipedia.org/wiki/Fresnel_integral }, + title = { Fresnel Integral }, + date = { 2022-05-13 } +} + +@online{fresnel:schale, + url = { https://www.youtube.com/watch?v=D3tdW9l1690 }, + title = { A Strange Map Projection (Euler Spiral) - Numberphile }, + date = { 2022-05-14 } +} diff --git a/buch/papers/fresnel/teil0.tex b/buch/papers/fresnel/teil0.tex index 5e9fdaf..85b8bf7 100644 --- a/buch/papers/fresnel/teil0.tex +++ b/buch/papers/fresnel/teil0.tex @@ -1,22 +1,101 @@ % -% einleitung.tex -- Beispiel-File für die Einleitung +% teil0.tex -- Definition % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{fresnel:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{fresnel:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Definition\label{fresnel:section:teil0}} +\rhead{Definition} +Die Funktion $e^{x^2}$ hat bekanntermassen keine elementare Stammfunktion, +weshalb die Fehlerfunktion als Stammfunktion definiert wurde. +Die Funktionen $\cos x^2$ und $\sin x^2$ sind eng mit $e^{x^2}$ +verwandt, es ist daher nicht überraschend, dass sie ebenfalls +keine elementare Stammfunktionen haben. +Dies rechtfertigt die Definition der Fresnel-Integrale als neue spezielle +Funktionen. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\begin{definition} +Die Funktionen +\begin{align*} +C(x) &= \int_0^x \cos\biggl(\frac{\pi}2 t^2\biggr)\,dt +\\ +S(x) &= \int_0^x \sin\biggl(\frac{\pi}2 t^2\biggr)\,dt +\end{align*} +heissen die Fresnel-Integrale. +\end{definition} +Der Faktor $\frac{\pi}2$ ist einigermassen willkürlich, man könnte +daher noch allgemeiner die Funktionen +\begin{align*} +C_a(x) &= \int_0^x \cos(at^2)\,dt +\\ +S_a(x) &= \int_0^x \sin(at^2)\,dt +\end{align*} +definieren, so dass die Funktionen $C(x)$ und $S(x)$ der Fall +$a=\frac{\pi}2$ werden, also +\[ +\begin{aligned} +C(x) &= C_{\frac{\pi}2}(x), +& +S(x) &= S_{\frac{\pi}2}(x). +\end{aligned} +\] +Durch eine Substitution $t=bs$ erhält man +\begin{align*} +C_a(x) +&= +\int_0^x \cos(at^2)\,dt += +b +\int_0^{\frac{x}b} \cos(ab^2s^2)\,ds += +b +C_{ab^2}\biggl(\frac{x}b\biggr) +\\ +S_a(x) +&= +\int_0^x \sin(at^2)\,dt += +b +\int_0^{\frac{x}b} \sin(ab^2s^2)\,ds += +b +S_{ab^2}\biggl(\frac{x}b\biggr). +\end{align*} +Indem man $ab^2=\frac{\pi}2$ setzt, also +\[ +b += +\sqrt{\frac{\pi}{2a}} +, +\] +kann man die Funktionen $C_a(x)$ und $S_a(x)$ durch $C(x)$ und $S(x)$ +ausdrücken: +\begin{align} +C_a(x) +&= +\sqrt{\frac{\pi}{2a}} +C\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr) +&&\text{und}& +S_a(x) +&= +\sqrt{\frac{\pi}{2a}} +S\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr). +\label{fresnel:equation:arg} +\end{align} +Im Folgenden werden wir meistens nur den Fall $a=1$, also die Funktionen +$C_1(x)$ und $S_1(x)$ betrachten, da in diesem Fall die Formeln einfacher +werden. +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/fresnelgraph.pdf} +\caption{Graph der Funktionen $C(x)$ ({\color{red}rot}) +und $S(x)$ ({\color{blue}blau}) +\label{fresnel:figure:plot}} +\end{figure} +Die Abbildung~\ref{fresnel:figure:plot} zeigt die Graphen der +Funktion $C(x)$ und $S(x)$. diff --git a/buch/papers/fresnel/teil1.tex b/buch/papers/fresnel/teil1.tex index a2df138..c716cd7 100644 --- a/buch/papers/fresnel/teil1.tex +++ b/buch/papers/fresnel/teil1.tex @@ -1,55 +1,205 @@ % -% teil1.tex -- Beispiel-File für das Paper +% teil1.tex -- Euler-Spirale % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 -\label{fresnel:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{fresnel:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +\section{Euler-Spirale +\label{fresnel:section:eulerspirale}} +\rhead{Euler-Spirale} +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/eulerspirale.pdf} +\caption{Die Eulerspirale ist die Kurve mit der Parameterdarstellung +$x\mapsto (C(x),S(x))$, sie ist rot dargestellt. +Sie windet sich unendlich oft um die beiden Punkte $(\pm\frac12,\pm\frac12)$. +\label{fresnel:figure:eulerspirale}} +\end{figure} +Ein besseres Verständnis für die beiden Funktionen $C(x)$ und $S(x)$ +als die Darstellung~\ref{fresnel:figure:plot} ermöglicht die +Abbildung~\ref{fresnel:figure:eulerspirale}, die die beiden Funktionen +als die $x$- und $y$-Koordinaten der Parameterdarstellung einer Kurve +zeigt. +Sie heisst die {\em Euler-Spirale}. +Die Spirale scheint sich für $x\to\pm\infty$ um die Punkte +$(\pm\frac12,\pm\frac12)$ zu winden. -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/pfad.pdf} +\caption{Pfad zur Berechnung der Grenzwerte $C_1(\infty)$ und +$S_1(\infty)$ mit Hilfe des Cauchy-Integralsatzes +\label{fresnel:figure:pfad}} +\end{figure} -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. -Et harum quidem rerum facilis est et expedita distinctio -\ref{fresnel:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{fresnel:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\begin{satz} +Die Grenzwerte der Fresnel-Integrale für $x\to\pm\infty$ sind +\[ +\lim_{x\to\pm\infty} C(x) += +\lim_{x\to\pm\infty} S(x) += +\frac12. +\] +\end{satz} +\begin{proof}[Beweis] +Die komplexe Funktion +\( +f(z) = e^{-z^2} +\) +ist eine ganze Funktion, das Integral über einen geschlossenen +Pfad in der komplexen Ebene verschwindet daher. +Wir verwenden den Pfad in Abbildung~\ref{fresnel:figure:pfad} +bestehend aus den drei Segmenten $\gamma_1$ entlang der reellen +Achse von $0$ bis $R$, dem Kreisbogen $\gamma_2$ um $0$ mit Radius $R$ +und $\gamma_3$ mit der Parametrisierung $t\mapsto te^{i\pi/4}$. + +Das Teilintegral über $\gamma_1$ ist +\[ +\lim_{R\to\infty} +\int_{\gamma_1} e^{-z^2}\,dz += +\int_0^\infty e^{-t^2}\,dt += +\frac{\sqrt{\pi}}2. +\] +Das Integral über $\gamma_3$ ist +\begin{align*} +\lim_{R\to\infty} +\int_{\gamma_3} +e^{-z^2}\,dz +&= +-\int_0^\infty \exp(-t^2 e^{i\pi/2}) e^{i\pi/4}\,dt += +- +\int_0^\infty e^{-it^2}\,dt\, +e^{i\pi/4} +\\ +&= +-e^{i\pi/4}\int_0^\infty \cos t^2 - i \sin t^2\,dt +\\ +&= +-\frac{1}{\sqrt{2}}(1+i) +\bigl( +C_1(\infty) +-i +S_1(\infty) +\bigr) +\\ +&= +-\frac{1}{\sqrt{2}} +\bigl( +C_1(\infty)+S_1(\infty) ++ +i(C_1(\infty)-S_1(\infty)) +\bigr), +\end{align*} +wobei wir +\[ +C_1(\infty) = \lim_{R\to\infty} C_1(R) +\qquad\text{und}\qquad +S_1(\infty) = \lim_{R\to\infty} S_1(R) +\] +abgekürzt haben. +Das Integral über das Segment $\gamma_2$ lässt sich +mit der Parametrisierung +\( +\gamma_2(t) += +Re^{it} += +R(\cos t + i\sin t) +\) +wie folgt +abschätzen: +\begin{align*} +\biggl|\int_{\gamma_2} e^{-z^2} \,dz\biggr| +&= +\biggl| +\int_0^{\frac{\pi}4} +\exp(-R^2(\cos 2t + i\sin 2t)) iR e^{it}\,dt +\biggr| +\\ +&\le +R +\int_0^{\frac{\pi}4} +e^{-R^2\cos 2t} +\,dt +\le +R +\int_0^{\frac{\pi}4} +e^{-R^2(1-\frac{4}{\pi}t)} +\,dt. +\intertext{Dabei haben wir $\cos 2t\ge 1-\frac{4}\pi t$ verwendet. +Mit dieser Vereinfachung kann das Integral ausgewertet werden und +ergibt} +&= +Re^{-R^2} +\int_0^{\frac{\pi}4} +e^{R^2\frac{\pi}4t} +\,dt += +Re^{-R^2} +\biggl[ +\frac{4}{\pi R^2} +e^{R^2\frac{\pi}4t} +\biggr]_0^{\frac{\pi}4} += +\frac{4}{\pi R} +e^{-R^2}(e^{R^2}-1) += +\frac{4}{\pi R} +(1-e^{-R^2}) +\to 0 +\end{align*} +für $R\to \infty$. +Im Grenzwert $R\to \infty$ kann der Teil $\gamma_2$ des Pfades +vernachlässigt werden. + +Das Integral über den geschlossenen Pfad $\gamma$ verschwindet. +Da der Teil $\gamma_2$ keine Rolle spielt, müssen sich die +Integrale über $\gamma_1$ und $\gamma_3$ wegheben, also +\begin{align*} +0 += +\int_\gamma e^{-z^2}\,dz +&= +\int_{\gamma_1} e^{-z^2}\,dz ++ +\int_{\gamma_2} e^{-z^2}\,dz ++ +\int_{\gamma_3} e^{-z^2}\,dz +\\ +&\to +\frac{\sqrt{\pi}}2 +-\frac{1}{\sqrt{2}}(C_1(\infty)+S_1(\infty)) +-\frac{i}{\sqrt{2}}(C_1(\infty)-S_1(\infty)). +\end{align*} +Der Imaginärteil ist $C_1(\infty)-S_1(\infty)$, da er verschwinden +muss, folgt $C_1(\infty)=S_1(\infty)$. +Nach Multlikation mit $\sqrt{2}$ folgt aus der Tatsache, dass auch +der Realteil verschwinden muss +\[ +\sqrt{\frac{\pi}{2}} = C_1(\infty)+S_1(\infty) +\qquad +\Rightarrow +\qquad +C_1(\infty) += +S_1(\infty) += +\frac12 +\sqrt{ +\frac{\pi}{2} +}. +\] +Aus +\eqref{fresnel:equation:arg} +erhält man dann auch die Grenzwerte +\[ +C(\infty)=S(\infty)=\frac12. +\qedhere +\] +\end{proof} diff --git a/buch/papers/fresnel/teil2.tex b/buch/papers/fresnel/teil2.tex index 701c3ee..ec8c896 100644 --- a/buch/papers/fresnel/teil2.tex +++ b/buch/papers/fresnel/teil2.tex @@ -3,38 +3,177 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 -\label{fresnel:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\section{Klothoide +\label{fresnel:section:klothoide}} +\rhead{Klothoide} +In diesem Abschnitt soll gezeigt werden, dass die Krümmung der +Euler-Spirale proportional zur vom Nullpunkt aus gemessenen Bogenlänge +ist. + +\begin{definition} +Eine ebene Kurve, deren Krümmung proportionale zur Kurvenlänge ist, +heisst {\em Klothoide}. +\end{definition} + +Die Klothoide wird zum Beispiel im Strassenbau für Autobahnkurven +verwendet. +Fährt man mit konstanter Geschwindigkeit entlang einer Klothoide, +muss man die Krümmung mit konstaner Geschwindigkeit ändern, +also das Lenkrad mit konstanter Geschwindigkeit drehen. +Dies ermöglicht eine ruhige Fahrweise. + +\subsection{Krümmung einer ebenen Kurve} +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/kruemmung.pdf} +\caption{Berechnung der Krümmung einer ebenen Kurve. +\label{fresnel:figure:kruemmung}} +\end{figure} +Abbildung~\ref{fresnel:figure:kruemmung} erinnert daran, dass der +Bogen eines Kreises vom Radius $r$, entlang dem sich die Richtung +der Tangente um $\Delta\varphi$ ändert, die Länge +$\Delta s = r\Delta\varphi$. +Die Krümmung ist der Kehrwert des Krümmungsradius, daraus kann +man ablesen, dass +\[ +\kappa = \frac{1}{r} = \frac{\Delta \varphi}{\Delta s}. +\] +Für eine beliebige ebene Kurve ist daher die Krümmung +\[ +\kappa = \frac{d\varphi}{ds}. +\] + +\subsection{Krümmung der Euler-Spirale} +Wir betrachten jetzt die Euler-Spirale mit der Parametrisierung +$\gamma(s) = (C_1(s),S_1(s))$. +Zunächst stellen wir fest, dass die Länge der Tangente +\[ +\dot{\gamma}(s) += +\frac{d\gamma}{ds} += +\begin{pmatrix} +\dot{C}_1(s)\\ +\dot{S}_1(s) +\end{pmatrix} += +\begin{pmatrix} +\cos s^2\\ +\sin s^2 +\end{pmatrix} +\qquad\Rightarrow\qquad +|\dot{\gamma}(s)| += +\sqrt{\cos^2s^2+\sin^2s^2} += +1. +\] +Insbesondere ist der Parameter $s$ der Kurve $\gamma(s)$ die +Bogenlänge. + +Der zu $\dot{\gamma}(s)$ gehörige Polarwinkel kann aus dem Vergleich +mit einem Vektor mit bekanntem Polarwinkel $\varphi$ abgelesen werden: +\[ +\begin{pmatrix} +\cos \varphi\\ +\sin \varphi +\end{pmatrix} += +\dot{\gamma}(s) += +\begin{pmatrix} +\cos s^2\\\sin s^2 +\end{pmatrix}, +\] +der Polarwinkel +ist daher $\varphi = s^2$. +Die Krümmung ist die Ableitung des Polarwinkels nach $s$, also +\[ +\kappa += +\frac{d\varphi}{ds} += +\frac{ds^2}{ds} += +2s, +\] +sie ist somit proportional zur Bogenlänge $s$. +Damit folgt, dass die Euler-Spirale eine Klothoide ist. + +\subsection{Eine Kugel schälen} +\begin{figure} +\centering +\includegraphics[width=\textwidth]{papers/fresnel/images/schale.pdf} +\caption{Schält man eine einen Streifen konstanter Breite beginnend am +Äquator von einer Kugel ab und breitet ihn in der Ebene aus, entsteht +eine Klothoide. +\label{fresnel:figure:schale}} +\end{figure} +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/apfel.pdf} +\caption{Klothoide erhalten durch Abschälen eines Streifens von einem +Apfel (vgl.~Abbildung~\ref{fresnel:figure:schale}) +\label{fresnel:figure:apfel}} +\end{figure} +Schält man einen Streifen konstanter Breite beginnend parallel zum Äquator +von einer Kugel ab und breitet ihn in die Ebene aus, entsteht eine +Approximation einer Klothoide. +Abbildung~\ref{fresnel:figure:schale} zeigt blau den abgeschälten Streifen, +Abbildung~\ref{fresnel:figure:apfel} zeigt das Resultat dieses Versuches +an einem Apfel, das Youtube-Video \cite{fresnel:schale} des +Numberphile-Kanals illustriert das Problem anhand eines aufblasbaren +Globus. + +Windet sich die Kurve in Abbildung~\ref{fresnel:figure:schale} $n$ +mal um die vertikale Achse, bevor sie den Nordpol erreicht, dann kann +die Kurve mit der Funktion +\[ +\gamma(t) += +\begin{pmatrix} +\cos(t) \cos(t/n) \\ +\sin(t) \cos(t/n) \\ +\sin(t/n) +\end{pmatrix} +\] +parametrisiert werden. +Der Tangentialvektor +\[ +\dot{\gamma}(t) += +\begin{pmatrix} +-\sin(t)\cos(t/n) - \cos(t)\sin(t/n)/n \\ +\cos(t)\cos(t/n) - \sin(t)\sin(t/n)/n \\ +\cos(t/n)/n +\end{pmatrix} +\] +hat die Länge +\[ +| \dot{\gamma}(t) |^2 += +\frac{1}{n^2} ++ +\cos^2\frac{t}{n}. +\] +Die Ableitung der Bogenlänge ist daher +\[ +\dot{s}(t) += +\sqrt{ +\frac{1}{n^2} ++ +\cos^2\frac{t}{n} +}. +\] + + +Der Krümmungsradius des blauen Streifens, der die Kugel im Punkt $P$ bei +geographischer $\vartheta$ berührt, hat die Länge der Tangente, die +die Kugel im Punkt $P$ berührt und im Punkt $Q$ durch die Achse der +Kugel geht (Abbildung~\ref{fresnel:figure:schale}). +Die Krümmung in Abhängigkeit von $\vartheta$ ist daher $\tan\vartheta$. + + diff --git a/buch/papers/fresnel/teil3.tex b/buch/papers/fresnel/teil3.tex index d4f15f6..ceddbe0 100644 --- a/buch/papers/fresnel/teil3.tex +++ b/buch/papers/fresnel/teil3.tex @@ -3,38 +3,110 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 -\label{fresnel:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Numerische Berechnung der Fresnel-Integrale +\label{fresnel:section:numerik}} +\rhead{Numerische Berechnung} +Die Fresnel-Integrale können mit verschiedenen Methoden effizient berechnet +werden. -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\subsection{Komplexe Fehlerfunktionen} +Es wurde schon darauf hingewiesen, dass der Integrand der Fresnel-Integrale +mit $e^{t^2}$ verwandt ist. +Tatsächlich kann gezeigt werden dass sich die Fresnel-Integrale mit +Hilfe der komplexen Fehlerfunktion als +\[ +\left. +\begin{matrix} +S_1(z) +\\ +C_1(z) +\end{matrix} +\; +\right\} += +\frac{1\pm i}4\biggl( +\operatorname{erf}\biggl(\frac{1+i}2\sqrt{\pi}z\biggr) +\mp +\operatorname{erf}\biggl(\frac{1-i}2\sqrt{\pi}z\biggr) +\biggr) +\] +ausdrücken lassen \cite{fresnel:fresnelC}. +Diese Darstellung ist jedoch für die numerische Berechnung nur +beschränkt nützlich, weil die meisten Bibliotheken für die Fehlerfunktion +diese nur für reelle Argument auszuwerten gestatten. + +\subsection{Als Lösung einer Differentialgleichung} +Da die Fresnel-Integrale die sehr einfachen Differentialgleichungen +\[ +C'(x) = \cos \biggl(\frac{\pi}2 x^2\biggr) +\qquad\text{und}\qquad +S'(x) = \sin \biggl(\frac{\pi}2 x^2\biggr) +\] +erfüllen, kann man eine Methode zur numerischen Lösung von +Differentialgleichung verwenden. +Die Abbildungen~\ref{fresnel:figure:plot} und \ref{fresnel:figure:eulerspirale} +wurden auf diese Weise erzeugt. + +\subsection{Taylor-Reihe integrieren} +Die Taylorreihen +\begin{align*} +\cos x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} x^{2k} +&&\text{und}& +\sin x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k+1} +\intertext{% +der trigonometrischen Funktionen werden durch Einsetzen von $x=t^2$ +zu} +\cos t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} t^{4k} +&&\text{und}& +\sin t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} t^{4k+2}. +\intertext{% +Die Fresnel-Integrale $C_1(x)$ und $S_1(x)$ können daher durch +termweise Integration mit Hilfe der Reihen} +C_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} \frac{x^{4k+1}}{4k+1} +&&\text{und}& +S_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} \frac{x^{4k+3}}{4k+3} +\end{align*} +berechnet werden. +Diese Reihen sind insbesondere für kleine Werte von $x$ sehr +schnell konvergent. + +\subsection{Hypergeometrische Reihen} +Aus der Reihenentwicklung kann jetzt auch eine Darstellung der +Fresnel-Integrale durch hypergeometrische Reihen gefunden werden +\cite{fresnel:fresnelC}. +Es ergibt sich +\begin{align*} +S(z) +&= +\frac{\pi z^3}{6} +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac34\\\frac32,\frac74\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr) +\\ +C(z) +&= +z +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac14\\\frac12,\frac54\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr). +\end{align*} diff --git a/buch/papers/kreismembran/main.tex b/buch/papers/kreismembran/main.tex index 67b436c..f6000a1 100644 --- a/buch/papers/kreismembran/main.tex +++ b/buch/papers/kreismembran/main.tex @@ -3,34 +3,16 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:kreismembran}} -\lhead{Thema} +\chapter{Schwingungen einer kreisförmigen Membran\label{chapter:kreismembran}} +\lhead{Schwingungen einer kreisförmigen Membran} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +\chapterauthor{Andrea Mozzini Vellen und Tim Tönz} \input{papers/kreismembran/teil0.tex} \input{papers/kreismembran/teil1.tex} \input{papers/kreismembran/teil2.tex} \input{papers/kreismembran/teil3.tex} +\input{papers/kreismembran/teil4.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/kreismembran/references.bib b/buch/papers/kreismembran/references.bib index 0b6a683..acf8f90 100644 --- a/buch/papers/kreismembran/references.bib +++ b/buch/papers/kreismembran/references.bib @@ -4,6 +4,25 @@ % (c) 2020 Autor, Hochschule Rapperswil % +@online{kreismembran:Duden:Membran, + title = {Duden:Membran}, + url = {https://www.duden.de/rechtschreibung/Membran}, + date = {2022-07-20}, + year = {2022}, + month = {7}, + day = {20} +} + +@online{kreismembran:wellengleichung_herleitung, + title = {Derivation of the 2D Wave Equation}, + author = {Dr. Christopher Lum}, + url = {https://www.youtube.com/watch?v=KAS7JBztw8E&t=0s}, + date = {2022-07-20}, + year = {2022}, + month = {7}, + day = {20} +} + @online{kreismembran:bibtex, title = {BibTeX}, url = {https://de.wikipedia.org/wiki/BibTeX}, @@ -24,7 +43,7 @@ } @article{kreismembran:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, + author = { Tabea Méndez and Andreas Müller }, title = { Noncommutative harmonic analysis and image registration }, journal = { Appl. Comput. Harmon. Anal.}, year = 2019, @@ -33,3 +52,33 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@book{lokenath_debnath_integral_2015, + edition = {Third Edition}, + title = {Integral Tansforms and Their Applications}, + publisher = {{CRC} Press}, + author = {{Lokenath Debnath} and Dambaru Bhatta}, + date = {2015}, +} + +@thesis{nishanth_p_vibrations_2018, + title = {Vibrations of a Circular Membrane - Some Undergraduadte Exercises}, + type = {phdthesis}, + author = {{Nishanth P.} and {Udayanandan K. M.}}, + date = {2018}, +} + +@thesis{prof_dr_horst_knorrer_kreisformige_2013, + title = {Kreisförmige Membranen}, + institution = {{ETHZ}}, + type = {phdthesis}, + author = {{Prof. Dr. Horst Knörrer}}, + date = {2013}, +} + +@thesis{kreismembran:membrane_vs_thin_plate, + title = {Modeling and Control of SPIDER Satellite Components}, + institution = {{faculty of the Virginia Polytechnic Institute and State University}}, + type = {Dissertation}, + author = {{Eric John Ruggiero Doctor of Philosophy In Mechanical Engineering}}, + date = {2005}, +}
\ No newline at end of file diff --git a/buch/papers/kreismembran/teil0.tex b/buch/papers/kreismembran/teil0.tex index e4b1711..bb8188d 100644 --- a/buch/papers/kreismembran/teil0.tex +++ b/buch/papers/kreismembran/teil0.tex @@ -3,20 +3,80 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{kreismembran:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{kreismembran:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Einleitung\label{kreismembran:section:teil0}} +\rhead{Membran} +Eine Membran oder selten ein Schwingblatt ist laut Duden \cite{kreismembran:Duden:Membran} ein "dünnes Blättchen aus Metall, Papier o. Ä., das durch seine Schwingungsfähigkeit geeignet ist, Schallwellen zu übertragen ...". +Ein dünnes Blättchen aus Metall zeig jedoch nicht die selben dynamischen Eigenschaften auf, wie ein gespanntes Stück Papier. +Beschreibt man das dynamische Verhalten, muss zwischen einer dünnen Platte und einer Membrane unterschieden werden \cite{kreismembran:membrane_vs_thin_plate}. +Eine dünne Platte zum Beispiel aus Metall, wirkt selbst entgegen ihrer Deformation sobald sie gekrümmt wird. +Eine Membran auf der anderen Seite besteht aus einem Material welches sich ohne Kraftaufwand verbiegen lässt wie zum Beispiel Papier. +Bevor Papier als schwingende Membran betrachtet werden kann wird jedoch noch eine Spannung $ T $ benötigt welche das Material daran hindert aus der Ruhelage gebracht zu werden. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +Ein geläufiges Beispiel einer Kreismembran ist eine runde Trommel. +Sie besteht herkömmlicher weise aus einem Leder (Fell), welches auf einen offenen Zylinder (Zargen) aufgespannt wird. +Das Leder alleine erzeugt nach einem Aufschlag keine hörbaren Schwingungen. +Sobald das Fell jedoch über den Zargen gespannt wird, kann das Fell auf verschiedensten weisen weiter schwingen, was für den Klang der Trommel verantwortlich ist. +Wie genau diese Schwingungen untersucht werden können wird in der Folgenden Arbeit Diskutiert. +\paragraph{Annahmen} +Um die Wellengleichung herzuleiten \cite{kreismembran:wellengleichung_herleitung}, muss ein Modell einer Membran definiert werden. +Das untersuchte Modell einer Membrane Erfüllt folgende Eigenschaften: +\begin{enumerate}[i] + \item Die Membran ist homogen. + Dies bedeutet, dass die Membran über die ganze Fläche die selbe Dichte $ \rho $ und Elastizität hat. + Durch die konstante Elastizität ist die ganze Membran unter gleichmässiger Spannung $ T $. + \item Die Membran ist perfekt flexibel. + Daraus folgt, dass die Membran ohne Kraftaufwand verbogen werden kann. + Die Membran ist dadurch nicht allein stehend schwing-fähig, hierzu muss sie gespannt werden mit einer Kraft $ T $. + \item Die Membran kann sich nur in Richtung ihrer Normalen in kleinem Ausmass Auslenken. + Auslenkungen in der ebene der Membran sind nicht möglich. + \item Die Membran erfährt keine Art von Dämpfung. + Die Membran wird also nicht durch ihr umliegendes Medium abgebremst noch erfährt sie Wärmeverluste durch Deformation. + Die resultierende Schwingung wird daher nicht gedämpft sein. + +\end{enumerate} + +\subsection{Wellengleichung} Um die Wellengleichung einer Membran herzuleiten wird vorerst eine schwingende Saite betrachtet. +Es lohnt sich das Verhalten einer Saite zu beschreiben da eine Saite das selbe Verhalten wie eine Membran aufweist mit dem Unterschied einer fehlenden Dimension. +Die Verbindung zwischen Membran und Saite ist intuitiv ersichtlich, stellt man sich einen Querschnitt einer Trommel vor. +%Wie analog zur Membran kann eine Saite erst unter Spannung schwingen. + +Abbildung \ref{TODO} ist ein infinitesimales Stück einer Saite mit Länge $ dx $ skizziert. +Wie für die Membran ist die Annahme iii gültig, keine Bewegung in die Richtung $ \hat{x} $. +Um dies zu erfüllen muss der Punkt $ P_1 $ gleich stark in Richtung $ -\hat{x} $ gezogen werden wie der Punkt $ P_2 $ in Richtung $ \hat{x} $ gezogen wird. Ist $ T_1 $ die Kraft welche mit Winkel $ \alpha $ auf Punkt $ P_1 $ wirkt sowie $ T_2 $ und $ \beta$ das analoge für Punkt $ P_2 $ ist, so können die Kräfte +\begin{equation}\label{kreismembran:eq:no_translation} + T_1 \cos \alpha = T_2 \cos \beta = T +\end{equation} +gleichgesetzt werden. +Das dynamische verhalten der senkrechten Auslenkung $ u(x,t) $ muss das newtonsche Gesetz +\begin{equation*} + \sum F = m \cdot a +\end{equation*} +befolgen. Die senkrecht wirkenden Kräfte werden mit $ T_1 $ und $ T_2 $ ausgedrückt, die Masse als Funktion der Dichte $ \rho $ und die Beschleunigung in Form der zweiten Ableitung als +\begin{equation*} + T_2 \sin \beta - T_1 \sin \alpha = \rho dx \frac{\partial^2 u}{\partial t^2} . +\end{equation*} +Die Gleichung wird durch $ T $ dividiert, wobei $ T $ nach \ref{kreismembran:eq:no_translation} geschickt gewählt wird. Somit kann +\begin{equation*} + \frac{T_2 \sin \beta}{T_2 \cos \beta} - \frac{T_1 \sin \alpha}{T_1 \cos \alpha} = \frac{\rho dx}{T} \frac{\partial^2 u}{\partial t^2} +\end{equation*} +vereinfacht als +\begin{equation*} + \tan \beta - \tan \alpha = \frac{\rho dx}{T} \frac{\partial^2 u}{\partial t^2} +\end{equation*} +geschrieben werden. +Der $ \tan \alpha $ entspricht der örtlichen Ableitung von $ u(x,t) $ an der Stelle $ x_0 $ und analog der $ \tan \beta $ für die Stelle $ x_0 + dx $. +Die Gleichung wird dadurch zu +\begin{equation*} + \frac{\partial u}{\partial x} \big\vert_{x_0 + dx} - \frac{\partial u}{\partial x} \big\vert_{x_0} = \frac{\rho dx}{T} \frac{\partial^2 u}{\partial t^2}. +\end{equation*} +Durch die Division mit $ dx $ entsteht +\begin{equation*} + \frac{1}{dx} \bigg[\frac{\partial u}{\partial x} \big\vert_{x_0 + dx} - \frac{\partial u}{\partial x} \big\vert_{x_0}\bigg] = \frac{\rho}{T}\frac{\partial^2 u}{\partial t^2}. +\end{equation*} +Auf der Linken Seite der Gleichung wird die Differenz der Steigungen durch die Intervall-Länge geteilt, in anderen Worten die zweite Ableitung von $ u(x,t) $ nach $ x $ berechnet. Der Term $ \frac{\rho}{T} $ wird mit $ c^2 $ ersetzt, da der Bruch für eine gegebene Membran eine positive Konstante sein muss. Somit resultiert die, in der Literatur gebräuchliche Form +\begin{equation} + \frac{1}{c^2}\frac{\partial^2u}{\partial t^2} = \Delta u. +\end{equation} +In dieser Form ist die Gleichung auch gültig für eine Membran. Für den Fall einer Membran muss lediglich die Ableitung in zwei Dimensionen gerechnet werden.
\ No newline at end of file diff --git a/buch/papers/kreismembran/teil1.tex b/buch/papers/kreismembran/teil1.tex index b715075..39ca598 100644 --- a/buch/papers/kreismembran/teil1.tex +++ b/buch/papers/kreismembran/teil1.tex @@ -2,54 +2,120 @@ % teil1.tex -- Beispiel-File für das Paper % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{kreismembran:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{kreismembran:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{kreismembran:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{kreismembran:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\section{Lösungsmethode 1: Separationsmethode + \label{kreismembran:section:teil1}} +\rhead{Lösungsmethode 1: Separationsmethode} +An diesem Punkt bleibt also nur noch die Lösung der partiellen Differentialgleichung. In diesem Kapitel wird sie mit Hilfe der Separationsmethode gelöst. + +\subsection{Aufgabestellung\label{sub:aufgabestellung}} +Wie im vorherigen Abschnitt gezeigt, lautet die partielle Differentialgleichung, die die Schwingungen einer Membran beschreibt: +\begin{equation*} + \frac{1}{c^2}\frac{\partial^2u}{\partial t^2} = \Delta u. +\end{equation*} +Da es sich um eine Kreisscheibe handelt, werden Polarkoordinaten verwendet, so dass sich der Laplaceoperator +\begin{equation*} + \Delta + = + \frac{\partial^2}{\partial r^2} + + + \frac1r + \frac{\partial}{\partial r} + + + \frac{1}{r 2} + \frac{\partial^2}{\partial\varphi^2} + \label{buch:pde:kreis:laplace} +\end{equation*} +ergibt. + +Es wird eine runde elastische Membran berücksichtigt, die das Gebiet $\Omega$ abdeckt und am Rand $\Gamma$ befestigt ist. +Es wird daher davon ausgegangen, dass die Membran aus einem homogenen Material von vernachlässigbarer Dicke gefertigt ist. +Die Membran kann verformt werden, aber innere elastische Kräfte wirken den Verformungen entgegen. Es wirken keine äusseren Kräfte. Es handelt sich somit von einer kreisförmligen eingespannten homogenen schwingenden Membran. + +Daher ist die Membranabweichung im Punkt $(r,\varphi)$ $\in$ $\overline{\rm \Omega}$ zum Zeitpunkt $t$: +\begin{align*} + u: \overline{\rm \Omega} \times \mathbb{R}_{\geq 0} &\longrightarrow \mathbb{R}\\ + (r,\varphi,t) &\longmapsto u(r,\varphi,t) +\end{align*} +Da die Membran am Rand befestigt ist, kann es keine Schwingungen geben, so dass die \textit{Dirichlet-Randbedingung} \cite{prof_dr_horst_knorrer_kreisformige_2013} +\begin{equation*} + u\big|_{\Gamma} = 0 \quad \text{für} \quad 0 \leq \varphi \leq 2\pi,\quad t \geq 0 +\end{equation*} +gilt. + +Um eine eindeutige Lösung bestimmen zu können, werden die folgenden Anfangsbedingungen festgelegt: +\begin{align*} + u(r,\varphi, 0) &= f(r,\varphi)\\ + u_t(r,\varphi, 0) &= g(r,\varphi). +\end{align*} + +\subsection{Lösung\label{sub:lösung1}} +\subsubsection{Ansatz der Separation der Variablen\label{subsub:ansatz_separation}} +Daher muss an dieser Stelle von einer Separation der Variablen ausgegangen werden: +\begin{equation*} + u(r,\varphi, t) = F(r)G(\varphi)T(t) +\end{equation*} +Dank der Randbedingungen kann also gefordert werden, dass $F(R)=0$ ist, und natürlich, dass $G(\varphi)$ $2\pi$ periodisch ist. Eingesetzt in der Differenzialgleichung ergibt sich: +\begin{equation*} + \frac{1}{c^2}\frac{T''(t)}{T(t)}=\frac{F''(r)}{F(r)}+\frac{1}{r}\frac{F'(r)}{F(r)}+\frac{1}{r^2}\frac{G''(\varphi)}{G(\varphi)}. +\end{equation*} +Da die linke Seite nur von $t$ und die rechte Seite nur von $r$ und $\varphi$ abhängt, müssen sie gleich einer reellen Zahl sein. Aus physikalischen Gründen suchen wir nach Lösungen, die weder exponentiell in der Zeit wachsen noch exponentiell abklingen. Dies bedeutet, dass die Konstante negativ sein muss, also schreibt man $k=-k^2$. Daraus ergeben sich die folgenden zwei Gleichungen: +\begin{align*} + T''(t) + c^2\kappa^2T(t) &= 0\\ + r^2\frac{F''(r)}{F(r)} + r \frac{F'(r)}{F(r)} +\kappa^2 r^2 &= - \frac{G''(\varphi)}{G(\varphi)}. +\end{align*} +In der zweiten Gleichung hängt die linke Seite nur von $r$ ab, während die rechte Seite nur von $\varphi$ abhängt. Sie müssen also wiederum gleich einer reellen Zahl $\nu$ sein. Also das: +\begin{align*} + r^2F''(r) + rF'(r) + (\kappa^2 r^2 - \nu)F(r) &= 0 \\ + G''(\varphi) &= \nu G(\varphi). +\end{align*} + +\subsubsection{Lösung für $G(\varphi)$\label{subsub:lösung_G}} +Da für die Zweite Gelichung Lösungen von Schwingungen erwartet werden, für die $G''(\varphi)=-\omega^2 G(\varphi)$ gilt, schreibt die gemeinsame Konstante als $-\nu^2$, was die Formeln später vereinfacht. Also: +\begin{equation*} + G(\varphi) = C_n \cos(\varphi) + D_n \sin(\varphi) + \label{eq:cos_sin_überlagerung} +\end{equation*} + +\subsubsection{Lösung für $F(r)$\label{subsub:lösung_F}} +Die Gleichung für $F$ hat die Gestalt +\begin{align} + r^2F''(r) + rF'(r) + (\kappa^2 r^2 - n^2)F(r) = 0 + \label{eq:2nd_degree_PDE} +\end{align} +Wir bereits in Kapitel \ref{buch:differntialgleichungen:section:bessel} gezeigt, sind die Besselfunktionen +\begin{equation*} + J_{\nu}(x) = r^\nu \displaystyle\sum_{m=0}^{\infty} \frac{(-1)^m x^{2m}}{2^{2m+\nu}m! \Gamma (\nu + m+1)} +\end{equation*} +Lösungen der Besselschen Differenzialgleichung +\begin{equation*} + x^2 y'' + xy' + (x^2 - \nu^2)y = 0 +\end{equation*} +Die Funktionen $F(r) = J_n(\kappa r)$ lösen also die Differentialgleichung \eqref{eq:2nd_degree_PDE}. Die +Randbedingung $F(R)=0$ impliziert, dass $\kappa R$ eine Nullstelle der Besselfunktion +$J_n$ sein muss. Man kann zeigen, dass die Besselfunktionen $J_n, n \geq 0$, alle unendlich +viele Nullstellen +\begin{equation*} + \alpha_{1n} < \alpha_{2n} < ... +\end{equation*} +haben, und dass $\underset{\substack{m\to\infty}}{\text{lim}} \alpha_{mn}=\infty$. Somit ergibt sich, dass $\kappa = \frac{\alpha_{mn}}{R}$ für ein $m\geq 1$, und dass +\begin{equation*} + F(r) = J_n (\kappa_{mn}r) \quad \text{mit} \quad \kappa_{mn}=\frac{\alpha_{mn}}{R} +\end{equation*} + +\subsubsection{Lösung für $T(t)$\label{subsub:lösung_T}} +Die Differenzialgleichung $T''(t) + c^2\kappa^2T(t) = 0$, wird auf ähnliche Weise gelöst wie $G(\varphi)$. + +\subsubsection{Zusammenfassung der Lösungen\label{subsub:zusammenfassung_lösungen}} +Durch Überlagerung aller Ergebnisse erhält man die Lösung +\begin{align} + u(r, \varphi, t) = \displaystyle\sum_{m=1}^{\infty}\displaystyle\sum_{n=0}^{\infty} J_n (k_{mn}r)[a_{mn}\cos(n\varphi) + b_{mn}\sin(n\varphi)](n\varphi)[c_{mn}\cos(c \kappa_{mn} t)+d_{mn}\sin(c \kappa_{mn} t)] + \label{eq:lösung_endliche_generelle} +\end{align} + +Dabei sind $m$ und $n$ ganze Zahlen, wobei $m$ für die Anzahl der Knotenkreise und $n$ +für die Anzahl der Knotenlinien steht. Es gibt bestimmte Bereiche auf der Membran, in denen es keine Bewegung oder Vibration gibt. Wenn der nicht schwingende Bereich ein Kreis ist, nennt man ihn einen Knotenkreis, und wenn er eine Linie ist, nennt man ihn ebenfalls eine Knotenlinie. $Jn(\kappa_{mn}r)$ ist die Besselfunktion $n$-ter Ordnung, wobei $\kappa mn$ die Wellenzahl und $r$ der Radius ist. $a_{mn}$ und $b_{mn}$ sind die zu bestimmenden Konstanten. + + +An diesem Punkt stellte sich die Frage, ob es möglich wäre, die partielle Differentialgleichung mit einer anderen Methode als der der Trennung der Variablen zu lösen. Nach einer kurzen Recherche wurde festgestellt, dass die beste Methode die Transformationsmethode ist, genauer gesagt die Anwendung der Hankel-Transformation. Im nächsten Kapitel wird daher diese Integraltransformation vorgestellt und entwickelt, und es wird erläutert, warum sie für diese Art von Problem geeignet ist. diff --git a/buch/papers/kreismembran/teil2.tex b/buch/papers/kreismembran/teil2.tex index 7ed217f..6efda49 100644 --- a/buch/papers/kreismembran/teil2.tex +++ b/buch/papers/kreismembran/teil2.tex @@ -1,40 +1,116 @@ % -% teil2.tex -- Beispiel-File für teil2 -% % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 -\label{kreismembran:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\section{Die Hankel Transformation \label{kreismembran:section:teil2}} +\rhead{Die Hankel Transformation} + +Hermann Hankel (1839--1873) war ein deutscher Mathematiker, der für seinen Beitrag zur mathematischen Analysis und insbesondere für die nach ihm benannte Transformation bekannt ist. +Diese Transformation tritt bei der Untersuchung von Funktionen auf, die nur von der Entfernung des Ursprungs abhängen. +Er studierte auch Funktionen, jetzt Hankel- oder Bessel- Funktionen genannt, der dritten Art. +Die Hankel-Transformation, die die Bessel-Funktion enthält, taucht natürlich bei achsensymmetrischen Problemen auf, die in zylindrischen Polarkoordinaten formuliert sind. +In diesem Abschnitt werden die Theorie der Transformation und einige Eigenschaften der Grundoperationen erläutert. + +\subsubsection{Hankel-Transformation \label{subsub:hankel_tansformation}} +Wir führen die Definition der Hankel-Transformation \cite{lokenath_debnath_integral_2015} aus der zweidimensionalen Fourier-Transformation und ihrer Umkehrung ein, die durch: +\begin{align} + \mathscr{F}\{f(x,y)\} & = F(k,l)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i( \bm{\kappa}\cdot \mathbf{r})}f(x,y) \; dx dy,\label{equation:fourier_transform}\\ + \mathscr{F}^{-1}\{F(x,y)\} & = f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(\bm{\kappa}\cdot \mathbf{r}))}F(k,l) \; dx dy \label{equation:inv_fourier_transform} +\end{align} +wo $\mathbf{r}=(x,y)$ und $\bm{\kappa}=(k,l)$. Polarkoordinaten sind für diese Art von Problemen am besten geeignet, mit $(x,y)=r(\cos\theta,\sin\theta)$ und $(k,l)=\kappa(\cos\phi,\sin\phi)$ findet man $\bm{\kappa}\cdot\mathbf{r}=\kappa r(\cos(\theta-\phi))$ und danach: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}r \; dr \int_{0}^{2\pi}e^{-ikr\cos(\theta-\phi)}f(r,\theta) \; d\phi. + \label{equation:F_ohne_variable_wechsel} +\end{align} +Dann wird angenommen dass, $f(r,\theta)=e^{in\theta}f(r)$, was keine strenge Einschränkung ist, und es wird eine Änderung der Variabeln vorgenommen $\theta-\phi=\alpha-\frac{\pi}{2}$, um \eqref{equation:F_ohne_variable_wechsel} zu reduzieren: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}rf(r) \; dr \int_{\phi_{0}}^{2\pi+\phi_{0}}e^{in(\phi-\frac{\pi}{2})+i(n\alpha-kr\sin\alpha)} \; d\alpha, + \label{equation:F_ohne_bessel} +\end{align} +wo $\phi_{0}=(\frac{\pi}{2}-\phi)$. + +Unter Verwendung der Integraldarstellung der Besselfunktion vom Ordnung $n$ \eqref{buch:fourier:eqn:bessel-integraldarstellung} +\begin{equation*} + J_n(\kappa r)=\frac{1}{2\pi}\int_{\phi_{0}}^{2\pi + \phi_{0}}e^{i(n\alpha-\kappa r \sin \alpha)} \; d\alpha + \label{equation:bessel_n_ordnung} +\end{equation*} +\eqref{equation:F_ohne_bessel} wird sie zu: +\begin{align} + F(k,\phi)&=e^{in(\phi-\frac{\pi}{2})}\int_{0}^{\infty}rJ_n(\kappa r) f(r) \; dr \nonumber \\ + &=e^{in(\phi-\frac{\pi}{2})}\tilde{f}_n(\kappa), + \label{equation:F_mit_bessel_step_2} +\end{align} +wo $\tilde{f}_n(\kappa)$ ist die \textit{Hankel-Transformation} von $f(r)$ und ist formell definiert durch: +\begin{align} + \mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)=\int_{0}^{\infty}rJ_n(\kappa r) f(r) \; dr. + \label{equation:hankel} +\end{align} + +\subsubsection{Inverse Hankel-Transformation \label{subsub:inverse_hankel_tansformation}} +Ähnlich verhält es sich mit der inversen Fourier Transformation in Form von polaren Koordinaten unter der Annahme $f(r,\theta)=e^{in\theta}f(r)$ mit \eqref{equation:F_mit_bessel_step_2}, wird die inverse Fourier Transformation \eqref{equation:inv_fourier_transform}: + +\begin{align*} + e^{in\theta}f(r)&=\frac{1}{2\pi}\int_{0}^{\infty}\kappa \; d\kappa \int_{0}^{2\pi}e^{i\kappa r \cos (\theta - \phi)}F(\kappa,\phi) \; d\phi \\ + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) \; d\kappa \int_{0}^{2\pi}e^{in(\phi - \frac{\pi}{2})- i\kappa r \cos (\theta - \phi)} \; d\phi, +\end{align*} +was durch den Wechsel der Variablen $\theta-\phi=-(\alpha+\frac{\pi}{2})$ und $\theta_0=-(\theta+\frac{\pi}{2})$, + +\begin{align*} + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) \; d\kappa \int_{\theta_0}^{2\pi+\theta_0}e^{in(\theta + \alpha - i\kappa r \sin\alpha)} \; d\alpha \\ + &= e^{in\theta}\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) \; d\kappa, +\end{align*} + +von \eqref{equation:bessel_n_ordnung} also ist, die inverse \textit{Hankel-Transformation} so definiert: +\begin{align} + \mathscr{H}^{-1}_n\{\tilde{f}_n(\kappa)\}=f(r)=\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) \; d\kappa. + \label{equation:inv_hankel} +\end{align} + +Anstelle von $\tilde{f}_n(\kappa)$, wird häufig für die Hankel-Transformation verwendet, indem die Ordnung angegeben wird. +\eqref{equation:hankel} und \eqref{equation:inv_hankel} Integralen existieren für eine grosse Klasse von Funktionen, die normalerweise in physikalischen Anwendungen benötigt werden. +Alternativ kann auch die berühmte Hankel-Transformationsformel verwendet werden, + +\begin{align*} + f(r) = \int_{0}^{\infty}\kappa J_n(\kappa r) \; d\kappa \int_{0}^{\infty} p J_n(\kappa p)f(p) \; dp, + \label{equation:hankel_integral_formula} +\end{align*} +um die Hankel-Transformation \eqref{equation:hankel} und ihre Inverse \eqref{equation:inv_hankel} zu definieren. +Insbesondere die Hankel-Transformation der nullten Ordnung ($n=0$) und der ersten Ordnung ($n=1$) sind häufig nützlich, um Lösungen für Probleme mit der Laplace Gleichung in einer achsensymmetrischen zylindrischen Geometrie zu finden. + +\subsection{Operative Eigenschaften der Hankel-Transformation\label{sub:op_properties_hankel}} +In diesem Kapitel werden die operativen Eigenschaften der Hankel-Transformation aufgeführt. Der Beweis für ihre Gültigkeit wird jedoch nicht analysiert. + +\begin{satz}{Skalierung:} + Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: + + \begin{equation*} + \mathscr{H}_n\{f(ar)\}=\frac{1}{a^{2}}\tilde{f}_n \left(\frac{\kappa}{a}\right), \quad a>0. + \end{equation*} +\end{satz} + +\begin{satz}{Persevalsche Relation (Skalarprodukt bleibt erhalten):} +Wenn $\tilde{f}(\kappa)=\mathscr{H}_n\{f(r)\}$ und $\tilde{g}(\kappa)=\mathscr{H}_n\{g(r)\}$, dann: + +\begin{equation*} + \int_{0}^{\infty}rf(r)g(r) \; dr = \int_{0}^{\infty}\kappa\tilde{f}(\kappa)\tilde{g}(\kappa) \; d\kappa. +\end{equation*} +\end{satz} + +\begin{satz}{Hankel-Transformationen von Ableitungen:} +Wenn $\tilde{f}_n(\kappa)=\mathscr{H}_n\{f(r)\}$, dann: + +\begin{align*} + &\mathscr{H}_n\{f'(r)\}=\frac{\kappa}{2n}\left[(n-1)\tilde{f}_{n+1}(\kappa)-(n+1)\tilde{f}_{n-1}(\kappa)\right], \quad n\geq1, \\ + &\mathscr{H}_1\{f'(r)\}=-\kappa \tilde{f}_0(\kappa), +\end{align*} +bereitgestellt dass $[rf(r)]$ verschwindet als $r\to0$ und $r\to\infty$. +\end{satz} + +\begin{satz} +Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: +\begin{equation*} + \mathscr{H}_n \left\{ \left( \nabla^2 - \frac{n^2}{r^2} f(r)\right)\right\}= \mathscr{H}_n\left\{\frac{1}{r}\frac{d}{dr}\left(r\frac{df}{dr}\right) - \frac{n^2}{r^2}f(r)\right\}=-\kappa^2\tilde{f}_{n}(\kappa), +\end{equation*} +bereitgestellt dass $rf'(r)$ und $rf(r)$ verschwinden für $r\to0$ und $r\to\infty$. +\end{satz} diff --git a/buch/papers/kreismembran/teil3.tex b/buch/papers/kreismembran/teil3.tex index 73dee0f..7d5648a 100644 --- a/buch/papers/kreismembran/teil3.tex +++ b/buch/papers/kreismembran/teil3.tex @@ -3,38 +3,82 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Lösungsmethode 2: Transformationsmethode \label{kreismembran:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\rhead{Lösungsmethode 2: Transformationsmethode} +Die Hankel-Transformation wird dann zur Lösung der Differentialgleichung verwendet. Es müssen jedoch einige Änderungen an dem Problem vorgenommen werden, damit es mit den Annahmen übereinstimmt, die für die Verwendung der Hankel-Transformation erforderlich sind. Das heisst, dass die Funktion $u$ nur von der Entfernung zum Ausgangspunkt abhängt. + +\subsubsection{Transformation und Reduktion auf eine algebraische Gleichung\label{subsub:transf_reduktion}} +Führt man also das Konzept einer unendlichen und achsensymmetrischen Membran ein: +\begin{equation*} + \frac{\partial^2u}{\partial t^2} + = + c^2 \left(\frac{\partial^2 u}{\partial r^2} + + + \frac{1}{r} + \frac{\partial u}{\partial r} \right), \quad 0<r<\infty, \quad t>0 + \label{eq:PDE_inf_membane} +\end{equation*} + +\begin{align} + u(r,0)=f(r), \quad u_t(r,0) = g(r), \quad \text{für} \quad 0<r<\infty + \label{eq:PDE_inf_membane_RB} +\end{align} + +Mit Anwendung der Hankel-Transformation nullter Ordnung in Abhängigkeit von $r$ auf die Gleichungen \eqref{eq:PDE_inf_membane} und \eqref{eq:PDE_inf_membane_RB}: + +\begin{align} + \tilde{u}(\kappa,t)=\int_{0}^{\infty}r J_0(\kappa r)u(r,t) \; dr, +\end{align} +bekommt man: + +\begin{equation*} + \frac{d^2 \tilde{u}}{dt^2} + c^2\kappa^2\tilde{u}=0, +\end{equation*} + +\begin{equation*} + \tilde{u}(\kappa,0)=\tilde{f}(\kappa), \quad + \tilde{u}_t(\kappa,0)=\tilde{g}(\kappa). +\end{equation*} +Die allgemeine Lösung für diese Transformation lautet, wie in Gleighung \eqref{eq:cos_sin_überlagerung} gesehen, wie folgt + +\begin{equation*} + \tilde{u}(\kappa,t)=\tilde{f}(\kappa)\cos(c\kappa t) + \frac{1}{c\kappa}\tilde{g}(\kappa)\sin(c\kappa t). +\end{equation*} +Wendet man an nun die inverse Hankel-Transformation an, so erhält man die formale Lösung + +\begin{align} + u(r,t)=\int_{0}^{\infty}\kappa\tilde{f}(\kappa)\cos(c\kappa t) J_0(\kappa r) \; d\kappa +\frac{1}{c}\int_{0}^{\infty}\tilde{g}(\kappa)\sin(c\kappa t)J_0(\kappa r) \; d\kappa. + \label{eq:formale_lösung} +\end{align} + +\subsubsection{Erfüllung der Anfangsbedingungen\label{subsub:erfüllung_AB}} +Es wird in Folgenden davon ausgegangen, dass sich die Membran verformt und zum Zeitpunkt $t=0$ freigegeben wird + +\begin{equation*} + u(r,0)=f(r)=Aa(r^2 + a^2)^{-\frac{1}{2}}, \quad u_t(r,0)=g(r)=0 +\end{equation*} +so dass $\tilde{g}(\kappa)\equiv 0$ und +\begin{equation*} + \tilde{f}(\kappa)=Aa\int_{0}^{\infty}r(a^2 + r^2)^{-\frac{1}{2}} J_0 (\kappa r) \; dr=\frac{Aa}{\kappa}e^{-a\kappa} +\end{equation*} +Die formale Lösung \eqref{eq:formale_lösung} lautet also +\begin{align*} + u(r,t)&=Aa\int_{0}^{\infty}e^{-a\kappa} J_0(\kappa r)\cos(c\kappa t) \; dk=AaRe\int_{0}^{\infty}e^{-\kappa(a+ict)} J_0(\kappa r) \; dk\\ + &=AaRe\left\{r^2+\left(a+ict\right)^2\right\}^{-\frac{1}{2}} +\end{align*} + +Nimmt man jedoch die allgemeine Lösung mit Summationen, + +\begin{align} + u(r, t) = \displaystyle\sum_{m=1}^{\infty} J_0 (k_{m}r)[a_{m}\cos(c \kappa_{m} t)+b_{m}\sin(c \kappa_{m} t)] + \label{eq:lösung_unendliche_generelle} +\end{align} +kann man die Lösungsmethoden 1 und 2 vergleichen. + +\subsection{Vergleich der Analytischen Lösungen +\label{kreismembran:vergleich}} +Bei der Analyse der Gleichungen \eqref{eq:lösung_endliche_generelle} und \eqref{eq:lösung_unendliche_generelle} fällt sofort auf, dass die Gleichung \eqref{eq:lösung_unendliche_generelle} nicht mehr von $m$ und $n$ abhängt, sondern nur noch von $n$ \cite{nishanth_p_vibrations_2018}. Das macht Sinn, denn $n$ beschreibt die Anzahl der Knotenlinien, und in einer unendlichen Membran gibt es keine. Tatsächlich werden $a_{m0}$, $b_{m0}$ und $\kappa_{m0}$ in $a_m$, $b_m$ bzw. $\kappa_m$ umbenannt. Die beiden Termen $\cos(n\varphi)$ und $\sin(n\varphi)$ verschwinden ebenfalls, da für $n=0$ der $\cos(n\varphi)$ gleich 1 und der $\sin(n \varphi)$ gleich 0 ist. +Die Funktion hängt also nicht mehr von der Besselfunktionen $n$-ter Ordnung ab, sondern nur von der $0$-ter Ordnung. diff --git a/buch/papers/kreismembran/teil4.tex b/buch/papers/kreismembran/teil4.tex new file mode 100644 index 0000000..c124354 --- /dev/null +++ b/buch/papers/kreismembran/teil4.tex @@ -0,0 +1,16 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Lösungsmethode 3: Simulation + \label{kreismembran:section:teil4}} +\paragraph{TODO Einleitung} + +Um numerisch das Verhalten einer Membran zu ermitteln, muss eine numerische Darstellung definiert werden. +Die Membran wird hier in Form der Matrix $ A $ digitalisiert. +Jedes Element $ A_{ij} $ steh für die Auslenkung der Membran $ u(x,y,t) $ an der Stelle $ \{x,y\}=\{i,j\} $. +Die zeitliche Dimension wird in Form des Array $ X[] $ aus $ v \times A $ Matrizen dargestellt. +Das Element auf Zeile $ i $, Spalte $ j $ der $ w $-ten Matrix von $ X[] $ also $ X[w]_{ij} $ entspricht der Auslenkung $ u(i,j,w) $. + +\paragraph{title}
\ No newline at end of file diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex new file mode 100644 index 0000000..b2f227e --- /dev/null +++ b/buch/papers/kugel/applications.tex @@ -0,0 +1,9 @@ +% vim:ts=2 sw=2 et spell: + +\section{Applications} + +\subsection{Electroencephalography (EEG)} + +\subsection{Measuring Gravitational Fields} + +\subsection{Quantisation of Angular Momentum} diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile new file mode 100644 index 0000000..4226dab --- /dev/null +++ b/buch/papers/kugel/images/Makefile @@ -0,0 +1,30 @@ +# +# Makefile -- build images +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: curvature.jpg spherecurve.jpg + +curvature.inc: curvgraph.m + octave curvgraph.m + +curvature.png: curvature.pov curvature.inc + povray +A0.1 +W1920 +H1080 +Ocurvature.png curvature.pov + +curvature.jpg: curvature.png + convert curvature.png -density 300 -units PixelsPerInch curvature.jpg + +spherecurve2.inc: spherecurve.m + octave spherecurve.m + +spherecurve.png: spherecurve.pov spherecurve.inc + povray +A0.1 +W1080 +H1080 +Ospherecurve.png spherecurve.pov + +spherecurve.jpg: spherecurve.png + convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg + +spherecurve: spherecurve.cpp + g++ -o spherecurve -g -Wall -O spherecurve.cpp + +spherecurve.inc: spherecurve + ./spherecurve diff --git a/buch/papers/kugel/images/curvature.maxima b/buch/papers/kugel/images/curvature.maxima new file mode 100644 index 0000000..6313642 --- /dev/null +++ b/buch/papers/kugel/images/curvature.maxima @@ -0,0 +1,6 @@ + +f: exp(-r^2/sigma^2)/sigma; +laplacef: ratsimp(diff(r * diff(f,r), r) / r); +f: exp(-r^2/(2*sigma^2))/(sqrt(2)*sigma); +laplacef: ratsimp(diff(r * diff(f,r), r) / r); + diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov new file mode 100644 index 0000000..3b15d77 --- /dev/null +++ b/buch/papers/kugel/images/curvature.pov @@ -0,0 +1,139 @@ +// +// curvature.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.09; + +camera { + location <10, 10, -40> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(<-3.1,0,0>, <3.1,0,0>, 0.01, White) +arrow(<0,-1,0>, <0,1,0>, 0.01, White) +arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White) + +#include "curvature.inc" + +#declare sigma = 1; +#declare s = 1.4; +#declare N0 = 0.4; +#declare funktion = function(r) { + (exp(-r*r/(sigma*sigma)) / sigma + - + exp(-r*r/(2*sigma*sigma)) / (sqrt(2)*sigma)) / N0 +}; +#declare hypot = function(xx, yy) { sqrt(xx*xx+yy*yy) }; + +#declare Funktion = function(x,y) { funktion(hypot(x+s,y)) - funktion(hypot(x-s,y)) }; +#macro punkt(xx,yy) + <xx, Funktion(xx, yy), yy> +#end + +#declare griddiameter = 0.006; +union { + #declare xmin = -3; + #declare xmax = 3; + #declare ymin = -2; + #declare ymax = 2; + + + #declare xstep = 0.2; + #declare ystep = 0.02; + #declare xx = xmin; + #while (xx < xmax + xstep/2) + #declare yy = ymin; + #declare P = punkt(xx, yy); + #while (yy < ymax - ystep/2) + #declare yy = yy + ystep; + #declare Q = punkt(xx, yy); + sphere { P, griddiameter } + cylinder { P, Q, griddiameter } + #declare P = Q; + #end + sphere { P, griddiameter } + #declare xx = xx + xstep; + #end + + #declare xstep = 0.02; + #declare ystep = 0.2; + #declare yy = ymin; + #while (yy < ymax + ystep/2) + #declare xx = xmin; + #declare P = punkt(xx, yy); + #while (xx < xmax - xstep/2) + #declare xx = xx + xstep; + #declare Q = punkt(xx, yy); + sphere { P, griddiameter } + cylinder { P, Q, griddiameter } + #declare P = Q; + #end + sphere { P, griddiameter } + #declare yy = yy + ystep; + #end + + pigment { + color rgb<0.8,0.8,0.8> + } + finish { + metallic + specular 0.8 + } +} + diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m new file mode 100644 index 0000000..75effd6 --- /dev/null +++ b/buch/papers/kugel/images/curvgraph.m @@ -0,0 +1,140 @@ +# +# curvature.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +global N; +N = 10; + +global sigma2; +sigma2 = 1; + +global s; +s = 1.4; + +global cmax; +cmax = 0.9; +global cmin; +cmin = -0.9; + +global Cmax; +global Cmin; +Cmax = 0; +Cmin = 0; + +xmin = -3; +xmax = 3; +xsteps = 200; +hx = (xmax - xmin) / xsteps; + +ymin = -2; +ymax = 2; +ysteps = 200; +hy = (ymax - ymin) / ysteps; + +function retval = f0(r) + global sigma2; + retval = exp(-r^2/sigma2)/sqrt(sigma2) - exp(-r^2/(2*sigma2))/(sqrt(2*sigma2)); +end + +global N0; +N0 = f0(0) +N0 = 0.4; + +function retval = f1(x,y) + global N0; + retval = f0(hypot(x, y)) / N0; +endfunction + +function retval = f(x, y) + global s; + retval = f1(x+s, y) - f1(x-s, y); +endfunction + +function retval = curvature0(r) + global sigma2; + retval = ( + -4*(sigma2-r^2)*exp(-r^2/sigma2) + + + (2*sigma2-r^2)*exp(-r^2/(2*sigma2)) + ) / (sigma2^(5/2)); +endfunction + +function retval = curvature1(x, y) + retval = curvature0(hypot(x, y)); +endfunction + +function retval = curvature(x, y) + global s; + retval = curvature1(x+s, y) - curvature1(x-s, y); +endfunction + +function retval = farbe(x, y) + global Cmax; + global Cmin; + global cmax; + global cmin; + c = curvature(x, y); + if (c < Cmin) + Cmin = c + endif + if (c > Cmax) + Cmax = c + endif + u = (c - cmin) / (cmax - cmin); + if (u > 1) + u = 1; + endif + if (u < 0) + u = 0; + endif + color = [ u, 0.5, 1-u ]; + color = color/max(color); + color(1,4) = c/2; + retval = color; +endfunction + +function dreieck(fn, A, B, C) + fprintf(fn, "\ttriangle {\n"); + fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", A(1,1), A(1,3), A(1,2)); + fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", B(1,1), B(1,3), B(1,2)); + fprintf(fn, "\t <%.4f,%.4f,%.4f>\n", C(1,1), C(1,3), C(1,2)); + fprintf(fn, "\t}\n"); +endfunction + +function viereck(fn, punkte) + color = farbe(mean(punkte(:,1)), mean(punkte(:,2))); + fprintf(fn, " mesh {\n"); + dreieck(fn, punkte(1,:), punkte(2,:), punkte(3,:)); + dreieck(fn, punkte(2,:), punkte(3,:), punkte(4,:)); + fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> } // %.4f\n", + color(1,1), color(1,2), color(1,3), color(1,4)); + fprintf(fn, " }\n"); +endfunction + +fn = fopen("curvature.inc", "w"); +punkte = zeros(4,3); +for ix = (0:xsteps-1) + x = xmin + ix * hx; + punkte(1,1) = x; + punkte(2,1) = x; + punkte(3,1) = x + hx; + punkte(4,1) = x + hx; + for iy = (0:ysteps-1) + y = ymin + iy * hy; + punkte(1,2) = y; + punkte(2,2) = y + hy; + punkte(3,2) = y; + punkte(4,2) = y + hy; + for i = (1:4) + punkte(i,3) = f(punkte(i,1), punkte(i,2)); + endfor + viereck(fn, punkte); + end +end +#fprintf(fn, " finish { metallic specular 0.5 }\n"); +fclose(fn); + +printf("Cmax = %.4f\n", Cmax); +printf("Cmin = %.4f\n", Cmin); diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp new file mode 100644 index 0000000..8ddf5e5 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.cpp @@ -0,0 +1,292 @@ +/* + * spherecurve.cpp + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +#include <cstdio> +#include <cstdlib> +#include <cmath> +#include <string> +#include <iostream> + +inline double sqr(double x) { return x * x; } + +/** + * \brief Class for 3d vectors (also used as colors) + */ +class vector { + double X[3]; +public: + vector() { X[0] = X[1] = X[2] = 0; } + vector(double a) { X[0] = X[1] = X[2] = a; } + vector(double x, double y, double z) { + X[0] = x; X[1] = y; X[2] = z; + } + vector(double theta, double phi) { + double s = sin(theta); + X[0] = cos(phi) * s; + X[1] = sin(phi) * s; + X[2] = cos(theta); + } + vector(const vector& other) { + for (int i = 0; i < 3; i++) { + X[i] = other.X[i]; + } + } + vector operator+(const vector& other) const { + return vector(X[0] + other.X[0], + X[1] + other.X[1], + X[2] + other.X[2]); + } + vector operator*(double l) const { + return vector(X[0] * l, X[1] * l, X[2] * l); + } + double operator*(const vector& other) const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += X[i] * other.X[i]; + } + return s; + } + double norm() const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += sqr(X[i]); + } + return sqrt(s); + } + vector normalize() const { + double l = norm(); + return vector(X[0]/l, X[1]/l, X[2]/l); + } + double max() const { + return std::max(X[0], std::max(X[1], X[2])); + } + double l0norm() const { + double l = 0; + for (int i = 0; i < 3; i++) { + if (fabs(X[i]) > l) { + l = fabs(X[i]); + } + } + return l; + } + vector l0normalize() const { + double l = l0norm(); + vector result(X[0]/l, X[1]/l, X[2]/l); + return result; + } + const double& operator[](int i) const { return X[i]; } + double& operator[](int i) { return X[i]; } +}; + +/** + * \brief Derived 3d vector class implementing color + * + * The constructor in this class converts a single value into a + * color on a suitable gradient. + */ +class color : public vector { +public: + static double utop; + static double ubottom; + static double green; +public: + color(double u) { + u = (u - ubottom) / (utop - ubottom); + if (u > 1) { + u = 1; + } + if (u < 0) { + u = 0; + } + u = pow(u,2); + (*this)[0] = u; + (*this)[1] = green * u * (1 - u); + (*this)[2] = 1-u; + double l = l0norm(); + for (int i = 0; i < 3; i++) { + (*this)[i] /= l; + } + } +}; + +double color::utop = 12; +double color::ubottom = -31; +double color::green = 0.5; + +/** + * \brief Surface model + * + * This class contains the definitions of the functions to plot + * and the parameters to + */ +class surfacefunction { + static vector axes[6]; + + double _a; + double _A; + + double _umin; + double _umax; +public: + double a() const { return _a; } + double A() const { return _A; } + + double umin() const { return _umin; } + double umax() const { return _umax; } + + surfacefunction(double a, double A) : _a(a), _A(A), _umin(0), _umax(0) { + } + + double f(double z) { + return A() * exp(a() * (sqr(z) - 1)); + } + + double g(double z) { + return -f(z) * 2*a() * ((2*a()*sqr(z) + (3-2*a()))*sqr(z) - 1); + } + + double F(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += f(axes[i] * v); + } + return s / 6; + } + + double G(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += g(axes[i] * v); + } + return s / 6; + } +protected: + color farbe(const vector& v) { + double u = G(v); + if (u < _umin) { + _umin = u; + } + if (u > _umax) { + _umax = u; + } + return color(u); + } +}; + +static double phi = (1 + sqrt(5)) / 2; +static double sl = sqrt(sqr(phi) + 1); +vector surfacefunction::axes[6] = { + vector( 0. , -1./sl, phi/sl ), + vector( 0. , 1./sl, phi/sl ), + vector( 1./sl, phi/sl, 0. ), + vector( -1./sl, phi/sl, 0. ), + vector( phi/sl, 0. , 1./sl ), + vector( -phi/sl, 0. , 1./sl ) +}; + +/** + * \brief Class to construct the plot + */ +class surface : public surfacefunction { + FILE *outfile; + + int _phisteps; + int _thetasteps; + double _hphi; + double _htheta; +public: + int phisteps() const { return _phisteps; } + int thetasteps() const { return _thetasteps; } + double hphi() const { return _hphi; } + double htheta() const { return _htheta; } + void phisteps(int s) { _phisteps = s; _hphi = 2 * M_PI / s; } + void thetasteps(int s) { _thetasteps = s; _htheta = M_PI / s; } + + surface(const std::string& filename, double a, double A) + : surfacefunction(a, A) { + outfile = fopen(filename.c_str(), "w"); + phisteps(400); + thetasteps(200); + } + + ~surface() { + fclose(outfile); + } + +private: + void triangle(const vector& v0, const vector& v1, const vector& v2) { + fprintf(outfile, " mesh {\n"); + vector c = (v0 + v1 + v2) * (1./3.); + vector color = farbe(c.normalize()); + vector V0 = v0 * (1 + F(v0)); + vector V1 = v1 * (1 + F(v1)); + vector V2 = v2 * (1 + F(v2)); + fprintf(outfile, "\ttriangle {\n"); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V0[0], V0[2], V0[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V1[0], V1[2], V1[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>\n", + V2[0], V2[2], V2[1]); + fprintf(outfile, "\t}\n"); + fprintf(outfile, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color[0], color[1], color[2]); + fprintf(outfile, "\tfinish { metallic specular 0.5 }\n"); + fprintf(outfile, " }\n"); + } + + void northcap() { + vector v0(0, 0, 1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // northcap i = %d\n", i); + vector v1(htheta(), (i - 1) * hphi()); + vector v2(htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void southcap() { + vector v0(0, 0, -1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // southcap i = %d\n", i); + vector v1(M_PI - htheta(), (i - 1) * hphi()); + vector v2(M_PI - htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void zone() { + for (int j = 1; j < thetasteps() - 1; j++) { + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // zone j = %d, i = %d\n", + j, i); + vector v0( j * htheta(), (i-1) * hphi()); + vector v1((j+1) * htheta(), (i-1) * hphi()); + vector v2( j * htheta(), i * hphi()); + vector v3((j+1) * htheta(), i * hphi()); + triangle(v0, v1, v2); + triangle(v1, v2, v3); + } + } + } +public: + void draw() { + northcap(); + southcap(); + zone(); + } +}; + +/** + * \brief main function + */ +int main(int argc, char *argv[]) { + surface S("spherecurve.inc", 5, 10); + color::green = 1.0; + S.draw(); + std::cout << "umin: " << S.umin() << std::endl; + std::cout << "umax: " << S.umax() << std::endl; + return EXIT_SUCCESS; +} diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m new file mode 100644 index 0000000..99d5c9a --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.m @@ -0,0 +1,160 @@ +# +# spherecurve.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global a; +a = 5; +global A; +A = 10; + +phisteps = 400; +hphi = 2 * pi / phisteps; +thetasteps = 200; +htheta = pi / thetasteps; + +function retval = f(z) + global a; + global A; + retval = A * exp(a * (z^2 - 1)); +endfunction + +function retval = g(z) + global a; + retval = -f(z) * 2 * a * (2 * a * z^4 + (3 - 2*a) * z^2 - 1); + # 2 + # - a 2 4 2 2 a z + #(%o6) - %e (4 a z + (6 a - 4 a ) z - 2 a) %e +endfunction + +phi = (1 + sqrt(5)) / 2; + +global axes; +axes = [ + 0, 0, 1, -1, phi, -phi; + 1, -1, phi, phi, 0, 0; + phi, phi, 0, 0, 1, 1; +]; +axes = axes / (sqrt(phi^2+1)); + +function retval = kugel(theta, phi) + retval = [ + cos(phi) * sin(theta); + sin(phi) * sin(theta); + cos(theta) + ]; +endfunction + +function retval = F(v) + global axes; + s = 0; + for i = (1:6) + z = axes(:,i)' * v; + s = s + f(z); + endfor + retval = s / 6; +endfunction + +function retval = F2(theta, phi) + v = kugel(theta, phi); + retval = F(v); +endfunction + +function retval = G(v) + global axes; + s = 0; + for i = (1:6) + s = s + g(axes(:,i)' * v); + endfor + retval = s / 6; +endfunction + +function retval = G2(theta, phi) + v = kugel(theta, phi); + retval = G(v); +endfunction + +function retval = cnormalize(u) + utop = 11; + ubottom = -30; + retval = (u - ubottom) / (utop - ubottom); + if (retval > 1) + retval = 1; + endif + if (retval < 0) + retval = 0; + endif +endfunction + +global umin; +umin = 0; +global umax; +umax = 0; + +function color = farbe(v) + global umin; + global umax; + u = G(v); + if (u < umin) + umin = u; + endif + if (u > umax) + umax = u; + endif + u = cnormalize(u); + color = [ u, 0.5, 1-u ]; + color = color/max(color); +endfunction + +function dreieck(fn, v0, v1, v2) + fprintf(fn, " mesh {\n"); + c = (v0 + v1 + v2) / 3; + c = c / norm(c); + color = farbe(c); + v0 = v0 * (1 + F(v0)); + v1 = v1 * (1 + F(v1)); + v2 = v2 * (1 + F(v2)); + fprintf(fn, "\ttriangle {\n"); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v0(1,1), v0(3,1), v0(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>\n", v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "\t}\n"); + fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color(1,1), color(1,2), color(1,3)); + fprintf(fn, "\tfinish { metallic specular 0.5 }\n"); + fprintf(fn, " }\n"); +endfunction + +fn = fopen("spherecurve2.inc", "w"); + + for i = (1:phisteps) + # Polkappe nord + v0 = [ 0; 0; 1 ]; + v1 = kugel(htheta, (i-1) * hphi); + v2 = kugel(htheta, i * hphi); + fprintf(fn, " // i = %d\n", i); + dreieck(fn, v0, v1, v2); + + # Polkappe sued + v0 = [ 0; 0; -1 ]; + v1 = kugel(pi-htheta, (i-1) * hphi); + v2 = kugel(pi-htheta, i * hphi); + dreieck(fn, v0, v1, v2); + endfor + + for j = (1:thetasteps-2) + for i = (1:phisteps) + v0 = kugel( j * htheta, (i-1) * hphi); + v1 = kugel((j+1) * htheta, (i-1) * hphi); + v2 = kugel( j * htheta, i * hphi); + v3 = kugel((j+1) * htheta, i * hphi); + fprintf(fn, " // i = %d, j = %d\n", i, j); + dreieck(fn, v0, v1, v2); + dreieck(fn, v1, v2, v3); + endfor + endfor + +fclose(fn); + +umin +umax diff --git a/buch/papers/kugel/images/spherecurve.maxima b/buch/papers/kugel/images/spherecurve.maxima new file mode 100644 index 0000000..1e9077c --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.maxima @@ -0,0 +1,13 @@ +/* + * spherecurv.maxima + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +f: exp(-a * sin(theta)^2); + +g: ratsimp(diff(sin(theta) * diff(f, theta), theta)/sin(theta)); +g: subst(z, cos(theta), g); +g: subst(sqrt(1-z^2), sin(theta), g); +ratsimp(g); + +f: ratsimp(subst(sqrt(1-z^2), sin(theta), f)); diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov new file mode 100644 index 0000000..b1bf4b8 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.pov @@ -0,0 +1,73 @@ +// +// curvature.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.13; + +camera { + location <10, 10, -40> + look_at <0, 0, 0> + right x * imagescale + up y * imagescale +} + +light_source { + <-10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(<-2.7,0,0>, <2.7,0,0>, 0.03, White) +arrow(<0,-2.7,0>, <0,2.7,0>, 0.03, White) +arrow(<0,0,-2.7>, <0,0,2.7>, 0.03, White) + +#include "spherecurve.inc" + diff --git a/buch/papers/kugel/introduction.tex b/buch/papers/kugel/introduction.tex new file mode 100644 index 0000000..5b09e9c --- /dev/null +++ b/buch/papers/kugel/introduction.tex @@ -0,0 +1,35 @@ +% vim:ts=2 sw=2 et spell tw=78: + +\section{Introduction} + +This chapter of the book is devoted to the sef of functions called +\emph{spherical harmonics}. However, before we dive into the topic, we want to +make a few preliminary remarks to avoid ``upsetting'' a certain type of +reader. Specifically, we would like to specify that the authors of this +chapter not mathematicians but engineers, and therefore the text will not be +always complete with sound proofs after every claim. Instead we will go +through the topic in a more intuitive way including rigorous proofs only if +they are enlightening or when they are very short. Where no proofs are given +we will try to give an intuition for why it is true. + +That being said, when talking about spherical harmonics one could start by +describing their name. The latter may be a cause of some confusion because of +the misleading translations in other languages. In German the name for this +set of functions is ``Kugelfunktionen'', which puts the emphasis only on the +spherical context, whereas the English name ``spherical harmonics'' also +contains the \emph{harmonic} part hinting at Fourier theories and harmonic +analysis in general. + +The structure of this chapter is organized in the following way. First, we +will quickly go through some fundamental linear algebra and Fourier theory to +refresh a few important concepts. In principle, we could have written the +whole thing starting from a much more abstract level without much preparation, +but then we would have lost some of the beauty that comes from the +appreciation of the power of some surprisingly simple ideas. Then once the +basics are done, we can explore the main topic of spherical harmonics which as +we will see arises from the eigenfunctions of the Laplacian operator in +spherical coordinates. Finally, after studying what we think are the most +beautiful and interesting properties of the spherical harmonics, to conclude +this journey we will present a few real-world applications, which are of +course most of interest for engineers. + diff --git a/buch/papers/kugel/main.tex b/buch/papers/kugel/main.tex index 06368af..98d9cb2 100644 --- a/buch/papers/kugel/main.tex +++ b/buch/papers/kugel/main.tex @@ -1,39 +1,20 @@ -% +% vim:ts=2 sw=2 et: % main.tex -- Paper zum Thema <kugel> % % (c) 2020 Hochschule Rapperswil % -\chapter{Recurrence Relations for Spherical Harmonics in Quantum Mechanics\label{chapter:kugel}} -\lhead{Recurrence Relations in Quantum Mechanics} +\begin{otherlanguage}{english} +\chapter{Spherical Harmonics\label{chapter:kugel}} +\lhead{Spherical Harmonics} \begin{refsection} \chapterauthor{Manuel Cattaneo, Naoki Pross} -\begin{verbatim} - -Ideas and current research goals --------------------------------- - -- Recurrence relations for spherical harmonics -- Associated Legendre polynomials -- Rodrigues' type formula aka Rodrigues' formula -- Applications: - * Quantization of angular momentum - * Gravitational field measurements (NASA ebb and flow, ESA goce) - * Literally anything that needs basis functions on the surface of a sphere - -Literature ----------- - -- Nichtkommutative Bildverarbeitung, T. Mendez, p57+ -- Linear Algebra Done Right, S. Axler, p212,221,231,237 -- Introduction to Quantum Mechanics, D. J. Griffith, p201+ -- Seminar Quantenmechanik, A. Müller, p101,106,114,121 -- Introduction to Partial Differential Equations, J. Oliver, p510+ -- Partial Differential Equations in Engineering Problems, K. Miller, p175,190 - -\end{verbatim} - +\input{papers/kugel/introduction} +\input{papers/kugel/preliminaries} +\input{papers/kugel/spherical-harmonics} +\input{papers/kugel/applications} \printbibliography[heading=subbibliography] \end{refsection} +\end{otherlanguage} diff --git a/buch/papers/kugel/preliminaries.tex b/buch/papers/kugel/preliminaries.tex new file mode 100644 index 0000000..03cd421 --- /dev/null +++ b/buch/papers/kugel/preliminaries.tex @@ -0,0 +1,346 @@ +% vim:ts=2 sw=2 et spell tw=78: + +\section{Preliminaries} + +The purpose of this section is to dust off some concepts that will become +important later on. This will enable us to be able to get a richer and more +general view of the topic than just liming ourselves to a specific example. + +\subsection{Vectors and inner product spaces} + +We shall start with a few fundamentals of linear algebra. We will mostly work +with complex numbers, but for the sake of generality we will do what most +textbook do, and write \(\mathbb{K}\) instead of \(\mathbb{C}\) since the +theory works the same when we replace \(\mathbb{K}\) with the real +numbers \(\mathbb{R}\). + +\begin{definition}[Vector space] + \label{kugel:def:vector-space} \nocite{axler_linear_2014} + A \emph{vector space} over a field \(\mathbb{K}\) is a set \(V\) with an + addition on \(V\) and a multiplication on \(V\) such that the following + properties hold: + \begin{enumerate}[(a)] + \item (Commutativity) \(u + v = v + u\) for all \(u, v \in V\); + \item (Associativity) \((u + v) + w = u + (v + w)\) and \((ab)v = a(bv)\) + for all \(u, v, w \in V\) and \(a, b \in \mathbb{K}\); + \item (Additive identity) There exists an element \(0 \in V\) such that + \(v + 0 = v\) for all \(v \in V\); + \item (Additive inverse) For every \(v \in V\), there exists a \(w \in V\) + such that \(v + w = 0\); + \item (Multiplicative identity) \(1 v = v\) for all \(v \in V\); + \item (Distributive properties) \(a(u + v) = au + av\) and \((a + b)v = av + + bv\) for all \(a, b \in \mathbb{K}\) and all \(u,v \in V\). + \end{enumerate} +\end{definition} + +\begin{definition}[Dot product] + \label{kugel:def:dot-product} + In the vector field \(\mathbb{K}^n\) the scalar or dot product between two + vectors \(u, v \in \mathbb{K}^n\) is + \( + u \cdot v + = u_1 \overline{v}_1 + u_2 \overline{v}_2 + \cdots + u_n \overline{v}_n + = \sum_{i=1}^n u_i \overline{v}_i. + \) +\end{definition} + +\texttt{TODO: Text here.} + +\begin{definition}[Span] +\end{definition} + +\texttt{TODO: Text here.} + +\begin{definition}[Linear independence] +\end{definition} + + +\texttt{TODO: Text here.} + +\begin{definition}[Basis] +\end{definition} + +\texttt{TODO: Text here.} + +\begin{definition}[Inner product] + \label{kugel:def:inner-product} \nocite{axler_linear_2014} + The \emph{inner product} on \(V\) is a function that takes each ordered pair + \((u, v)\) of elements of \(V\) to a number \(\langle u, v \rangle \in + \mathbb{K}\) and has the following properties: + \begin{enumerate}[(a)] + \item (Positivity) \(\langle v, v \rangle \geq 0\) for all \(v \in V\); + \item (Definiteness) \(\langle v, v \rangle = 0\) iff \(v = 0\); + \item (Additivity) \( + \langle u + v, w \rangle = + \langle u, w \rangle + \langle v, w \rangle + \) for all \(u, v, w \in V\); + \item (Homogeneity) \( + \langle \lambda u, v \rangle = + \lambda \langle u, v \rangle + \) for all \(\lambda \in \mathbb{K}\) and all \(u, v \in V\); + \item (Conjugate symmetry) + \(\langle u, v \rangle = \overline{\langle v, u \rangle}\) for all + \(u, v \in V\). + \end{enumerate} +\end{definition} + +This newly introduced inner product is thus a generalization of the scalar +product that does not explicitly depend on rows or columns of vectors. This +has the interesting consequence that anything that behaves according to the +rules given in definition \ref{kugel:def:inner-product} \emph{is} an inner +product. For example if we say that the vector space \(V = \mathbb{R}^n\), +then the dot product defined in definition \ref{kugel:def:dot-product} +\( + u \cdot v = u_1 \overline{v}_1 + u_2 \overline{v}_2 + \cdots + u_n \overline{v}_n +\) +is an inner product in \(V\), and the two are said to form an \emph{inner +product space}. + +\begin{definition}[Inner product space] + \nocite{axler_linear_2014} + An inner product space is a vector space \(V\) equipped with an inner + product on \(V\). +\end{definition} + +How about a more interesting example: the set of continuous complex valued +functions on the interval \([0; 1]\) can behave like vectors. Functions can +be added, subtracted, multiplied with scalars, are associative and there is +even the identity element (zero function \(f(x) = 0\)), so we can create an +inner product +\[ + \langle f, g \rangle = \int_0^1 f(x) \overline{g(x)} \, dx, +\] +which will indeed satisfy all of the rules for an inner product (in fact this +is called the Hermitian inner product\nocite{allard_mathematics_2009}). If +this last step sounds too good to be true, you are right, because it is not +quite so simple. The problem that we have swept under the rug here is +convergence, which any student who took an analysis class will know is a +rather hairy question. We will not need to go too much into the details since +formally discussing convergence is definitely beyond the scope of this text, +however, for our purposes we will still need to dig a little deeper for a few +more paragraph. + +\subsection{Convergence} + +In the last section we hinted that we can create ``infinite-dimensional'' +vector spaces using functions as vectors, and inner product spaces by +integrating the product of two functions of said vector space. However, there +is a problem with convergence which twofold: the obvious problem is that the +integral of the inner product may not always converge, while the second is a +bit more subtle and will be discussed later. The inner product that does +not converge is a problem because we want a \emph{norm}. + +\begin{definition}[\(L^2\) Norm] + \nocite{axler_linear_2014} + The norm of a vector \(v\) of an inner product space is a number + denoted as \(\| v \|\) that is computed by \(\| v \| = \sqrt{\langle v, v + \rangle}\). +\end{definition} + +In \(\mathbb{R}^n\) with the dot product (Euclidian space) the norm is the +geometric length of a vector, while in a more general inner product space the +norm can be thought of as a more abstract measure of ``length''. In any case +it is rather important that the expression \(\sqrt{\langle v, v \rangle}\), +which when using functions \(f: \mathbb{R} \to \mathbb{C}\) becomes +\[ + \sqrt{\langle f, f \rangle} = + \sqrt{\int_\mathbb{R} f(x) \overline{f(x)} \, dx} = + \sqrt{\int_\mathbb{R} |f(x)|^2 \, dx}, +\] +always exists. So, to fix this problems we do what mathematicians do best: +make up the solution. Since the integrand under the square root is always the +square of the magnitude, we can just specify that the functions must be +\emph{absolutely square integrable}. To be more compact it is common to just +write \(f \in L^2\), where \(L^2\) denotes the set of absolutely square +integrable functions. + +Now we can tackle the second (much more difficult) problem of convergence +mentioned at the beginning. Using the technical jargon, we need that our inner +product space is what is called a \emph{complete metric space}, which just +means that we can measure distances. For the more motivated readers although +not really necessary we can also give a more formal definition, the others can +skip to the next section. + +\begin{definition}[Metric space] + \nocite{tao_analysis_2016} + A metric space \((X, d)\) is a space \(X\) of objects (called points), + together with a distance function or metric \(d: X \times X \to [0, + +\infty)\), which associates to each pair \(x, y\) of points in \(X\) a + non-negative real number \(d(x, y) \geq 0\). Furthermore, the metric must + satisfy the following four axioms: + \begin{enumerate}[(a)] + \item For any \(x\in X\), we have \(d(x, x) = 0\). + \item (Positivity) For any \emph{distinct} \(x, y \in X\), we have + \(d(x,y) > 0\). + \item (Symmetry) For any \(x,y \in X\), we have \(d(x, y) = d(y, x)\). + \item (Triangle inequality) For any \(x, y, z \in X\) we have + \(d(x, z) \leq d(x, y) + d(y, z)\). + \end{enumerate} +\end{definition} + +As is seen in the definition metric spaces are a very abstract concept and +rely on rather weak statements, which makes them very general. Now, the more +intimidating part is the \emph{completeness} which is defined as follows. + +\begin{definition}[Complete metric space] + \label{kugel:def:complete-metric-space} + A metric space \((X, d)\) is said to be \emph{complete} iff every Cauchy + sequence in \((X, d)\) is convergent in \((X, d)\). +\end{definition} + +To fully explain definition \ref{kugel:def:complete-metric-space} it would +take a few more pages, which would get a bit too heavy. So instead we will +give an informal explanation through an counterexample to get a feeling of +what is actually happening. Cauchy sequences is a rather fancy name for a +sequence for example of numbers that keep changing, but in a such a way that +at some point the change keeps getting smaller (the infamous +\(\varepsilon-\delta\) definition). For example consider the sequence of +numbers +\[ + 1, + 1.4, + 1.41, + 1.414, + 1.4142, + 1.41421, + \ldots +\] +in the metric space \((\mathbb{Q}, d)\) with \(d(x, y) = |x - y|\). Each +element of this sequence can be written with some fraction in \(\mathbb{Q}\), +but in \(\mathbb{R}\) the sequence is converging towards the number +\(\sqrt{2}\). However, \(\sqrt{2} \notin \mathbb{Q}\). Since we can find a +sequence of fractions whose distance's limit is not in \(\mathbb{Q}\), the +metric space \((\mathbb{Q}, d)\) is \emph{not} complete. Conversely, +\((\mathbb{R}, d)\) is a complete metric space since \(\sqrt{2} \in +\mathbb{R}\). + +Of course the analogy above also applies to vectors, i.e. if in an inner +product space \(V\) over a field \(\mathbb{K}\) all sequences of vectors have +a distance that is always in \(\mathbb{K}\), then \(V\) is also a complete +metric space. In the jargon, this particular case is what is known as a +Hilbert space, after the incredibly influential German mathematician David +Hilbert. + +\begin{definition}[Hilbert space] + A Hilbert space is a vector space \(H\) with an inner product \(\langle f, g + \rangle\) and a norm \(\sqrt{\langle f, f \rangle}\) defined such that \(H\) + turns into a complete metric space. +\end{definition} + +\subsection{Orthogonal basis and Fourier series} + +Now we finally have almost everything we need to get into the domain of +Fourier theory from the perspective of linear algebra. However, we still need +to briefly discuss the matter of orthogonality\footnote{See chapter +\ref{buch:chapter:orthogonalitaet} for more on orthogonality.} and +periodicity. Both should be very straightforward and already well known. + +\begin{definition}[Orthogonality and orthonormality] + \label{kugel:def:orthogonality} + In an inner product space \(V\) two vectors \(u, v \in V\) are said to be + \emph{orthogonal} if \(\langle u, v \rangle = 0\). Further, if both \(u\) + and \(v\) are of unit length, i.e. \(\| u \| = 1\) and \(\| v \| = 1\), then + they are said to be ortho\emph{normal}. +\end{definition} + +\begin{definition}[1-periodic function and \(C(\mathbb{R}/\mathbb{Z}; \mathbb{C})\)] + A function is said to be 1-periodic if \(f(x + 1) = f(x)\). The set of + 1-periodic function from the real to the complex + numbers is denoted by \(C(\mathbb{R}/\mathbb{Z}; \mathbb{C})\). +\end{definition} + +In the definition above the notation \(\mathbb{R}/\mathbb{Z}\) was borrowed +from group theory, and is what is known as a quotient group; Not really +relevant for our discussion but still a ``good to know''. More importantly, it +is worth noting that we could have also defined more generally \(L\)-periodic +functions with \(L\in\mathbb{R}\), however, this would introduce a few ugly +\(L\)'s everywhere which are not really necessary (it will always be possible +to extend the theorems to \(\mathbb{R} / L\mathbb{Z}\)). Thus, we will +continue without the \(L\)'s, and to simplify the language unless specified +otherwise ``periodic'' will mean 1-periodic. Having said that, we can +officially begin with the Fourier theory. + +\begin{lemma} + The subset of absolutely square integrable functions in + \(C(\mathbb{R}/\mathbb{Z}; \mathbb{C})\) together with the Hermitian inner + product + \[ + \langle f, g \rangle = \int_{[0; 1)} f(x) \overline{g(x)} \, dx + \] + form a Hilbert space. +\end{lemma} +\begin{proof} + It is not too difficult to show that the functions in \(C(\mathbb{R} / + \mathbb{Z}; \mathbb{C})\) are well behaved and form a vector space. Thus, + what remains is that the norm needs to form a complete metric space. + However, this follows from the fact that we defined the functions to be + absolutely square integrable\footnote{For the curious on why, it is because + \(L^2\) is what is known as a \emph{compact metric space}, and compact + metric spaces are always complete (see \cite{eck_metric_2022, + tao_analysis_2016}). To explain compactness and the relationship between + compactness and completeness is definitely beyond the goals of this text.}. +\end{proof} + +This was probably not a very satisfactory proof since we brushed off a lot of +details by referencing other theorems. However, the main takeaway should be +that we have ``constructed'' this new Hilbert space of functions in a such a +way that from now on we will not have to worry about the details of +convergence. + +\begin{lemma} + \label{kugel:lemma:exp-1d} + The set of functions \(E_n(x) = e^{i2\pi nx}\) on the interval + \([0; 1)\) with \(n \in \mathbb{Z} \) are orthonormal. +\end{lemma} +\begin{proof} + We need to show that \(\langle E_m, E_n \rangle\) equals 1 when \(m = n\) + and zero otherwise. This is a straightforward computation: We start by + unpacking the notation to get + \[ + \langle E_m, E_n \rangle + = \int_0^1 e^{i2\pi mx} e^{- i2\pi nx} \, dx + = \int_0^1 e^{i2\pi (m - n)x} \, dx, + \] + then inside the integrand we can see that when \(m = n\) we have \(e^0 = 1\) and + thus \( \int_0^1 dx = 1, \) while when \(m \neq n\) we can just say that we + have a new non-zero integer + \(k := m - n\) and + \[ + \int_0^1 e^{i2\pi kx} \, dx + = \frac{e^{i2\pi k} - e^{0}}{i2\pi k} + = \frac{1 - 1}{i2\pi k} + = 0 + \] + as desired. \qedhere +\end{proof} + +\begin{definition}[Spectrum] +\end{definition} + +\begin{theorem}[Fourier Theorem] + \[ + \lim_{N \to \infty} \left \| + f(x) - \sum_{n = -N}^N \hat{f}(n) E_n(x) + \right \|_2 = 0 + \] +\end{theorem} + +\begin{lemma} + The set of functions \(E_{m, n}(\xi, \eta) = e^{i2\pi m\xi}e^{i2\pi n\eta}\) + on the square \([0; 1)^2\) with \(m, n \in \mathbb{Z} \) are orthonormal. +\end{lemma} +\begin{proof} + The proof is almost identical to lemma \ref{kugel:lemma:exp-1d}, with the + only difference that the inner product is given by + \[ + \langle E_{m,n}, E_{m', n'} \rangle + = \iint_{[0;1)^2} + E_{m, n}(\xi, \eta) \overline{E_{m', n'} (\xi, \eta)} + \, d\xi d\eta + .\qedhere + \] +\end{proof} + +\subsection{Laplacian operator} + +\subsection{Eigenvalue Problem} diff --git a/buch/papers/kugel/references.bib b/buch/papers/kugel/references.bib index 013da60..b74c5cd 100644 --- a/buch/papers/kugel/references.bib +++ b/buch/papers/kugel/references.bib @@ -1,35 +1,195 @@ -% -% references.bib -- Bibliography file for the paper kugel -% -% (c) 2020 Autor, Hochschule Rapperswil -% - -@online{kugel:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{kugel:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{kugel:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} + +@article{carvalhaes_surface_2015, + title = {The surface Laplacian technique in {EEG}: Theory and methods}, + volume = {97}, + issn = {01678760}, + url = {https://linkinghub.elsevier.com/retrieve/pii/S0167876015001749}, + doi = {10.1016/j.ijpsycho.2015.04.023}, + shorttitle = {The surface Laplacian technique in {EEG}}, + pages = {174--188}, + number = {3}, + journaltitle = {International Journal of Psychophysiology}, + shortjournal = {International Journal of Psychophysiology}, + author = {Carvalhaes, Claudio and de Barros, J. Acacio}, + urldate = {2022-05-16}, + date = {2015-09}, + langid = {english}, + file = {Submitted Version:/Users/npross/Zotero/storage/SN4YUNQC/Carvalhaes and de Barros - 2015 - The surface Laplacian technique in EEG Theory and.pdf:application/pdf}, +} + +@video{minutephysics_better_2021, + title = {A Better Way To Picture Atoms}, + url = {https://www.youtube.com/watch?v=W2Xb2GFK2yc}, + abstract = {Thanks to Google for sponsoring a portion of this video! +Support {MinutePhysics} on Patreon: http://www.patreon.com/minutephysics + +This video is about using Bohmian trajectories to visualize the wavefunctions of hydrogen orbitals, rendered in 3D using custom python code in Blender. + +{REFERENCES} +A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I +David Bohm, Physical Review, Vol 85 No. 2, January 15, 1952 + +Speakable and Unspeakable in Quantum Mechanics +J. S. Bell + +Trajectory construction of Dirac evolution +Peter Holland + +The de Broglie-Bohm Causal Interpretation of Quantum Mechanics and its Application to some Simple Systems by Caroline Colijn + +Bohmian Trajectories as the Foundation of Quantum Mechanics +http://arxiv.org/abs/0912.2666v1 + +The Pilot-Wave Perspective on Quantum Scattering and Tunneling +http://arxiv.org/abs/1210.7265v2 + +A Quantum Potential Description of One-Dimensional Time-Dependent Scattering From Square Barriers and Square Wells +Dewdney, Foundations of Physics, {VoL} 12, No. 1, 1982 + +Link to Patreon Supporters: http://www.minutephysics.com/supporters/ + +{MinutePhysics} is on twitter - @minutephysics +And facebook - http://facebook.com/minutephysics + +Minute Physics provides an energetic and entertaining view of old and new problems in physics -- all in a minute! + +Created by Henry Reich}, + author = {{minutephysics}}, + urldate = {2022-05-19}, + date = {2021-05-19}, +} + +@article{ries_role_2013, + title = {Role of the lateral prefrontal cortex in speech monitoring}, + volume = {7}, + issn = {1662-5161}, + url = {http://journal.frontiersin.org/article/10.3389/fnhum.2013.00703/abstract}, + doi = {10.3389/fnhum.2013.00703}, + journaltitle = {Frontiers in Human Neuroscience}, + shortjournal = {Front. Hum. Neurosci.}, + author = {Riès, Stephanie K. and Xie, Kira and Haaland, Kathleen Y. and Dronkers, Nina F. and Knight, Robert T.}, + urldate = {2022-05-16}, + date = {2013}, + file = {Full Text:/Users/npross/Zotero/storage/W7KTJB8E/Riès et al. - 2013 - Role of the lateral prefrontal cortex in speech mo.pdf:application/pdf}, +} + +@online{saylor_academy_atomic_2012, + title = {Atomic Orbitals and Their Energies}, + url = {http://saylordotorg.github.io/text_general-chemistry-principles-patterns-and-applications-v1.0/s10-05-atomic-orbitals-and-their-ener.html}, + author = {{Saylor Academy}}, + urldate = {2022-05-30}, + date = {2012}, + file = {Atomic Orbitals and Their Energies:/Users/npross/Zotero/storage/LJ8DM3YI/s10-05-atomic-orbitals-and-their-ener.html:text/html}, +} + +@inproceedings{schmitz_using_2012, + location = {Santa Clara, {CA}, {USA}}, + title = {Using spherical harmonics for modeling antenna patterns}, + isbn = {978-1-4577-1155-8 978-1-4577-1153-4 978-1-4577-1154-1}, + url = {http://ieeexplore.ieee.org/document/6175298/}, + doi = {10.1109/RWS.2012.6175298}, + eventtitle = {2012 {IEEE} Radio and Wireless Symposium ({RWS})}, + pages = {155--158}, + booktitle = {2012 {IEEE} Radio and Wireless Symposium}, + publisher = {{IEEE}}, + author = {Schmitz, Arne and Karolski, Thomas and Kobbelt, Leif}, + urldate = {2022-05-16}, + date = {2012-01}, +} + +@online{allard_mathematics_2009, + title = {Mathematics 203-204 - Basic Analysis I-{II}}, + url = {https://services.math.duke.edu/~wka/math204/}, + author = {Allard, William K.}, + urldate = {2022-07-25}, + date = {2009}, + file = {Mathematics 203-204 - Basic Analysis I-II:/Users/npross/Zotero/storage/LJISXBCM/math204.html:text/html}, +} + +@book{olver_introduction_2013, + location = {New York, {NY}}, + title = {Introduction to partial differential equations}, + isbn = {978-3-319-02098-3}, + publisher = {Springer Science+Business Media, {LLC}}, + author = {Olver, Peter J.}, + date = {2013}, +} + +@book{miller_partial_2020, + location = {Mineola, New York}, + title = {Partial differential equations in engineering problems}, + isbn = {978-0-486-84329-2}, + abstract = {"Requiring only an elementary knowledge of ordinary differential equations, this concise text begins by deriving common partial differential equations associated with vibration, heat flow, electricity, and elasticity. The treatment discusses and applies the techniques of Fourier analysis to these equations and extends the discussion to the Fourier integral. Final chapters discuss Legendre, Bessel, and Mathieu functions and the general structure of differential operators"--}, + publisher = {Dover Publications, Inc}, + author = {Miller, Kenneth S.}, + date = {2020}, + keywords = {Differential equations, Partial}, +} + +@book{asmar_complex_2018, + location = {Cham}, + title = {Complex analysis with applications}, + isbn = {978-3-319-94062-5}, + series = {Undergraduate texts in mathematics}, + pagetotal = {494}, + publisher = {Springer}, + author = {Asmar, Nakhlé H. and Grafakos, Loukas}, + date = {2018}, + doi = {10.1007/978-3-319-94063-2}, + file = {Table of Contents PDF:/Users/npross/Zotero/storage/G2Q2RDFU/Asmar and Grafakos - 2018 - Complex analysis with applications.pdf:application/pdf}, +} + +@book{adkins_ordinary_2012, + location = {New York}, + title = {Ordinary differential equations}, + isbn = {978-1-4614-3617-1}, + series = {Undergraduate texts in mathematics}, + pagetotal = {799}, + publisher = {Springer}, + author = {Adkins, William A. and Davidson, Mark G.}, + date = {2012}, + keywords = {Differential equations}, +} + +@book{griffiths_introduction_2015, + title = {Introduction to electrodynamics}, + isbn = {978-93-325-5044-5}, + author = {Griffiths, David J}, + date = {2015}, + note = {{OCLC}: 965197645}, +} + +@book{tao_analysis_2016, + title = {Analysis 2}, + isbn = {978-981-10-1804-6}, + url = {https://doi.org/10.1007/978-981-10-1804-6}, + author = {Tao, Terence}, + urldate = {2022-07-25}, + date = {2016}, + note = {{OCLC}: 965325026}, +} + +@book{axler_linear_2015, + location = {Cham}, + title = {Linear Algebra Done Right}, + isbn = {978-3-319-11079-0 978-3-319-11080-6}, + url = {https://link.springer.com/10.1007/978-3-319-11080-6}, + series = {Undergraduate Texts in Mathematics}, + publisher = {Springer International Publishing}, + author = {Axler, Sheldon}, + urldate = {2022-07-25}, + date = {2015}, + langid = {english}, + doi = {10.1007/978-3-319-11080-6}, + file = {Submitted Version:/Users/npross/Zotero/storage/3Y8MX74N/Axler - 2015 - Linear Algebra Done Right.pdf:application/pdf}, } +@online{eck_metric_2022, + title = {Metric Spaces: Completeness}, + url = {https://math.hws.edu/eck/metric-spaces/completeness.html}, + titleaddon = {Math 331: Foundations of Analysis}, + author = {Eck, David J.}, + urldate = {2022-08-01}, + date = {2022}, + file = {Metric Spaces\: Completeness:/Users/npross/Zotero/storage/5JYEE8NF/completeness.html:text/html}, +}
\ No newline at end of file diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex new file mode 100644 index 0000000..6b23ce5 --- /dev/null +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -0,0 +1,13 @@ +% vim:ts=2 sw=2 et spell: + +\section{Spherical Harmonics} + +\subsection{Eigenvalue Problem in Spherical Coordinates} + +\subsection{Properties} + +\subsection{Recurrence Relations} + +\section{Series Expansions in \(C(S^2)\)} + +\nocite{olver_introduction_2013} diff --git a/buch/papers/laguerre/Makefile b/buch/papers/laguerre/Makefile index 606d7e1..85a1b83 100644 --- a/buch/papers/laguerre/Makefile +++ b/buch/papers/laguerre/Makefile @@ -3,7 +3,42 @@ # # (c) 2020 Prof Dr Andreas Mueller # +IMGFOLDER := images +PRESFOLDER := presentation -images: - @echo "no images to be created in laguerre" +FIGURES := \ + images/targets.pdf \ + images/rel_error_complex.pdf \ + images/estimates.pdf \ + images/integrand.pdf \ + images/integrand_exp.pdf \ + images/laguerre_poly.pdf \ + images/rel_error_mirror.pdf \ + images/rel_error_range.pdf \ + images/rel_error_shifted.pdf \ + images/rel_error_simple.pdf \ + images/gammaplot.pdf +.PHONY: all +all: images presentation + +.PHONY: images +images: $(FIGURES) + +.PHONY: presentation +presentation: $(PRESFOLDER)/presentation.pdf + +images/%.pdf images/%.pgf: scripts/%.py scripts/gamma_approx.py + python3 $< + +images/gammaplot.pdf: images/gammaplot.tex images/gammapaths.tex + cd $(IMGFOLDER) && latexmk -quiet -pdf gammaplot.tex + +$(PRESFOLDER)/%.pdf: $(PRESFOLDER)/%.tex $(FIGURES) + cd $(PRESFOLDER) && latexmk -quiet -pdf $(<F) + +.PHONY: clean +clean: + rm $(FIGURES) + cd $(IMGFOLDER) && latexmk -C + cd $(PRESFOLDER) && latexmk -C
\ No newline at end of file diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index 1eb5034..39b5d6f 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -9,8 +9,5 @@ dependencies-laguerre = \ papers/laguerre/references.bib \ papers/laguerre/definition.tex \ papers/laguerre/eigenschaften.tex \ - papers/laguerre/quadratur.tex \ - papers/laguerre/transformation.tex \ - papers/laguerre/wasserstoff.tex - - + papers/laguerre/quadratur.tex \ + papers/laguerre/gamma.tex diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index 5f6d8bd..61549e0 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -3,46 +3,196 @@ % % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % -\section{Definition -\label{laguerre:section:definition}} -\rhead{Definition} +\section{Herleitung% + % \section{Einleitung + % \section{Definition + \label{laguerre:section:definition}} +\rhead{Definition}% +In einem ersten Schritt möchten wir die Laguerre-Polynome +aus der Laguerre-\-Differentialgleichung herleiten. +Zudem werden wir die Lösung auf die assoziierten Laguerre-Polynome ausweiten. +Im Anschluss soll dann noch die Orthogonalität dieser Polynome bewiesen werden. +\subsection{Assoziierte Laguerre-Differentialgleichung} +Die assoziierte Laguerre-Differentialgleichung ist gegeben durch \begin{align} - x y''(x) + (1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:dgl} +x y''(x) + (\nu + 1 - x) y'(x) + n y(x) += +0 +, \quad +n \in \mathbb{N} +, \quad +x \in \mathbb{R} +\label{laguerre:dgl} +. \end{align} +Spannenderweise wurde die assoziierte Laguerre-Differentialgleichung +zuerst von Yacovlevich Sonine (1849 - 1915) beschrieben, +aber aufgrund ihrer Ähnlichkeit nach Laguerre benannt. +Die klassische Laguerre-Diffentialgleichung erhält man, wenn $\nu = 0$. +\subsection{Potenzreihenansatz% +\label{laguerre:subsection:potenzreihenansatz}} +Hier wird die assoziierte Laguerre-Differentialgleichung verwendet, +weil die Lösung mit derselben Methode berechnet werden kann. +Zusätzlich erhält man aber die Lösung für den allgmeinen Fall. +Wir stellen die Vermutung auf, +dass die Lösungen orthogonale Polynome sind. +Die Orthogonalität der Lösung werden wir im +Abschnitt~\ref{laguerre:subsection:orthogonal} beweisen. +Zur Lösung von \eqref{laguerre:dgl} verwenden wir aufgrund +der getroffenen Vermutungen einen Potenzreihenansatz. +Der Potenzreihenansatz ist gegeben als +% Da wir bereits wissen, +% dass die Lösung orthogonale Polynome sind, +% erscheint dieser Ansatz sinnvoll. +\begin{align*} +y(x) + & = +\sum_{k=0}^\infty a_k x^k +% \\ +. +\end{align*} +Für die 1. und 2. Ableitungen erhalten wir +\begin{align*} +y'(x) + & = +\sum_{k=1}^\infty k a_k x^{k-1} += +\sum_{k=0}^\infty (k+1) a_{k+1} x^k +\\ +y''(x) + & = +\sum_{k=2}^\infty k (k-1) a_k x^{k-2} += +\sum_{k=1}^\infty (k+1) k a_{k+1} x^{k-1} +. +\end{align*} + +\subsection{Lösen der Laguerre-Differentialgleichung} +Setzt man nun den Potenzreihenansatz in +\eqref{laguerre:dgl} +%die Differentialgleichung +ein, +% erhält man +resultiert +\begin{align*} +\sum_{k=1}^\infty (k+1) k a_{k+1} x^k ++ +(\nu + 1)\sum_{k=0}^\infty (k+1) a_{k+1} x^k +- +\sum_{k=0}^\infty k a_k x^k ++ +n \sum_{k=0}^\infty a_k x^k + & = +0 \\ +\sum_{k=1}^\infty +\left[ (k+1) k a_{k+1} + (\nu + 1)(k+1) a_{k+1} - k a_k + n a_k \right] x^k + & = +0. +\end{align*} +Daraus lässt sich die Rekursionsbeziehung \begin{align} - L_n(x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n \\ - k - \end{pmatrix} - x^k - \label{laguerre:polynom} +a_{k+1} + & = +\frac{k-n}{(k+1) (k + \nu + 1)} a_k +\label{laguerre:rekursion} \end{align} - +ableiten. +Für ein konstantes $n$ erhalten wir als Potenzreihenlösung ein Polynom vom Grad +$n$, +denn für $k=n$ wird $a_{n+1} = 0$ und damit auch $a_{n+2}=a_{n+3}=\ldots=0$. +Aus %der Rekursionsbeziehung +\eqref{laguerre:rekursion} ist zudem ersichtlich, +dass $a_0 \neq 0$ beliebig gewählt werden kann. +Wählen wir nun $a_0 = 1$, dann folgt für die Koeffizienten +% $a_1, a_2, a_3$ +\begin{align*} +a_1 += +-\frac{n}{1 \cdot (\nu + 1)} +, & & +a_2 += +\frac{(n-1)n}{1 \cdot 2 \cdot (\nu + 1)(\nu + 2)} +, & & +a_3 += +-\frac{(n-2)(n-1)n}{1 \cdot 2 \cdot 3 \cdot (\nu + 1)(\nu + 2)(\nu + 3)} +\end{align*} +und allgemein +\begin{align*} +k + & \leq +n: + & +a_k + & = +(-1)^k \frac{n!}{(n-k)!} \frac{1}{k!(\nu + 1)_k} += +\frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} +\\ +k & >n: + & +a_k + & = +0 +. +\end{align*} +Die Koeffizienten wechseln also für $k \leq n$ das Vorzeichen. +Somit erhalten wir für $\nu = 0$ die Laguerre-Polynome \begin{align} - x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:generell_dgl} +L_n(x) += +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +\label{laguerre:polynom} \end{align} - +und mit $\nu \in \mathbb{R}$ die assoziierten Laguerre-Polynome \begin{align} - L_n^\alpha (x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n + \alpha \\ - n - k - \end{pmatrix} - x^k - \label{laguerre:polynom} +L_n^\nu(x) += +\sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. +\label{laguerre:allg_polynom} \end{align} +Die Laguerre-Polynome von Grad $0$ bis $7$ sind in +Abbildung~\ref{laguerre:fig:polyeval} dargestellt. +\begin{figure} +\centering +% \scalebox{0.8}{\input{papers/laguerre/images/laguerre_poly.pgf}} +\includegraphics[width=0.9\textwidth]{papers/laguerre/images/laguerre_poly.pdf} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\label{laguerre:fig:polyeval} +\end{figure} + +\subsection{Analytische Fortsetzung} +Durch die analytische Fortsetzung können wir zudem noch die zweite Lösung der +Differentialgleichung erhalten. +Laut \eqref{buch:funktionentheorie:singularitäten:eqn:w1} hat die Lösung +die Form +\begin{align*} +\Xi_n(x) += +L_n(x) \log(x) + \sum_{k=1}^\infty d_k x^k +. +\end{align*} +Eine Herleitung dazu lässt sich im +Abschnitt \ref{buch:funktionentheorie:subsection:dglsing} +im ersten Teil des Buches finden. +Nach einigen aufwändigen Rechnungen, +% die am besten ein Computeralgebrasystem übernimmt, +die den Rahmen dieses Kapitels sprengen würden, +erhalten wir +\begin{align*} +\Xi_n += +L_n(x) \log(x) ++ +\sum_{k=1}^n \frac{(-1)^k}{k!} \binom{n}{k} +(\alpha_{n-k} - \alpha_n - 2 \alpha_k)x^k ++ +(-1)^n \sum_{k=1}^\infty \frac{(k-1)!n!}{((n+k)!)^2} x^{n+k}, +\end{align*} +wobei $\alpha_0 = 0$ und $\alpha_k =\sum_{i=1}^k i^{-1}$, +$\forall k \in \mathbb{N}$. +% https://www.math.kit.edu/iana1/lehre/hm3phys2012w/media/laguerre.pdf +% http://www.physics.okayama-u.ac.jp/jeschke_homepage/E4/kapitel4.pdf diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex index b7597e5..b007c2d 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -3,6 +3,186 @@ % % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % -\section{Eigenschaften -\label{laguerre:section:eigenschaften}} -\rhead{Eigenschaften}
\ No newline at end of file +\subsection{Orthogonalität% +\label{laguerre:subsection:orthogonal}} +\rhead{Orthogonalität}% +Im Abschnitt~\ref{laguerre:subsection:potenzreihenansatz} +haben wir die Behauptung aufgestellt, +dass die Laguerre-Polynome orthogonal sind. +Zu dieser Behauptung möchten wir nun einen Beweis liefern. +% +Um die Orthogonalität von Funktionen zu zeigen, +bieten sich folgende Möglichkeiten an: +\begin{enumerate} +\item Identifizieren der Funktion als Eigenfunktion eines Skalarproduktes +mit einem selbstadjungierten Operator. +Dafür muss aber zuerst bewiesen werden, +dass der verwendete Operator selbstadjungiert ist. +Die Theorie dazu findet sich in den +Abschnitten~\ref{buch:orthogonal:section:orthogonale-polynome-und-dgl} und +\ref{buch:orthogonalitaet:section:bessel}. +\item Umformen der Differentialgleichung in die Form der +Sturm-Liouville-Differentialgleichung, +denn für dieses verallgemeinerte Problem +ist die Orthogonalität bereits bewiesen. +Die Theorie dazu findet sich im Abschnitt~\ref{buch:integrale:subsection:sturm-liouville-problem}. +\end{enumerate} + +% \subsubsection{Plan} +\subsubsection{Idee} +Für den Beweis der Orthogonalität der Laguerre-Polynome möchten +wir den zweiten Ansatz über das Sturm-Liouville-Problem verwenden. +% Dazu müssen wir die Laguerre-Differentialgleichung~\eqref{laguerre:dgl} +% in die Form der Sturm-Liouville-Differentialgleichung bringen. +Allerdings möchten wir nicht die Laguerre-Differentialgleichung +in die richtige Form bringen, +sondern den Laguerre-Operator +\begin{align} +\Lambda += +x \frac{d}{dx^2} + (\nu + 1 -x) \frac{d}{dx} +\label{laguerre:lagop} +. +\end{align} +Da es sich beim Sturm-Liouville-Problem um ein Eigenwertproblem handelt, +kann die Orthogonalität äquivalent über denn Sturm-Liouville-Operator +\begin{align} +S += +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right). +\label{laguerre:slop} +\end{align} +bewiesen werden. +Dazu müssen wir die Operatoren einander gleichsetzen. + +% Wenn wir \eqref{laguerre:dgl} in ein +% Sturm-Liouville-Problem umwandeln können, haben wir bewiesen, dass es sich +% bei den Laguerre-Polynomen um orthogonale Polynome handelt (siehe +% Abschnitt~\ref{buch:integrale:subsection:sturm-liouville-problem}). +% Der Beweis kann äquivalent auch über den Sturm-Liouville-Operator +% \begin{align} +% S +% = +% \frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right). +% \label{laguerre:slop} +% \end{align} +% und den Laguerre-Operator +% \begin{align} +% \Lambda +% = +% x \frac{d}{dx^2} + (\nu + 1 -x) \frac{d}{dx} +% \end{align} +% erhalten werden, +% indem wir diese Operatoren einander gleichsetzen. + +\subsubsection{Umformen in Sturm-Liouville-Operator} +% Aus der Beziehung von +Setzen wir nun +\eqref{laguerre:lagop} und \eqref{laguerre:slop} +einander gleich +\begin{align} +S + & = +\Lambda +\nonumber +\\ +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right) + & = +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx} +\label{laguerre:sl-lag} +, +\end{align} +lässt sich sofort erkennen, dass $q(x) = 0$. +Ausserdem ist ersichtlich, dass $p(x)$ die Differentialgleichung +\begin{align*} +x \frac{dp}{dx} += +(\nu + 1 - x) p +\end{align*} +erfüllen muss. +Durch Separation erhalten wir dann +\begin{align*} +\int \frac{dp}{p} + & = +\int \frac{\nu + 1 - x}{x} \, dx += +\int \frac{\nu + 1}{x} \, dx - \int 1\, dx +\\ +\log p + & = +(\nu + 1)\log x - x + c +\\ +p(x) + & = +C x^{\nu + 1} e^{-x} +. +\end{align*} +Eingefügt in Gleichung~\eqref{laguerre:sl-lag} ergibt sich +\begin{align*} +\frac{C}{w(x)} +\left( +-x^{\nu+1} e^{-x} \frac{d^2}{dx^2} - +(\nu + 1 - x) x^{\nu} e^{-x} \frac{d}{dx} +\right) += +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx}. +\end{align*} +Mittels Koeffizientenvergleich kann nun abgelesen werden, +dass $w(x) = x^\nu e^{-x}$ und $C=-1$. %mit $\nu \geq 0$. +Die Gewichtsfunktion $w(x)$ wächst für $x\rightarrow-\infty$ sehr schnell an. +Ausserdem hat die Gewichtsfunktion $w(x)$ für negative $\nu$ einen Pol bei $x=0$, +daher ist die Laguerre-Gewichtsfunktion nur für den +Definitionsbereich $(0, \infty)$ geeignet. + +\subsubsection{Randbedingungen} +Bleibt nur noch sicherzustellen, dass die Randbedingungen +\begin{align} +k_0 y(0) + h_0 p(0)y'(0) + & = +0 +\label{laguerre:sllag_randa} +\\ +k_\infty y(\infty) + h_\infty p(\infty) y'(\infty) + & = +0 +\label{laguerre:sllag_randb} +\end{align} +mit $|k_i|^2 + |h_i|^2 \neq 0,\,\forall i \in \{0, \infty\}$, erfüllt sind. +% +Am linken Rand \eqref{laguerre:sllag_randa} kann $y(0) = 1$, $k_0 = 0$ und +$h_0 = 1$ verwendet werden, +was auch die Laguerre-Polynome ergeben haben. + +Für den rechten Rand ist die Bedingung \eqref{laguerre:sllag_randb} +\begin{align*} +\lim_{x \rightarrow \infty} p(x) y'(x) + & = +\lim_{x \rightarrow \infty} -x^{\nu + 1} e^{-x} y'(x) += +0 +\end{align*} +für beliebige Polynomlösungen erfüllt für $k_\infty=0$ und $h_\infty=1$. + +% Somit können wir schlussfolgern: +\begin{satz} +Die Laguerre-Polynome %($\nu=0$) +\eqref{laguerre:polynom} +% \begin{align*} +% L_n(x) +% = +% \sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +% \end{align*} +sind orthognale Polynome bezüglich des Skalarproduktes +im Intervall~$(0, \infty)$ mit der Gewichts\-funktion~$w(x)=e^{-x}$. +\end{satz} + +\begin{satz} +Die assoziierten Laguerre-Polynome \eqref{laguerre:allg_polynom} +% \begin{align*} +% L_n^\nu(x) +% = +% \sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. +% \end{align*} +sind orthogonale Polynome bezüglich des Skalarproduktes +im Intervall~$(0, \infty)$ mit der Gewichts\-funktion~$w(x)=x^\nu e^{-x}$. +\end{satz} diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex new file mode 100644 index 0000000..0cf17b9 --- /dev/null +++ b/buch/papers/laguerre/gamma.tex @@ -0,0 +1,607 @@ +% +% gamma.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Anwendung: Berechnung der + Gamma-Funktion% + \label{laguerre:section:quad-gamma}} +\rhead{Approximation der Gamma-Funktion}% +Die Gauss-Laguerre-Quadratur kann nun verwendet werden, +um exponentiell abfallende Funktionen im Definitionsbereich~$(0, \infty)$ +zu berechnen. +Dabei bietet sich zum Beispiel die Gamma-Funktion hervorragend an, +wie wir in den folgenden Abschnitten sehen werden. + +Im ersten Abschnitt~\ref{laguerre:subsection:gamma} möchten wir noch einmal +die wichtigsten Eigenschaften der Gamma-Funktion betrachten, +bevor wir dann im zweiten Abschnitt~\ref{laguerre:subsection:gauss-lag-gamma} +diese Eigenschaften nutzen werden, +damit wir die Gauss-Laguerre-Quadratur für die Gamma-Funktion +markant verbessern können. +% damit wir sie dann in einem nächsten Schritt verwenden können, +% um unsere Approximationsmethode zu verbessern +% Im zweiten Abschnitt~\ref{laguerre:subsection:gauss-lag-gamma} +% wenden wir dann die Gauss-Laguerre-Quadratur auf die Gamma-Funktion und +% erweitern die Methode + +\subsection{Gamma-Funktion% +\label{laguerre:subsection:gamma}} +Die Gamma-Funktion ist eine Erweiterung der Fakultät auf die reale und komplexe +Zahlenmenge. +Mehr Informationen zur Gamma-Funktion lassen sich im +Abschnitt~\ref{buch:rekursion:section:gamma} finden. +Die Definition~\ref{buch:rekursion:def:gamma} beschreibt die Gamma-Funktion als +Integral der Form +\begin{align} +\Gamma(z) + & = +\int_0^\infty x^{z-1} e^{-x} \, dx +, +\quad +\text{wobei } \operatorname{Re}(z) > 0 +\label{laguerre:gamma} +. +\end{align} +Der Term $e^{-x}$ im Integranden und der Integrationsbereich erfüllen +genau die Bedingungen der Gauss-Laguerre-Integration. +% Der Term $e^{-t}$ ist genau die Gewichtsfunktion der Laguerre-Integration und +% der Definitionsbereich passt ebenfalls genau für dieses Verfahren. +Weiter zu erwähnen ist, dass für die assoziierte Gauss-Laguerre-Integration die +Gewichtsfunktion $x^\nu e^{-x}$ exakt dem Integranden +für $\nu = z - 1$ entspricht. + +\subsubsection{Funktionalgleichung} +Die Gamma-Funktion besitzt die gleiche Rekursionsbeziehung wie die Fakultät, +nämlich +\begin{align} +\Gamma(z+1) += +z \Gamma(z) +\quad +\text{mit } +\Gamma(1) += +1 +. +\label{laguerre:gamma_funktional} +\end{align} + +\subsubsection{Reflektionsformel} +Die Reflektionsformel +\begin{align} +\Gamma(z) \Gamma(1 - z) += +\frac{\pi}{\sin \pi z} +,\quad +\text{für } +z \notin \mathbb{Z} +\label{laguerre:gamma_refform} +\end{align} +stellt eine Beziehung zwischen den zwei Punkten, +die aus der Spiegelung an der Geraden $\real z = 1/2$ hervorgehen, +her. +Dadurch lassen Werte der Gamma-Funktion sich für $z$ in der rechten Halbebene +leicht in die linke Halbebene übersetzen und umgekehrt. + +\subsection{Berechnung mittels +Gauss-Laguerre-Quadratur% +\label{laguerre:subsection:gauss-lag-gamma}} +In den vorherigen Abschnitten haben wir gesehen, +dass sich die Gamma-Funktion bestens für die Gauss-Laguerre-Quadratur +\begin{align*} +\int_0^\infty x^{z-1} e^{-x} \, dx += +\int_0^\infty f(x) w(x) \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\end{align*} +eignet. +Nun bieten sich uns zwei Optionen, +diese zu berechnen: +\begin{enumerate} +\item Wir verwenden die assoziierten Laguerre-Polynome $L_n^\nu(x)$ mit +$w(x) = x^\nu e^{-x}$, $\nu = z - 1$ und $f(x) = 1$. +% $f(x)=1$. +% \begin{align*} +% \int_0^\infty x^{z-1} e^{-x} \, dx +% = +% \int_0^\infty f(x) w(x) \, dx +% \quad +% \text{mit } +% w(x) +% = +% x^\nu e^{-x}, +% \nu +% = +% z - 1 +% \text{ und } +% f(x) = 1 +% . +% \end{align*} +\item Wir verwenden die Laguerre-Polynome $L_n(x)$ mit +$w(x) = e^{-x}$ und $f(x) = x^{z - 1}$. +% $f(x)=x^{z-1}$ +% \begin{align*} +% \int_0^\infty x^{z-1} e^{-x} \, dx +% = +% \int_0^\infty f(x) w(x) \, dx +% \quad +% \text{mit } +% w(x) +% = +% e^{-x} +% \text{ und } +% f(x) = x^{z - 1} +% . +% \end{align*} +\end{enumerate} +Die erste Variante wäre optimal auf das Problem angepasst, +allerdings müssten die Gewichte und Nullstellen für jedes $z$ +neu berechnet werden, +da sie per Definition von $z$ abhängen. +Dazu kommt, +dass die Berechnung der Gewichte $A_i$ nach +\cite{laguerre:Cassity1965AbcissasCA} +\begin{align*} +A_i += +\frac{ +\Gamma(n) \Gamma(n+\nu) +} +{ +(n+\nu) +\left[L_{n-1}^{\nu}(x_i)\right]^2 +} +\end{align*} +Evaluationen der Gamma-Funktion benötigen. +Somit ist diese Methode eindeutig nicht geeignet für unser Vorhaben. + +Bei der zweiten Variante benötigen wir keine Neuberechung der Gewichte +und Nullstellen für unterschiedliche $z$. +In \eqref{laguerre:quadratur_gewichte} ist ersichtlich, +dass die Gewichte einfach zu berechnen sind. +Auch die Nullstellen können vorgängig, +mittels eines geeigneten Verfahrens, +aus den Polynomen bestimmt werden. +Als problematisch könnte sich höchstens +die zu integrierende Funktion $f(x)=x^{z-1}$ für $|z| \gg 0$ erweisen. +Somit entscheiden wir uns aufgrund der vorherigen Punkte, +die zweite Variante weiterzuverfolgen. + +\subsubsection{Direkter Ansatz} +% +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_simple.pgf} +\includegraphics{papers/laguerre/images/rel_error_simple.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des direkten Ansatzes +für verschiedene reelle Werte von $z$ und Grade $n$ der +Laguerre-Polynome}% +\label{laguerre:fig:rel_error_simple} +\end{figure} +%. +Wenden wir also die Gauss-Laguerre-Quadratur aus +\eqref{laguerre:laguerrequadratur} auf die Gamma-Funktion +\eqref{laguerre:gamma} an, +ergibt sich +\begin{align} +\Gamma(z) +\approx +\sum_{i=1}^n x_i^{z-1} A_i +\label{laguerre:naive_lag} +. +\end{align} +Bevor wir die Gauss-Laguerre-Quadratur anwenden, +möchten wir als ersten Schritt eine Fehlerabschätzung durchführen. +Für den Fehlerterm \eqref{laguerre:lag_error} wird die $2n$-te Ableitung +der zu integrierenden Funktion $f(\xi)$ benötigt. +Für das Integral der Gamma-Funktion ergibt sich also +\begin{align*} +\frac{d^{2n}}{d\xi^{2n}} f(\xi) + & = +\frac{d^{2n}}{d\xi^{2n}} \xi^{z-1} +\\ + & = +(z - 2n)_{2n} \xi^{z - 2n - 1} +. +\end{align*} +Eingesetzt im Fehlerterm \eqref{laguerre:lag_error} resultiert +\begin{align} +R_n += +(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z-2n-1} +, +\label{laguerre:gamma_err_simple} +\end{align} +wobei $\xi$ ein geeigneter Wert im Intervall $(0, \infty)$ ist +und $n$ der Grad des verwendeten Laguerre-Polynoms. +Eine Fehlerabschätzung mit dem Fehlerterm stellt sich als unnütz heraus, +da $R_n$ für $z < 2n - 1$ bei $\xi \rightarrow 0$ eine Singularität aufweist +und für $z > 2n - 1$ bei $\xi \rightarrow \infty$ divergiert. +Nur für den unwahrscheinlichen Fall $ z = 2n - 1$ +wäre eine Fehlerabschätzung plausibel. + +Wenden wir nun also direkt die Gauss-Laguerre-Quadratur +auf die Gamma-Funktion an. +Dazu benötigen wir die Gewichte nach +\eqref{laguerre:quadratur_gewichte} +und als Stützstellen die Nullstellen des Laguerre-Polynomes $L_n$. +Evaluieren wir den relativen Fehler unserer Approximation zeigt sich ein +Bild wie in Abbildung~\ref{laguerre:fig:rel_error_simple}. +Man kann sehen, +wie der relative Fehler Nullstellen aufweist für ganzzahlige $z \leq 2n$. +Laut der Theorie der Gauss-Quadratur ist das auch zu erwarten, +da die Approximation via Gauss-Quadratur +exakt ist für zu integrierende Polynome mit Grad $\leq 2n-1$ und +der Integrand $x^{z-1}$ wird für $z \in \mathbb{N} \setminus \{0\}$ +zu einem Polynom . +% Hinzukommt, dass zudem von $z$ noch $1$ abgezogen wird im Exponenten. +Es ist ersichtlich, +dass sich für den Polynomgrad $n$ ein Intervall gibt, +in dem der relative Fehler minimal ist. +Links steigt der relative Fehler besonders stark an, +während er auf der rechten Seite zu konvergieren scheint. + +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_mirror.pgf} +\includegraphics{papers/laguerre/images/rel_error_mirror.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit Spiegelung negativer Realwerte +für verschiedene reelle Werte von $z$ und Grade $n$ der Laguerre-Polynome} +\label{laguerre:fig:rel_error_mirror} +\end{figure} + +Um die linke Hälfte in den Griff zu bekommen, +könnten wir die Reflektionsformel der Gamma-Funktion verwenden. +Spiegelt man nun $z$ mit negativem Realteil mittels der Reflektionsformel, +ergibt sich ein stabilerer Fehler in der linken Hälfte, +wie in Abbildung~\ref{laguerre:fig:rel_error_mirror}. +Die Spiegelung bringt nur für wenige Werte einen, +für praktische Anwendungen geeigneten, +relativen Fehler. +Wie wir aber in Abbildung~\ref{laguerre:fig:rel_error_simple} sehen konnten, +gibt es für jeden Polynomgrad $n$ ein Intervall $[a(n), a(n) + 1]$, +$a(n) \in \mathbb{Z}$, +in welchem der relative Fehler minimal ist. +Die Funktionalgleichung der Gamma-Funktion \eqref{laguerre:gamma_funktional} +könnte uns hier helfen, +das Problem in den Griff zu bekommen. + +\subsubsection{Analyse des Integranden} +Wie wir im vorherigen Abschnitt gesehen haben, +scheint der Integrand problematisch. +Darum möchten wir ihn jetzt analysieren, +damit wir ihn besser verstehen können. +Dies sollte es uns ermöglichen, +anschliessend geeignete Gegenmassnahmen zu entwickeln. + +% Dieser Abschnitt soll eine grafisches Verständnis dafür schaffen, +% wieso der Integrand so problematisch ist. +% Was das heisst sollte in Abbildung~\ref{laguerre:fig:integrand} +% und Abbildung~\ref{laguerre:fig:integrand_exp} grafisch dargestellt werden. +\begin{figure} +\centering +% \input{papers/laguerre/images/integrand.pgf} +\includegraphics{papers/laguerre/images/integrand.pdf} +%\vspace{-12pt} +\caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\label{laguerre:fig:integrand} +\end{figure} + +In Abbildung~\ref{laguerre:fig:integrand} ist der Integrand $x^z$ für +unterschiedliche Werte von $z$ dargestellt. +Dies entspricht der zu integrierenden Funktion $f(x)$ +der Gauss-Laguerre-Quadratur für die Gamma-Funktion. +Man erkennt, +dass für kleine $z$ sich ein singulärer Integrand ergibt +und auch für grosse $z$ wächst der Integrand sehr schnell an. +Das heisst, +die Ableitungen im Fehlerterm divergieren noch schneller +und das wirkt sich negativ auf die Genauigkeit der Approximation aus. +Somit lässt sich hier sagen, +dass kleine Exponenten um $0$ genauere Resultate liefern sollten. + +\begin{figure} +\centering +% \input{papers/laguerre/images/integrand_exp.pgf} +\includegraphics{papers/laguerre/images/integrand_exp.pdf} +%\vspace{-12pt} +\caption{Integrand $x^z e^{-x}$ mit unterschiedlichen Werten für $z$} +\label{laguerre:fig:integrand_exp} +\end{figure} + +In Abbildung~\ref{laguerre:fig:integrand_exp} fügen wir +die Dämpfung der Gewichtsfunktion $w(x)$ +der Gauss-Laguerre-Quadratur wieder hinzu +und erhalten so wieder den kompletten Integranden $x^{z} e^{-x}$ +der Gamma-Funktion. +Für negative $z$ ergeben sich immer noch Singularitäten, +wenn $x \rightarrow 0$. +Um $x = 1$ wächst der Term $x^z$ für positive $z$ +schneller als die Dämpfung $e^{-x}$, +aber für $x \rightarrow \infty$ geht der Integrand gegen $0$. +Das führt zu glockenförmigen Kurven, +die für grosse Exponenten $z$ nach der Stelle $x=1$ schnell anwachsen. +Zu grosse Exponenten $z$ sind also immer noch problematisch. +Kleine positive $z$ scheinen nun aber auch zulässig zu sein. +Damit formulieren wir die Vermutung, +dass $a(n)$, +welches das Intervall $[a(n), a(n) + 1]$ definiert, +in dem der relative Fehler minimal ist, +grösser als $0$ und kleiner als $2n-1$ ist. + +\subsubsection{Ansatz mit Verschiebungsterm} +% Mittels der Funktionalgleichung \eqref{laguerre:gamma_funktional} +% kann der Wert von $\Gamma(z)$ im Intervall $z \in [a,a+1]$, +% in dem der relative Fehler minimal ist, +% evaluiert werden und dann mit der Funktionalgleichung zurückverschoben werden. +Nun stellt sich die Frage, +ob die Approximation mittels Gauss-Laguerre-Quadratur verbessert werden kann, +wenn man das Problem in einem geeigneten Intervall $[a(n), a(n)+1]$, +$a(n) \in \mathbb{Z}$, +evaluiert und dann mit der +Funktionalgleichung \eqref{laguerre:gamma_funktional} zurückverschiebt. +Für dieses Vorhaben führen wir einen Verschiebungsterm $m \in \mathbb{Z}$ ein. +Passen wir \eqref{laguerre:naive_lag} +mit dem Verschiebungsterm $m$ +%,der $z$ and die Stelle $z_m = z + m$ verschiebt, +an, +ergibt sich +\begin{align} +\Gamma(z) +\approx +s(z, m) \sum_{i=1}^n x_i^{z + m - 1} A_i +% && +% \text{mit } +% s(z, m) +% = +% \begin{cases} +% \displaystyle +% \frac{1}{(z - m)_m} & \text{wenn } m \geq 0\\ +% (z + m)_{-m} & \text{wenn } m < 0 +% \end{cases} +% . +\label{laguerre:shifted_lag} +\end{align} +mit +\begin{align*} +s(z, m) += +\begin{cases} +\displaystyle +\frac{1}{(z)_m} & \text{wenn } m \geq 0 \\ +(z + m)_{-m} & \text{wenn } m < 0 +\end{cases} +. +\end{align*} + +\subsubsection{Finden der optimalen Berechnungsstelle} +Um die optimale Stelle $z^*(n) \in \left[a(n), a(n) + 1\right]$, +$z^*(n) \in \mathbb{R}$, +zu finden, +erweitern wir denn Fehlerterm \eqref{laguerre:gamma_err_simple} +und erhalten +\begin{align} +R_{n,m}(\xi) += +s(z, m) \cdot (z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z + m - 2n - 1} +,\quad +\text{für } +\xi \in (0, \infty) +\label{laguerre:gamma_err_shifted} +. +\end{align} +% +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/targets.pdf} +% %\vspace{-12pt} +\caption{$m^*$ in Abhängigkeit von $z$ und $n$} +\label{laguerre:fig:targets} +\end{figure} +% +Daraus formulieren wir das Optimierungproblem +\begin{align*} +m^* += +\operatorname*{argmin}_m \max_\xi R_{n,m}(\xi) +. +\end{align*} +Allerdings ist die Funktion $R_{n,m}(\xi)$ unbeschränkt und +hat die gleichen Probleme wie die Fehlerabschätzung des direkten Ansatzes. +Dazu müssten wir $\xi$ versuchen, +unter Kontrolle zu bringen, +was ein äussersts schwieriges Unterfangen zu sein scheint. +Da die Gauss-Quadratur aber sowieso +nur wirklich praktisch sinnvoll für kleine $n$ ist, +können die Intervalle $[a(n), a(n)+1]$ empirisch gesucht werden. + +Wir bestimmen nun die optimalen Verschiebungsterme empirisch +für $n = 1,\ldots, 12$ im Intervall $z \in (0, 1)$, +da $z$ sowieso mit den Term $m$ verschoben wird, +reicht es, +die $m^*$ nur in diesem Intervall zu analysieren. +In Abbildung~\ref{laguerre:fig:targets} sind die empirisch bestimmten $m^*$ +abhängig von $z$ und $n$ dargestellt. +In $n$-Richtung lässt sich eine klare lineare Abhängigkeit erkennen und +die Beziehung zu $z$ ist negativ, +d.h. wenn $z$ grösser ist, wird $m^*$ kleiner. +Allerdings ist die genaue Beziehung zu $z$ +aus dieser Grafik nicht offensichtlich, +aber sie scheint regelmässig zu sein. +Es lässt die Vermutung aufkommen, +dass die Restriktion von $m^* \in \mathbb{Z}$ Rundungsprobleme verursacht. +Wir versuchen, +dieses Problem via lineare Regression und geeignete Rundung zu beheben. +Den linearen Regressor +\begin{align*} +\hat{m} += +\alpha n + \beta +\end{align*} +machen wir nur abhängig von $n$, +in dem wir den Mittelwert $\overline{m}$ von $m^*$ über $z$ berechnen. + +\begin{figure} +\centering +% \input{papers/laguerre/images/estimates.pgf} +\includegraphics{papers/laguerre/images/estimates.pdf} +%\vspace{-12pt} +\caption{Schätzung Mittelwert von $m$ und Fehler} +\label{laguerre:fig:schaetzung} +\end{figure} + +In Abbildung~\ref{laguerre:fig:schaetzung} sind die Resultate +der linearen Regression aufgezeigt mit $\alpha = 1.34154$ und $\beta = +0.848786$. +Die lineare Beziehung ist ganz klar ersichtlich und der Fit scheint zu genügen. +Der optimale Verschiebungsterm kann nun mit +\begin{align*} +m^* +\approx +\lceil \hat{m} - z \rceil += +\lceil \alpha n + \beta - z \rceil +\end{align*} +% kann nun mit dem linearen Regressor und $z$ +gefunden werden. + +\subsubsection{Evaluation des Schätzers} +In einem ersten Schritt möchten wir analysieren, +wie gut die Abschätzung des optimalen Verschiebungsterms ist. +Dazu bestimmen wir den relativen Fehler für verschiedene Verschiebungsterme $m$ +in der Nähe von $m^*$ bei gegebenem Polynomgrad $n = 8$ für $z \in (0, 1)$. +In Abbildung~\ref{laguerre:fig:rel_error_shifted} sind die relativen Fehler +der Approximation dargestellt. +Man kann deutlich sehen, +dass der relative Fehler anwächst, +je weiter der Verschiebungsterm vom idealen Wert abweicht. +Zudem scheint der Schätzer den optimalen Verschiebungsterm gut zu bestimmen, +da der Schätzer zuerst der grünen Linie folgt und +dann beim Übergang auf die orange Linie wechselt. +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_shifted.pgf} +\includegraphics{papers/laguerre/images/rel_error_shifted.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit Verschiebungsterm +für verschiedene reelle Werte von $z$ und Verschiebungsterme $m$. +Das verwendete Laguerre-Polynom besitzt den Grad $n = 8$. +$m^*$ bezeichnet hier den optimalen Verschiebungsterm.} +\label{laguerre:fig:rel_error_shifted} +\end{figure} + +\subsubsection{Resultate} +Das Verfahren scheint für den Grad $n=8$ und $z \in (0,1)$ gut zu funktioneren. +Es stellt sich nun die Frage, +wie der relative Fehler sich für verschiedene $z$ und $n$ verhält. +In Abbildung~\ref{laguerre:fig:rel_error_range} sind die relativen Fehler für +unterschiedliche $n$ dargestellt. +Der relative Fehler scheint immer noch Nullstellen aufzuweisen +für ganzzahlige $z$. +Durch das Verschieben ergibt sich jetzt aber, +wie zu erwarten war, +ein periodischer relativer Fehler mit einer Periodendauer von $1$. +Zudem lässt sich erkennen, +dass der Fehler abhängig von der Ordnung $n$ +des verwendeten Laguerre-Polynoms ist. +Wenn der Grad $n$ um $1$ erhöht wird, +verbessert sich die Genauigkeit des Resultats um etwa eine signifikante Stelle. + +In Abbildung~\ref{laguerre:fig:rel_error_complex} +ist der Betrag des relativen Fehlers in der komplexen Ebene dargestellt. +Je stärker der Imaginäranteil von $z$ von $0$ abweicht, +umso schlechter wird die Genauigkeit der Approximation. +Das erstaunt nicht weiter, +da die Gauss-Quadratur eigentlich nur für reelle Zahlen definiert ist. +Wenn der Imaginäranteil von $z$ ungefähr $0$ ist, +lässt sich das gleiche Bild beobachten wie in +Abbildung~\ref{laguerre:fig:rel_error_range}. + +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_range.pgf} +\includegraphics{papers/laguerre/images/rel_error_range.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit optimalen Verschiebungsterm +für verschiedene reelle Werte von $z$ und Laguerre-Polynome vom Grad $n$} +\label{laguerre:fig:rel_error_range} +\end{figure} + +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/rel_error_complex.pdf} +%\vspace{-12pt} +\caption{Absolutwert des relativen Fehlers in der komplexen Ebene} +\label{laguerre:fig:rel_error_complex} +\end{figure} + +\subsubsection{Vergleich mit Lanczos-Methode} +Nun stellt sich die Frage, +wie das in diesem Abschnitt beschriebene Approximationsverfahren +der Gamma-Funktion sich gegenüber den üblichen Approximationsverfahren schlägt. +Eine häufig verwendete Methode ist die Lanczos-Approximation, +welche gegeben ist durch +\begin{align} +\Gamma(z + 1) +\approx +\sqrt{2\pi} \left( z + \sigma + \frac{1}{2} \right)^{z + 1/2} +e^{-(z + \sigma + 1/2)} \sum_{k=0}^n g_k H_k(z) +, +\end{align} +wobei +\begin{align*} +g_k = \frac{e^\sigma \varepsilon_k (-1)^k}{\sqrt{2\pi}} +\sum_{r=0}^k (-1)^r \, \binom{k}{r} \, (k)_r +\left( \frac{e}{r + \sigma + \frac{1}{2}}\right)^{r + 1/2} +, +\end{align*} +\begin{align*} +\varepsilon_k += +\begin{cases} +1 & \text{für } k = 0 \\ +2 & \text{sonst} +\end{cases} +\quad \text{und}\quad +H_k(z) += +\frac{(-1)^k (-z)_k}{(z+1)_k} +\end{align*} +mit $H_0 = 1$ und $\sum_0^n g_k = 1$ (siehe \cite{laguerre:lanczos}). +Diese Methode wurde zum Beispiel in +{\em GNU Scientific Library}, {\em Boost}, {\em CPython} und +{\em musl} implementiert. +Diese Methode erreicht für $n = 7$ typischerweise eine Genauigkeit von $13$ +korrekten, signifikanten Stellen für reelle Argumente. +Zum Vergleich: die vorgestellte Methode erreicht für $n = 7$ +eine minimale Genauigkeit von $6$ korrekten, signifikanten Stellen +für reelle Argumente. + +\subsubsection{Fazit} +% Das Resultat ist etwas enttäuschend, +Die Genauigkeit der vorgestellten Methode schneidet somit schlechter ab +als die Lanczos-Methode. +Dieser Erkenntnis kommt nicht ganz unerwartet, +% aber nicht unerwartet, +da die Lanczos-Methode spezifisch auf dieses Problem zugeschnitten ist und +unsere Methode eine erweiterte allgemeine Methode ist. +Allerdings besticht die vorgestellte Methode +durch ihre stark reduzierte Komplexität. % und Rechenaufwand. +% Was die Komplexität der Berechnungen im Betrieb angeht, +% ist die Gauss-Laguerre-Quadratur wesentlich ressourcensparender, +% weil sie nur aus $n$ Funktionsevaluationen, +% wenigen Multiplikationen und Additionen besteht. +Was den Rechenaufwand angeht, +benötigt die vorgestellte Methode, +für eine Genauigkeit von $n-1$ signifikanten Stellen, +nur $n$ Funktionsevaluationen +und wenige zusätzliche Multiplikationen und Additionen. +Demzufolge könnte diese Methode Anwendung in Systemen mit wenig Rechenleistung +und/oder knappen Energieressourcen finden. +Die vorgestellte Methode ist ein weiteres Beispiel dafür, +wie Verfahren durch die Kenntnis der Eigenschaften einer Funktion +verbessert werden können.
\ No newline at end of file diff --git a/buch/papers/laguerre/images/estimates.pdf b/buch/papers/laguerre/images/estimates.pdf Binary files differnew file mode 100644 index 0000000..fe48f47 --- /dev/null +++ b/buch/papers/laguerre/images/estimates.pdf diff --git a/buch/papers/laguerre/images/gammapaths.tex b/buch/papers/laguerre/images/gammapaths.tex new file mode 100644 index 0000000..efa0863 --- /dev/null +++ b/buch/papers/laguerre/images/gammapaths.tex @@ -0,0 +1,1024 @@ +\def\gammaplus{({\dx*0.0190},{\dy*52.0728}) + -- ({\dx*0.0200},{\dy*49.4422}) + -- ({\dx*0.0400},{\dy*24.4610}) + -- ({\dx*0.0600},{\dy*16.1457}) + -- ({\dx*0.0800},{\dy*11.9966}) + -- ({\dx*0.1000},{\dy*9.5135}) + -- ({\dx*0.1200},{\dy*7.8633}) + -- ({\dx*0.1400},{\dy*6.6887}) + -- ({\dx*0.1600},{\dy*5.8113}) + -- ({\dx*0.1800},{\dy*5.1318}) + -- ({\dx*0.2000},{\dy*4.5908}) + -- ({\dx*0.2200},{\dy*4.1505}) + -- ({\dx*0.2400},{\dy*3.7855}) + -- ({\dx*0.2600},{\dy*3.4785}) + -- ({\dx*0.2800},{\dy*3.2169}) + -- ({\dx*0.3000},{\dy*2.9916}) + -- ({\dx*0.3200},{\dy*2.7958}) + -- ({\dx*0.3400},{\dy*2.6242}) + -- ({\dx*0.3600},{\dy*2.4727}) + -- ({\dx*0.3800},{\dy*2.3383}) + -- ({\dx*0.4000},{\dy*2.2182}) + -- ({\dx*0.4200},{\dy*2.1104}) + -- ({\dx*0.4400},{\dy*2.0132}) + -- ({\dx*0.4600},{\dy*1.9252}) + -- ({\dx*0.4800},{\dy*1.8453}) + -- ({\dx*0.5000},{\dy*1.7725}) + -- ({\dx*0.5200},{\dy*1.7058}) + -- ({\dx*0.5400},{\dy*1.6448}) + -- ({\dx*0.5600},{\dy*1.5886}) + -- ({\dx*0.5800},{\dy*1.5369}) + -- ({\dx*0.6000},{\dy*1.4892}) + -- ({\dx*0.6200},{\dy*1.4450}) + -- ({\dx*0.6400},{\dy*1.4041}) + -- ({\dx*0.6600},{\dy*1.3662}) + -- ({\dx*0.6800},{\dy*1.3309}) + -- ({\dx*0.7000},{\dy*1.2981}) + -- ({\dx*0.7200},{\dy*1.2675}) + -- ({\dx*0.7400},{\dy*1.2390}) + -- ({\dx*0.7600},{\dy*1.2123}) + -- ({\dx*0.7800},{\dy*1.1875}) + -- ({\dx*0.8000},{\dy*1.1642}) + -- ({\dx*0.8200},{\dy*1.1425}) + -- ({\dx*0.8400},{\dy*1.1222}) + -- ({\dx*0.8600},{\dy*1.1031}) + -- ({\dx*0.8800},{\dy*1.0853}) + -- ({\dx*0.9000},{\dy*1.0686}) + -- ({\dx*0.9200},{\dy*1.0530}) + -- ({\dx*0.9400},{\dy*1.0384}) + -- ({\dx*0.9600},{\dy*1.0247}) + -- ({\dx*0.9800},{\dy*1.0119}) + -- ({\dx*1.0000},{\dy*1.0000}) + -- ({\dx*1.0200},{\dy*0.9888}) + -- ({\dx*1.0400},{\dy*0.9784}) + -- ({\dx*1.0600},{\dy*0.9687}) + -- ({\dx*1.0800},{\dy*0.9597}) + -- ({\dx*1.1000},{\dy*0.9514}) + -- ({\dx*1.1200},{\dy*0.9436}) + -- ({\dx*1.1400},{\dy*0.9364}) + -- ({\dx*1.1600},{\dy*0.9298}) + -- ({\dx*1.1800},{\dy*0.9237}) + -- ({\dx*1.2000},{\dy*0.9182}) + -- ({\dx*1.2200},{\dy*0.9131}) + -- ({\dx*1.2400},{\dy*0.9085}) + -- ({\dx*1.2600},{\dy*0.9044}) + -- ({\dx*1.2800},{\dy*0.9007}) + -- ({\dx*1.3000},{\dy*0.8975}) + -- ({\dx*1.3200},{\dy*0.8946}) + -- ({\dx*1.3400},{\dy*0.8922}) + -- ({\dx*1.3600},{\dy*0.8902}) + -- ({\dx*1.3800},{\dy*0.8885}) + -- ({\dx*1.4000},{\dy*0.8873}) + -- ({\dx*1.4200},{\dy*0.8864}) + -- ({\dx*1.4400},{\dy*0.8858}) + -- ({\dx*1.4600},{\dy*0.8856}) + -- ({\dx*1.4800},{\dy*0.8857}) + -- ({\dx*1.5000},{\dy*0.8862}) + -- ({\dx*1.5200},{\dy*0.8870}) + -- ({\dx*1.5400},{\dy*0.8882}) + -- ({\dx*1.5600},{\dy*0.8896}) + -- ({\dx*1.5800},{\dy*0.8914}) + -- ({\dx*1.6000},{\dy*0.8935}) + -- ({\dx*1.6200},{\dy*0.8959}) + -- ({\dx*1.6400},{\dy*0.8986}) + -- ({\dx*1.6600},{\dy*0.9017}) + -- ({\dx*1.6800},{\dy*0.9050}) + -- ({\dx*1.7000},{\dy*0.9086}) + -- ({\dx*1.7200},{\dy*0.9126}) + -- ({\dx*1.7400},{\dy*0.9168}) + -- ({\dx*1.7600},{\dy*0.9214}) + -- ({\dx*1.7800},{\dy*0.9262}) + -- ({\dx*1.8000},{\dy*0.9314}) + -- ({\dx*1.8200},{\dy*0.9368}) + -- ({\dx*1.8400},{\dy*0.9426}) + -- ({\dx*1.8600},{\dy*0.9487}) + -- ({\dx*1.8800},{\dy*0.9551}) + -- ({\dx*1.9000},{\dy*0.9618}) + -- ({\dx*1.9200},{\dy*0.9688}) + -- ({\dx*1.9400},{\dy*0.9761}) + -- ({\dx*1.9600},{\dy*0.9837}) + -- ({\dx*1.9800},{\dy*0.9917}) + -- ({\dx*2.0000},{\dy*1.0000}) + -- ({\dx*2.0200},{\dy*1.0086}) + -- ({\dx*2.0400},{\dy*1.0176}) + -- ({\dx*2.0600},{\dy*1.0269}) + -- ({\dx*2.0800},{\dy*1.0365}) + -- ({\dx*2.1000},{\dy*1.0465}) + -- ({\dx*2.1200},{\dy*1.0568}) + -- ({\dx*2.1400},{\dy*1.0675}) + -- ({\dx*2.1600},{\dy*1.0786}) + -- ({\dx*2.1800},{\dy*1.0900}) + -- ({\dx*2.2000},{\dy*1.1018}) + -- ({\dx*2.2200},{\dy*1.1140}) + -- ({\dx*2.2400},{\dy*1.1266}) + -- ({\dx*2.2600},{\dy*1.1395}) + -- ({\dx*2.2800},{\dy*1.1529}) + -- ({\dx*2.3000},{\dy*1.1667}) + -- ({\dx*2.3200},{\dy*1.1809}) + -- ({\dx*2.3400},{\dy*1.1956}) + -- ({\dx*2.3600},{\dy*1.2107}) + -- ({\dx*2.3800},{\dy*1.2262}) + -- ({\dx*2.4000},{\dy*1.2422}) + -- ({\dx*2.4200},{\dy*1.2586}) + -- ({\dx*2.4400},{\dy*1.2756}) + -- ({\dx*2.4600},{\dy*1.2930}) + -- ({\dx*2.4800},{\dy*1.3109}) + -- ({\dx*2.5000},{\dy*1.3293}) + -- ({\dx*2.5200},{\dy*1.3483}) + -- ({\dx*2.5400},{\dy*1.3678}) + -- ({\dx*2.5600},{\dy*1.3878}) + -- ({\dx*2.5800},{\dy*1.4084}) + -- ({\dx*2.6000},{\dy*1.4296}) + -- ({\dx*2.6200},{\dy*1.4514}) + -- ({\dx*2.6400},{\dy*1.4738}) + -- ({\dx*2.6600},{\dy*1.4968}) + -- ({\dx*2.6800},{\dy*1.5204}) + -- ({\dx*2.7000},{\dy*1.5447}) + -- ({\dx*2.7200},{\dy*1.5696}) + -- ({\dx*2.7400},{\dy*1.5953}) + -- ({\dx*2.7600},{\dy*1.6216}) + -- ({\dx*2.7800},{\dy*1.6487}) + -- ({\dx*2.8000},{\dy*1.6765}) + -- ({\dx*2.8200},{\dy*1.7051}) + -- ({\dx*2.8400},{\dy*1.7344}) + -- ({\dx*2.8600},{\dy*1.7646}) + -- ({\dx*2.8800},{\dy*1.7955}) + -- ({\dx*2.9000},{\dy*1.8274}) + -- ({\dx*2.9200},{\dy*1.8600}) + -- ({\dx*2.9400},{\dy*1.8936}) + -- ({\dx*2.9600},{\dy*1.9281}) + -- ({\dx*2.9800},{\dy*1.9636}) + -- ({\dx*3.0000},{\dy*2.0000}) + -- ({\dx*3.0200},{\dy*2.0374}) + -- ({\dx*3.0400},{\dy*2.0759}) + -- ({\dx*3.0600},{\dy*2.1153}) + -- ({\dx*3.0800},{\dy*2.1559}) + -- ({\dx*3.1000},{\dy*2.1976}) + -- ({\dx*3.1200},{\dy*2.2405}) + -- ({\dx*3.1400},{\dy*2.2845}) + -- ({\dx*3.1600},{\dy*2.3297}) + -- ({\dx*3.1800},{\dy*2.3762}) + -- ({\dx*3.2000},{\dy*2.4240}) + -- ({\dx*3.2200},{\dy*2.4731}) + -- ({\dx*3.2400},{\dy*2.5235}) + -- ({\dx*3.2600},{\dy*2.5754}) + -- ({\dx*3.2800},{\dy*2.6287}) + -- ({\dx*3.3000},{\dy*2.6834}) + -- ({\dx*3.3200},{\dy*2.7397}) + -- ({\dx*3.3400},{\dy*2.7976}) + -- ({\dx*3.3600},{\dy*2.8571}) + -- ({\dx*3.3800},{\dy*2.9183}) + -- ({\dx*3.4000},{\dy*2.9812}) + -- ({\dx*3.4200},{\dy*3.0459}) + -- ({\dx*3.4400},{\dy*3.1124}) + -- ({\dx*3.4600},{\dy*3.1807}) + -- ({\dx*3.4800},{\dy*3.2510}) + -- ({\dx*3.5000},{\dy*3.3234}) + -- ({\dx*3.5200},{\dy*3.3977}) + -- ({\dx*3.5400},{\dy*3.4742}) + -- ({\dx*3.5600},{\dy*3.5529}) + -- ({\dx*3.5800},{\dy*3.6338}) + -- ({\dx*3.6000},{\dy*3.7170}) + -- ({\dx*3.6200},{\dy*3.8027}) + -- ({\dx*3.6400},{\dy*3.8908}) + -- ({\dx*3.6600},{\dy*3.9814}) + -- ({\dx*3.6800},{\dy*4.0747}) + -- ({\dx*3.7000},{\dy*4.1707}) + -- ({\dx*3.7200},{\dy*4.2694}) + -- ({\dx*3.7400},{\dy*4.3711}) + -- ({\dx*3.7600},{\dy*4.4757}) + -- ({\dx*3.7800},{\dy*4.5833}) + -- ({\dx*3.8000},{\dy*4.6942}) + -- ({\dx*3.8200},{\dy*4.8083}) + -- ({\dx*3.8400},{\dy*4.9257}) + -- ({\dx*3.8600},{\dy*5.0466}) + -- ({\dx*3.8800},{\dy*5.1711}) + -- ({\dx*3.9000},{\dy*5.2993}) + -- ({\dx*3.9200},{\dy*5.4313}) + -- ({\dx*3.9400},{\dy*5.5673}) + -- ({\dx*3.9600},{\dy*5.7073}) + -- ({\dx*3.9800},{\dy*5.8515}) + -- ({\dx*4.0000},{\dy*6.0000}) + -- ({\dx*4.0200},{\dy*6.1530}) + -- ({\dx*4.0400},{\dy*6.3106}) + -- ({\dx*4.0600},{\dy*6.4730}) + -- ({\dx*4.0800},{\dy*6.6403}) + -- ({\dx*4.0810},{\dy*6.6488})} +\def\gammaone{({\dx*-0.9810},{\dy*-53.0814}) + -- ({\dx*-0.9800},{\dy*-50.4512}) + -- ({\dx*-0.9600},{\dy*-25.4802}) + -- ({\dx*-0.9400},{\dy*-17.1763}) + -- ({\dx*-0.9200},{\dy*-13.0397}) + -- ({\dx*-0.9000},{\dy*-10.5706}) + -- ({\dx*-0.8800},{\dy*-8.9355}) + -- ({\dx*-0.8600},{\dy*-7.7775}) + -- ({\dx*-0.8400},{\dy*-6.9182}) + -- ({\dx*-0.8200},{\dy*-6.2583}) + -- ({\dx*-0.8000},{\dy*-5.7386}) + -- ({\dx*-0.7800},{\dy*-5.3211}) + -- ({\dx*-0.7600},{\dy*-4.9809}) + -- ({\dx*-0.7400},{\dy*-4.7006}) + -- ({\dx*-0.7200},{\dy*-4.4678}) + -- ({\dx*-0.7000},{\dy*-4.2737}) + -- ({\dx*-0.6800},{\dy*-4.1114}) + -- ({\dx*-0.6600},{\dy*-3.9760}) + -- ({\dx*-0.6400},{\dy*-3.8636}) + -- ({\dx*-0.6200},{\dy*-3.7714}) + -- ({\dx*-0.6000},{\dy*-3.6969}) + -- ({\dx*-0.5800},{\dy*-3.6386}) + -- ({\dx*-0.5600},{\dy*-3.5950}) + -- ({\dx*-0.5400},{\dy*-3.5652}) + -- ({\dx*-0.5200},{\dy*-3.5487}) + -- ({\dx*-0.5000},{\dy*-3.5449}) + -- ({\dx*-0.4800},{\dy*-3.5538}) + -- ({\dx*-0.4600},{\dy*-3.5756}) + -- ({\dx*-0.4400},{\dy*-3.6105}) + -- ({\dx*-0.4200},{\dy*-3.6594}) + -- ({\dx*-0.4000},{\dy*-3.7230}) + -- ({\dx*-0.3800},{\dy*-3.8027}) + -- ({\dx*-0.3600},{\dy*-3.9004}) + -- ({\dx*-0.3400},{\dy*-4.0181}) + -- ({\dx*-0.3200},{\dy*-4.1590}) + -- ({\dx*-0.3000},{\dy*-4.3269}) + -- ({\dx*-0.2800},{\dy*-4.5267}) + -- ({\dx*-0.2600},{\dy*-4.7652}) + -- ({\dx*-0.2400},{\dy*-5.0514}) + -- ({\dx*-0.2200},{\dy*-5.3976}) + -- ({\dx*-0.2000},{\dy*-5.8211}) + -- ({\dx*-0.1800},{\dy*-6.3472}) + -- ({\dx*-0.1600},{\dy*-7.0135}) + -- ({\dx*-0.1400},{\dy*-7.8795}) + -- ({\dx*-0.1200},{\dy*-9.0442}) + -- ({\dx*-0.1000},{\dy*-10.6863}) + -- ({\dx*-0.0800},{\dy*-13.1627}) + -- ({\dx*-0.0600},{\dy*-17.3067}) + -- ({\dx*-0.0400},{\dy*-25.6183}) + -- ({\dx*-0.0200},{\dy*-50.5974}) + -- ({\dx*-0.0190},{\dy*-53.2279})} +\def\gammatwo{({\dx*-1.9810},{\dy*26.7952}) + -- ({\dx*-1.9800},{\dy*25.4804}) + -- ({\dx*-1.9600},{\dy*13.0001}) + -- ({\dx*-1.9400},{\dy*8.8538}) + -- ({\dx*-1.9200},{\dy*6.7915}) + -- ({\dx*-1.9000},{\dy*5.5635}) + -- ({\dx*-1.8800},{\dy*4.7529}) + -- ({\dx*-1.8600},{\dy*4.1815}) + -- ({\dx*-1.8400},{\dy*3.7599}) + -- ({\dx*-1.8200},{\dy*3.4386}) + -- ({\dx*-1.8000},{\dy*3.1881}) + -- ({\dx*-1.7800},{\dy*2.9894}) + -- ({\dx*-1.7600},{\dy*2.8301}) + -- ({\dx*-1.7400},{\dy*2.7015}) + -- ({\dx*-1.7200},{\dy*2.5976}) + -- ({\dx*-1.7000},{\dy*2.5139}) + -- ({\dx*-1.6800},{\dy*2.4473}) + -- ({\dx*-1.6600},{\dy*2.3952}) + -- ({\dx*-1.6400},{\dy*2.3559}) + -- ({\dx*-1.6200},{\dy*2.3280}) + -- ({\dx*-1.6000},{\dy*2.3106}) + -- ({\dx*-1.5800},{\dy*2.3029}) + -- ({\dx*-1.5600},{\dy*2.3045}) + -- ({\dx*-1.5400},{\dy*2.3151}) + -- ({\dx*-1.5200},{\dy*2.3346}) + -- ({\dx*-1.5000},{\dy*2.3633}) + -- ({\dx*-1.4800},{\dy*2.4012}) + -- ({\dx*-1.4600},{\dy*2.4490}) + -- ({\dx*-1.4400},{\dy*2.5073}) + -- ({\dx*-1.4200},{\dy*2.5770}) + -- ({\dx*-1.4000},{\dy*2.6593}) + -- ({\dx*-1.3800},{\dy*2.7556}) + -- ({\dx*-1.3600},{\dy*2.8679}) + -- ({\dx*-1.3400},{\dy*2.9986}) + -- ({\dx*-1.3200},{\dy*3.1508}) + -- ({\dx*-1.3000},{\dy*3.3283}) + -- ({\dx*-1.2800},{\dy*3.5365}) + -- ({\dx*-1.2600},{\dy*3.7819}) + -- ({\dx*-1.2400},{\dy*4.0737}) + -- ({\dx*-1.2200},{\dy*4.4243}) + -- ({\dx*-1.2000},{\dy*4.8510}) + -- ({\dx*-1.1800},{\dy*5.3790}) + -- ({\dx*-1.1600},{\dy*6.0461}) + -- ({\dx*-1.1400},{\dy*6.9118}) + -- ({\dx*-1.1200},{\dy*8.0752}) + -- ({\dx*-1.1000},{\dy*9.7148}) + -- ({\dx*-1.0800},{\dy*12.1877}) + -- ({\dx*-1.0600},{\dy*16.3271}) + -- ({\dx*-1.0400},{\dy*24.6330}) + -- ({\dx*-1.0200},{\dy*49.6053}) + -- ({\dx*-1.0190},{\dy*52.2354})} +\def\gammathree{({\dx*-2.9810},{\dy*-8.9887}) + -- ({\dx*-2.9800},{\dy*-8.5505}) + -- ({\dx*-2.9600},{\dy*-4.3919}) + -- ({\dx*-2.9400},{\dy*-3.0115}) + -- ({\dx*-2.9200},{\dy*-2.3259}) + -- ({\dx*-2.9000},{\dy*-1.9184}) + -- ({\dx*-2.8800},{\dy*-1.6503}) + -- ({\dx*-2.8600},{\dy*-1.4621}) + -- ({\dx*-2.8400},{\dy*-1.3239}) + -- ({\dx*-2.8200},{\dy*-1.2194}) + -- ({\dx*-2.8000},{\dy*-1.1386}) + -- ({\dx*-2.7800},{\dy*-1.0753}) + -- ({\dx*-2.7600},{\dy*-1.0254}) + -- ({\dx*-2.7400},{\dy*-0.9859}) + -- ({\dx*-2.7200},{\dy*-0.9550}) + -- ({\dx*-2.7000},{\dy*-0.9311}) + -- ({\dx*-2.6800},{\dy*-0.9132}) + -- ({\dx*-2.6600},{\dy*-0.9004}) + -- ({\dx*-2.6400},{\dy*-0.8924}) + -- ({\dx*-2.6200},{\dy*-0.8886}) + -- ({\dx*-2.6000},{\dy*-0.8887}) + -- ({\dx*-2.5800},{\dy*-0.8926}) + -- ({\dx*-2.5600},{\dy*-0.9002}) + -- ({\dx*-2.5400},{\dy*-0.9115}) + -- ({\dx*-2.5200},{\dy*-0.9264}) + -- ({\dx*-2.5000},{\dy*-0.9453}) + -- ({\dx*-2.4800},{\dy*-0.9682}) + -- ({\dx*-2.4600},{\dy*-0.9955}) + -- ({\dx*-2.4400},{\dy*-1.0276}) + -- ({\dx*-2.4200},{\dy*-1.0649}) + -- ({\dx*-2.4000},{\dy*-1.1080}) + -- ({\dx*-2.3800},{\dy*-1.1578}) + -- ({\dx*-2.3600},{\dy*-1.2152}) + -- ({\dx*-2.3400},{\dy*-1.2815}) + -- ({\dx*-2.3200},{\dy*-1.3581}) + -- ({\dx*-2.3000},{\dy*-1.4471}) + -- ({\dx*-2.2800},{\dy*-1.5511}) + -- ({\dx*-2.2600},{\dy*-1.6734}) + -- ({\dx*-2.2400},{\dy*-1.8186}) + -- ({\dx*-2.2200},{\dy*-1.9929}) + -- ({\dx*-2.2000},{\dy*-2.2050}) + -- ({\dx*-2.1800},{\dy*-2.4674}) + -- ({\dx*-2.1600},{\dy*-2.7991}) + -- ({\dx*-2.1400},{\dy*-3.2298}) + -- ({\dx*-2.1200},{\dy*-3.8091}) + -- ({\dx*-2.1000},{\dy*-4.6261}) + -- ({\dx*-2.0800},{\dy*-5.8595}) + -- ({\dx*-2.0600},{\dy*-7.9258}) + -- ({\dx*-2.0400},{\dy*-12.0750}) + -- ({\dx*-2.0200},{\dy*-24.5571}) + -- ({\dx*-2.0190},{\dy*-25.8719})} +\def\gammafour{({\dx*-3.9950},{\dy*8.3966}) + -- ({\dx*-3.9800},{\dy*2.1484}) + -- ({\dx*-3.9600},{\dy*1.1091}) + -- ({\dx*-3.9400},{\dy*0.7643}) + -- ({\dx*-3.9200},{\dy*0.5933}) + -- ({\dx*-3.9000},{\dy*0.4919}) + -- ({\dx*-3.8800},{\dy*0.4253}) + -- ({\dx*-3.8600},{\dy*0.3788}) + -- ({\dx*-3.8400},{\dy*0.3448}) + -- ({\dx*-3.8200},{\dy*0.3192}) + -- ({\dx*-3.8000},{\dy*0.2996}) + -- ({\dx*-3.7800},{\dy*0.2845}) + -- ({\dx*-3.7600},{\dy*0.2727}) + -- ({\dx*-3.7400},{\dy*0.2636}) + -- ({\dx*-3.7200},{\dy*0.2567}) + -- ({\dx*-3.7000},{\dy*0.2516}) + -- ({\dx*-3.6800},{\dy*0.2481}) + -- ({\dx*-3.6600},{\dy*0.2460}) + -- ({\dx*-3.6400},{\dy*0.2452}) + -- ({\dx*-3.6200},{\dy*0.2455}) + -- ({\dx*-3.6000},{\dy*0.2469}) + -- ({\dx*-3.5800},{\dy*0.2493}) + -- ({\dx*-3.5600},{\dy*0.2529}) + -- ({\dx*-3.5400},{\dy*0.2575}) + -- ({\dx*-3.5200},{\dy*0.2632}) + -- ({\dx*-3.5000},{\dy*0.2701}) + -- ({\dx*-3.4800},{\dy*0.2782}) + -- ({\dx*-3.4600},{\dy*0.2877}) + -- ({\dx*-3.4400},{\dy*0.2987}) + -- ({\dx*-3.4200},{\dy*0.3114}) + -- ({\dx*-3.4000},{\dy*0.3259}) + -- ({\dx*-3.3800},{\dy*0.3425}) + -- ({\dx*-3.3600},{\dy*0.3617}) + -- ({\dx*-3.3400},{\dy*0.3837}) + -- ({\dx*-3.3200},{\dy*0.4091}) + -- ({\dx*-3.3000},{\dy*0.4385}) + -- ({\dx*-3.2800},{\dy*0.4729}) + -- ({\dx*-3.2600},{\dy*0.5133}) + -- ({\dx*-3.2400},{\dy*0.5613}) + -- ({\dx*-3.2200},{\dy*0.6189}) + -- ({\dx*-3.2000},{\dy*0.6891}) + -- ({\dx*-3.1800},{\dy*0.7759}) + -- ({\dx*-3.1600},{\dy*0.8858}) + -- ({\dx*-3.1400},{\dy*1.0286}) + -- ({\dx*-3.1200},{\dy*1.2209}) + -- ({\dx*-3.1000},{\dy*1.4923}) + -- ({\dx*-3.0800},{\dy*1.9024}) + -- ({\dx*-3.0600},{\dy*2.5901}) + -- ({\dx*-3.0400},{\dy*3.9720}) + -- ({\dx*-3.0200},{\dy*8.1315}) + -- ({\dx*-3.0050},{\dy*33.1259})} +\def\gammafive{({\dx*-4.9990},{\dy*-8.3476}) + -- ({\dx*-4.9800},{\dy*-0.4314}) + -- ({\dx*-4.9600},{\dy*-0.2236}) + -- ({\dx*-4.9400},{\dy*-0.1547}) + -- ({\dx*-4.9200},{\dy*-0.1206}) + -- ({\dx*-4.9000},{\dy*-0.1004}) + -- ({\dx*-4.8800},{\dy*-0.0872}) + -- ({\dx*-4.8600},{\dy*-0.0779}) + -- ({\dx*-4.8400},{\dy*-0.0712}) + -- ({\dx*-4.8200},{\dy*-0.0662}) + -- ({\dx*-4.8000},{\dy*-0.0624}) + -- ({\dx*-4.7800},{\dy*-0.0595}) + -- ({\dx*-4.7600},{\dy*-0.0573}) + -- ({\dx*-4.7400},{\dy*-0.0556}) + -- ({\dx*-4.7200},{\dy*-0.0544}) + -- ({\dx*-4.7000},{\dy*-0.0535}) + -- ({\dx*-4.6800},{\dy*-0.0530}) + -- ({\dx*-4.6600},{\dy*-0.0528}) + -- ({\dx*-4.6400},{\dy*-0.0528}) + -- ({\dx*-4.6200},{\dy*-0.0531}) + -- ({\dx*-4.6000},{\dy*-0.0537}) + -- ({\dx*-4.5800},{\dy*-0.0544}) + -- ({\dx*-4.5600},{\dy*-0.0555}) + -- ({\dx*-4.5400},{\dy*-0.0567}) + -- ({\dx*-4.5200},{\dy*-0.0582}) + -- ({\dx*-4.5000},{\dy*-0.0600}) + -- ({\dx*-4.4800},{\dy*-0.0621}) + -- ({\dx*-4.4600},{\dy*-0.0645}) + -- ({\dx*-4.4400},{\dy*-0.0673}) + -- ({\dx*-4.4200},{\dy*-0.0704}) + -- ({\dx*-4.4000},{\dy*-0.0741}) + -- ({\dx*-4.3800},{\dy*-0.0782}) + -- ({\dx*-4.3600},{\dy*-0.0830}) + -- ({\dx*-4.3400},{\dy*-0.0884}) + -- ({\dx*-4.3200},{\dy*-0.0947}) + -- ({\dx*-4.3000},{\dy*-0.1020}) + -- ({\dx*-4.2800},{\dy*-0.1105}) + -- ({\dx*-4.2600},{\dy*-0.1205}) + -- ({\dx*-4.2400},{\dy*-0.1324}) + -- ({\dx*-4.2200},{\dy*-0.1467}) + -- ({\dx*-4.2000},{\dy*-0.1641}) + -- ({\dx*-4.1800},{\dy*-0.1856}) + -- ({\dx*-4.1600},{\dy*-0.2129}) + -- ({\dx*-4.1400},{\dy*-0.2485}) + -- ({\dx*-4.1200},{\dy*-0.2963}) + -- ({\dx*-4.1000},{\dy*-0.3640}) + -- ({\dx*-4.0800},{\dy*-0.4663}) + -- ({\dx*-4.0600},{\dy*-0.6380}) + -- ({\dx*-4.0400},{\dy*-0.9832}) + -- ({\dx*-4.0200},{\dy*-2.0228}) + -- ({\dx*-4.0010},{\dy*-41.6040})} +\def\gammasix{({\dx*-5.9998},{\dy*6.9470}) + -- ({\dx*-5.9800},{\dy*0.0721}) + -- ({\dx*-5.9600},{\dy*0.0375}) + -- ({\dx*-5.9400},{\dy*0.0260}) + -- ({\dx*-5.9200},{\dy*0.0204}) + -- ({\dx*-5.9000},{\dy*0.0170}) + -- ({\dx*-5.8800},{\dy*0.0148}) + -- ({\dx*-5.8600},{\dy*0.0133}) + -- ({\dx*-5.8400},{\dy*0.0122}) + -- ({\dx*-5.8200},{\dy*0.0114}) + -- ({\dx*-5.8000},{\dy*0.0108}) + -- ({\dx*-5.7800},{\dy*0.0103}) + -- ({\dx*-5.7600},{\dy*0.0099}) + -- ({\dx*-5.7400},{\dy*0.0097}) + -- ({\dx*-5.7200},{\dy*0.0095}) + -- ({\dx*-5.7000},{\dy*0.0094}) + -- ({\dx*-5.6800},{\dy*0.0093}) + -- ({\dx*-5.6600},{\dy*0.0093}) + -- ({\dx*-5.6400},{\dy*0.0094}) + -- ({\dx*-5.6200},{\dy*0.0095}) + -- ({\dx*-5.6000},{\dy*0.0096}) + -- ({\dx*-5.5800},{\dy*0.0098}) + -- ({\dx*-5.5600},{\dy*0.0100}) + -- ({\dx*-5.5400},{\dy*0.0102}) + -- ({\dx*-5.5200},{\dy*0.0105}) + -- ({\dx*-5.5000},{\dy*0.0109}) + -- ({\dx*-5.4800},{\dy*0.0113}) + -- ({\dx*-5.4600},{\dy*0.0118}) + -- ({\dx*-5.4400},{\dy*0.0124}) + -- ({\dx*-5.4200},{\dy*0.0130}) + -- ({\dx*-5.4000},{\dy*0.0137}) + -- ({\dx*-5.3800},{\dy*0.0145}) + -- ({\dx*-5.3600},{\dy*0.0155}) + -- ({\dx*-5.3400},{\dy*0.0166}) + -- ({\dx*-5.3200},{\dy*0.0178}) + -- ({\dx*-5.3000},{\dy*0.0192}) + -- ({\dx*-5.2800},{\dy*0.0209}) + -- ({\dx*-5.2600},{\dy*0.0229}) + -- ({\dx*-5.2400},{\dy*0.0253}) + -- ({\dx*-5.2200},{\dy*0.0281}) + -- ({\dx*-5.2000},{\dy*0.0316}) + -- ({\dx*-5.1800},{\dy*0.0358}) + -- ({\dx*-5.1600},{\dy*0.0413}) + -- ({\dx*-5.1400},{\dy*0.0483}) + -- ({\dx*-5.1200},{\dy*0.0579}) + -- ({\dx*-5.1000},{\dy*0.0714}) + -- ({\dx*-5.0800},{\dy*0.0918}) + -- ({\dx*-5.0600},{\dy*0.1261}) + -- ({\dx*-5.0400},{\dy*0.1951}) + -- ({\dx*-5.0200},{\dy*0.4029}) + -- ({\dx*-5.0002},{\dy*41.6525})} +\def\gammasinplus{({\dx*0.0190},{\dy*52.1325}) + -- ({\dx*0.0200},{\dy*49.5050}) + -- ({\dx*0.0400},{\dy*24.5863}) + -- ({\dx*0.0600},{\dy*16.3331}) + -- ({\dx*0.0800},{\dy*12.2453}) + -- ({\dx*0.1000},{\dy*9.8225}) + -- ({\dx*0.1200},{\dy*8.2314}) + -- ({\dx*0.1400},{\dy*7.1145}) + -- ({\dx*0.1600},{\dy*6.2930}) + -- ({\dx*0.1800},{\dy*5.6676}) + -- ({\dx*0.2000},{\dy*5.1786}) + -- ({\dx*0.2200},{\dy*4.7879}) + -- ({\dx*0.2400},{\dy*4.4701}) + -- ({\dx*0.2600},{\dy*4.2074}) + -- ({\dx*0.2800},{\dy*3.9874}) + -- ({\dx*0.3000},{\dy*3.8006}) + -- ({\dx*0.3200},{\dy*3.6401}) + -- ({\dx*0.3400},{\dy*3.5005}) + -- ({\dx*0.3600},{\dy*3.3776}) + -- ({\dx*0.3800},{\dy*3.2680}) + -- ({\dx*0.4000},{\dy*3.1692}) + -- ({\dx*0.4200},{\dy*3.0790}) + -- ({\dx*0.4400},{\dy*2.9955}) + -- ({\dx*0.4600},{\dy*2.9173}) + -- ({\dx*0.4800},{\dy*2.8433}) + -- ({\dx*0.5000},{\dy*2.7725}) + -- ({\dx*0.5200},{\dy*2.7039}) + -- ({\dx*0.5400},{\dy*2.6369}) + -- ({\dx*0.5600},{\dy*2.5709}) + -- ({\dx*0.5800},{\dy*2.5055}) + -- ({\dx*0.6000},{\dy*2.4402}) + -- ({\dx*0.6200},{\dy*2.3748}) + -- ({\dx*0.6400},{\dy*2.3090}) + -- ({\dx*0.6600},{\dy*2.2425}) + -- ({\dx*0.6800},{\dy*2.1752}) + -- ({\dx*0.7000},{\dy*2.1071}) + -- ({\dx*0.7200},{\dy*2.0380}) + -- ({\dx*0.7400},{\dy*1.9679}) + -- ({\dx*0.7600},{\dy*1.8969}) + -- ({\dx*0.7800},{\dy*1.8249}) + -- ({\dx*0.8000},{\dy*1.7520}) + -- ({\dx*0.8200},{\dy*1.6783}) + -- ({\dx*0.8400},{\dy*1.6039}) + -- ({\dx*0.8600},{\dy*1.5289}) + -- ({\dx*0.8800},{\dy*1.4534}) + -- ({\dx*0.9000},{\dy*1.3776}) + -- ({\dx*0.9200},{\dy*1.3017}) + -- ({\dx*0.9400},{\dy*1.2258}) + -- ({\dx*0.9600},{\dy*1.1501}) + -- ({\dx*0.9800},{\dy*1.0747}) + -- ({\dx*1.0000},{\dy*1.0000}) + -- ({\dx*1.0200},{\dy*0.9261}) + -- ({\dx*1.0400},{\dy*0.8531}) + -- ({\dx*1.0600},{\dy*0.7814}) + -- ({\dx*1.0800},{\dy*0.7110}) + -- ({\dx*1.1000},{\dy*0.6423}) + -- ({\dx*1.1200},{\dy*0.5755}) + -- ({\dx*1.1400},{\dy*0.5106}) + -- ({\dx*1.1600},{\dy*0.4480}) + -- ({\dx*1.1800},{\dy*0.3879}) + -- ({\dx*1.2000},{\dy*0.3304}) + -- ({\dx*1.2200},{\dy*0.2757}) + -- ({\dx*1.2400},{\dy*0.2240}) + -- ({\dx*1.2600},{\dy*0.1754}) + -- ({\dx*1.2800},{\dy*0.1302}) + -- ({\dx*1.3000},{\dy*0.0885}) + -- ({\dx*1.3200},{\dy*0.0503}) + -- ({\dx*1.3400},{\dy*0.0159}) + -- ({\dx*1.3600},{\dy*-0.0146}) + -- ({\dx*1.3800},{\dy*-0.0412}) + -- ({\dx*1.4000},{\dy*-0.0638}) + -- ({\dx*1.4200},{\dy*-0.0822}) + -- ({\dx*1.4400},{\dy*-0.0965}) + -- ({\dx*1.4600},{\dy*-0.1065}) + -- ({\dx*1.4800},{\dy*-0.1123}) + -- ({\dx*1.5000},{\dy*-0.1138}) + -- ({\dx*1.5200},{\dy*-0.1110}) + -- ({\dx*1.5400},{\dy*-0.1039}) + -- ({\dx*1.5600},{\dy*-0.0926}) + -- ({\dx*1.5800},{\dy*-0.0772}) + -- ({\dx*1.6000},{\dy*-0.0575}) + -- ({\dx*1.6200},{\dy*-0.0339}) + -- ({\dx*1.6400},{\dy*-0.0062}) + -- ({\dx*1.6600},{\dy*0.0254}) + -- ({\dx*1.6800},{\dy*0.0607}) + -- ({\dx*1.7000},{\dy*0.0996}) + -- ({\dx*1.7200},{\dy*0.1421}) + -- ({\dx*1.7400},{\dy*0.1879}) + -- ({\dx*1.7600},{\dy*0.2368}) + -- ({\dx*1.7800},{\dy*0.2888}) + -- ({\dx*1.8000},{\dy*0.3436}) + -- ({\dx*1.8200},{\dy*0.4010}) + -- ({\dx*1.8400},{\dy*0.4609}) + -- ({\dx*1.8600},{\dy*0.5229}) + -- ({\dx*1.8800},{\dy*0.5869}) + -- ({\dx*1.9000},{\dy*0.6527}) + -- ({\dx*1.9200},{\dy*0.7201}) + -- ({\dx*1.9400},{\dy*0.7887}) + -- ({\dx*1.9600},{\dy*0.8584}) + -- ({\dx*1.9800},{\dy*0.9289}) + -- ({\dx*2.0000},{\dy*1.0000}) + -- ({\dx*2.0200},{\dy*1.0714}) + -- ({\dx*2.0400},{\dy*1.1429}) + -- ({\dx*2.0600},{\dy*1.2142}) + -- ({\dx*2.0800},{\dy*1.2852}) + -- ({\dx*2.1000},{\dy*1.3555}) + -- ({\dx*2.1200},{\dy*1.4249}) + -- ({\dx*2.1400},{\dy*1.4933}) + -- ({\dx*2.1600},{\dy*1.5603}) + -- ({\dx*2.1800},{\dy*1.6258}) + -- ({\dx*2.2000},{\dy*1.6896}) + -- ({\dx*2.2200},{\dy*1.7514}) + -- ({\dx*2.2400},{\dy*1.8111}) + -- ({\dx*2.2600},{\dy*1.8685}) + -- ({\dx*2.2800},{\dy*1.9234}) + -- ({\dx*2.3000},{\dy*1.9757}) + -- ({\dx*2.3200},{\dy*2.0253}) + -- ({\dx*2.3400},{\dy*2.0719}) + -- ({\dx*2.3600},{\dy*2.1155}) + -- ({\dx*2.3800},{\dy*2.1560}) + -- ({\dx*2.4000},{\dy*2.1932}) + -- ({\dx*2.4200},{\dy*2.2272}) + -- ({\dx*2.4400},{\dy*2.2578}) + -- ({\dx*2.4600},{\dy*2.2851}) + -- ({\dx*2.4800},{\dy*2.3089}) + -- ({\dx*2.5000},{\dy*2.3293}) + -- ({\dx*2.5200},{\dy*2.3463}) + -- ({\dx*2.5400},{\dy*2.3599}) + -- ({\dx*2.5600},{\dy*2.3701}) + -- ({\dx*2.5800},{\dy*2.3770}) + -- ({\dx*2.6000},{\dy*2.3807}) + -- ({\dx*2.6200},{\dy*2.3812}) + -- ({\dx*2.6400},{\dy*2.3786}) + -- ({\dx*2.6600},{\dy*2.3731}) + -- ({\dx*2.6800},{\dy*2.3647}) + -- ({\dx*2.7000},{\dy*2.3537}) + -- ({\dx*2.7200},{\dy*2.3402}) + -- ({\dx*2.7400},{\dy*2.3242}) + -- ({\dx*2.7600},{\dy*2.3062}) + -- ({\dx*2.7800},{\dy*2.2861}) + -- ({\dx*2.8000},{\dy*2.2643}) + -- ({\dx*2.8200},{\dy*2.2409}) + -- ({\dx*2.8400},{\dy*2.2162}) + -- ({\dx*2.8600},{\dy*2.1903}) + -- ({\dx*2.8800},{\dy*2.1637}) + -- ({\dx*2.9000},{\dy*2.1364}) + -- ({\dx*2.9200},{\dy*2.1087}) + -- ({\dx*2.9400},{\dy*2.0810}) + -- ({\dx*2.9600},{\dy*2.0535}) + -- ({\dx*2.9800},{\dy*2.0264}) + -- ({\dx*3.0000},{\dy*2.0000}) + -- ({\dx*3.0200},{\dy*1.9746}) + -- ({\dx*3.0400},{\dy*1.9505}) + -- ({\dx*3.0600},{\dy*1.9280}) + -- ({\dx*3.0800},{\dy*1.9072}) + -- ({\dx*3.1000},{\dy*1.8886}) + -- ({\dx*3.1200},{\dy*1.8723}) + -- ({\dx*3.1400},{\dy*1.8587}) + -- ({\dx*3.1600},{\dy*1.8480}) + -- ({\dx*3.1800},{\dy*1.8404}) + -- ({\dx*3.2000},{\dy*1.8362}) + -- ({\dx*3.2200},{\dy*1.8356}) + -- ({\dx*3.2400},{\dy*1.8390}) + -- ({\dx*3.2600},{\dy*1.8464}) + -- ({\dx*3.2800},{\dy*1.8581}) + -- ({\dx*3.3000},{\dy*1.8744}) + -- ({\dx*3.3200},{\dy*1.8954}) + -- ({\dx*3.3400},{\dy*1.9213}) + -- ({\dx*3.3600},{\dy*1.9523}) + -- ({\dx*3.3800},{\dy*1.9885}) + -- ({\dx*3.4000},{\dy*2.0301}) + -- ({\dx*3.4200},{\dy*2.0773}) + -- ({\dx*3.4400},{\dy*2.1301}) + -- ({\dx*3.4600},{\dy*2.1886}) + -- ({\dx*3.4800},{\dy*2.2530}) + -- ({\dx*3.5000},{\dy*2.3234}) + -- ({\dx*3.5200},{\dy*2.3997}) + -- ({\dx*3.5400},{\dy*2.4821}) + -- ({\dx*3.5600},{\dy*2.5706}) + -- ({\dx*3.5800},{\dy*2.6652}) + -- ({\dx*3.6000},{\dy*2.7660}) + -- ({\dx*3.6200},{\dy*2.8729}) + -- ({\dx*3.6400},{\dy*2.9859}) + -- ({\dx*3.6600},{\dy*3.1051}) + -- ({\dx*3.6800},{\dy*3.2303}) + -- ({\dx*3.7000},{\dy*3.3616}) + -- ({\dx*3.7200},{\dy*3.4989}) + -- ({\dx*3.7400},{\dy*3.6421}) + -- ({\dx*3.7600},{\dy*3.7911}) + -- ({\dx*3.7800},{\dy*3.9459}) + -- ({\dx*3.8000},{\dy*4.1064}) + -- ({\dx*3.8200},{\dy*4.2724}) + -- ({\dx*3.8400},{\dy*4.4440}) + -- ({\dx*3.8600},{\dy*4.6209}) + -- ({\dx*3.8800},{\dy*4.8030}) + -- ({\dx*3.9000},{\dy*4.9903}) + -- ({\dx*3.9200},{\dy*5.1826}) + -- ({\dx*3.9400},{\dy*5.3799}) + -- ({\dx*3.9600},{\dy*5.5819}) + -- ({\dx*3.9800},{\dy*5.7887}) + -- ({\dx*4.0000},{\dy*6.0000}) + -- ({\dx*4.0200},{\dy*6.2158}) + -- ({\dx*4.0400},{\dy*6.4359}) + -- ({\dx*4.0600},{\dy*6.6603}) + -- ({\dx*4.0800},{\dy*6.8889}) + -- ({\dx*4.0810},{\dy*6.9005})} +\def\gammasinone{({\dx*-0.9810},{\dy*-53.1410}) + -- ({\dx*-0.9800},{\dy*-50.5140}) + -- ({\dx*-0.9600},{\dy*-25.6055}) + -- ({\dx*-0.9400},{\dy*-17.3637}) + -- ({\dx*-0.9200},{\dy*-13.2884}) + -- ({\dx*-0.9000},{\dy*-10.8796}) + -- ({\dx*-0.8800},{\dy*-9.3036}) + -- ({\dx*-0.8600},{\dy*-8.2033}) + -- ({\dx*-0.8400},{\dy*-7.3999}) + -- ({\dx*-0.8200},{\dy*-6.7941}) + -- ({\dx*-0.8000},{\dy*-6.3263}) + -- ({\dx*-0.7800},{\dy*-5.9586}) + -- ({\dx*-0.7600},{\dy*-5.6655}) + -- ({\dx*-0.7400},{\dy*-5.4296}) + -- ({\dx*-0.7200},{\dy*-5.2384}) + -- ({\dx*-0.7000},{\dy*-5.0827}) + -- ({\dx*-0.6800},{\dy*-4.9557}) + -- ({\dx*-0.6600},{\dy*-4.8523}) + -- ({\dx*-0.6400},{\dy*-4.7685}) + -- ({\dx*-0.6200},{\dy*-4.7012}) + -- ({\dx*-0.6000},{\dy*-4.6480}) + -- ({\dx*-0.5800},{\dy*-4.6072}) + -- ({\dx*-0.5600},{\dy*-4.5773}) + -- ({\dx*-0.5400},{\dy*-4.5573}) + -- ({\dx*-0.5200},{\dy*-4.5467}) + -- ({\dx*-0.5000},{\dy*-4.5449}) + -- ({\dx*-0.4800},{\dy*-4.5519}) + -- ({\dx*-0.4600},{\dy*-4.5677}) + -- ({\dx*-0.4400},{\dy*-4.5928}) + -- ({\dx*-0.4200},{\dy*-4.6279}) + -- ({\dx*-0.4000},{\dy*-4.6740}) + -- ({\dx*-0.3800},{\dy*-4.7325}) + -- ({\dx*-0.3600},{\dy*-4.8052}) + -- ({\dx*-0.3400},{\dy*-4.8944}) + -- ({\dx*-0.3200},{\dy*-5.0033}) + -- ({\dx*-0.3000},{\dy*-5.1359}) + -- ({\dx*-0.2800},{\dy*-5.2972}) + -- ({\dx*-0.2600},{\dy*-5.4942}) + -- ({\dx*-0.2400},{\dy*-5.7359}) + -- ({\dx*-0.2200},{\dy*-6.0350}) + -- ({\dx*-0.2000},{\dy*-6.4089}) + -- ({\dx*-0.1800},{\dy*-6.8830}) + -- ({\dx*-0.1600},{\dy*-7.4952}) + -- ({\dx*-0.1400},{\dy*-8.3052}) + -- ({\dx*-0.1200},{\dy*-9.4124}) + -- ({\dx*-0.1000},{\dy*-10.9953}) + -- ({\dx*-0.0800},{\dy*-13.4114}) + -- ({\dx*-0.0600},{\dy*-17.4941}) + -- ({\dx*-0.0400},{\dy*-25.7436}) + -- ({\dx*-0.0200},{\dy*-50.6602}) + -- ({\dx*-0.0190},{\dy*-53.2876})} +\def\gammasintwo{({\dx*-1.9810},{\dy*26.8549}) + -- ({\dx*-1.9800},{\dy*25.5432}) + -- ({\dx*-1.9600},{\dy*13.1254}) + -- ({\dx*-1.9400},{\dy*9.0411}) + -- ({\dx*-1.9200},{\dy*7.0402}) + -- ({\dx*-1.9000},{\dy*5.8725}) + -- ({\dx*-1.8800},{\dy*5.1211}) + -- ({\dx*-1.8600},{\dy*4.6073}) + -- ({\dx*-1.8400},{\dy*4.2416}) + -- ({\dx*-1.8200},{\dy*3.9745}) + -- ({\dx*-1.8000},{\dy*3.7759}) + -- ({\dx*-1.7800},{\dy*3.6268}) + -- ({\dx*-1.7600},{\dy*3.5146}) + -- ({\dx*-1.7400},{\dy*3.4305}) + -- ({\dx*-1.7200},{\dy*3.3681}) + -- ({\dx*-1.7000},{\dy*3.3229}) + -- ({\dx*-1.6800},{\dy*3.2916}) + -- ({\dx*-1.6600},{\dy*3.2715}) + -- ({\dx*-1.6400},{\dy*3.2607}) + -- ({\dx*-1.6200},{\dy*3.2578}) + -- ({\dx*-1.6000},{\dy*3.2616}) + -- ({\dx*-1.5800},{\dy*3.2715}) + -- ({\dx*-1.5600},{\dy*3.2868}) + -- ({\dx*-1.5400},{\dy*3.3072}) + -- ({\dx*-1.5200},{\dy*3.3327}) + -- ({\dx*-1.5000},{\dy*3.3633}) + -- ({\dx*-1.4800},{\dy*3.3993}) + -- ({\dx*-1.4600},{\dy*3.4412}) + -- ({\dx*-1.4400},{\dy*3.4896}) + -- ({\dx*-1.4200},{\dy*3.5456}) + -- ({\dx*-1.4000},{\dy*3.6103}) + -- ({\dx*-1.3800},{\dy*3.6854}) + -- ({\dx*-1.3600},{\dy*3.7727}) + -- ({\dx*-1.3400},{\dy*3.8749}) + -- ({\dx*-1.3200},{\dy*3.9951}) + -- ({\dx*-1.3000},{\dy*4.1374}) + -- ({\dx*-1.2800},{\dy*4.3070}) + -- ({\dx*-1.2600},{\dy*4.5109}) + -- ({\dx*-1.2400},{\dy*4.7583}) + -- ({\dx*-1.2200},{\dy*5.0617}) + -- ({\dx*-1.2000},{\dy*5.4387}) + -- ({\dx*-1.1800},{\dy*5.9148}) + -- ({\dx*-1.1600},{\dy*6.5279}) + -- ({\dx*-1.1400},{\dy*7.3376}) + -- ({\dx*-1.1200},{\dy*8.4433}) + -- ({\dx*-1.1000},{\dy*10.0238}) + -- ({\dx*-1.0800},{\dy*12.4364}) + -- ({\dx*-1.0600},{\dy*16.5145}) + -- ({\dx*-1.0400},{\dy*24.7583}) + -- ({\dx*-1.0200},{\dy*49.6681}) + -- ({\dx*-1.0190},{\dy*52.2951})} +\def\gammasinthree{({\dx*-2.9810},{\dy*-9.0483}) + -- ({\dx*-2.9800},{\dy*-8.6133}) + -- ({\dx*-2.9600},{\dy*-4.5173}) + -- ({\dx*-2.9400},{\dy*-3.1989}) + -- ({\dx*-2.9200},{\dy*-2.5746}) + -- ({\dx*-2.9000},{\dy*-2.2274}) + -- ({\dx*-2.8800},{\dy*-2.0184}) + -- ({\dx*-2.8600},{\dy*-1.8878}) + -- ({\dx*-2.8400},{\dy*-1.8057}) + -- ({\dx*-2.8200},{\dy*-1.7552}) + -- ({\dx*-2.8000},{\dy*-1.7264}) + -- ({\dx*-2.7800},{\dy*-1.7127}) + -- ({\dx*-2.7600},{\dy*-1.7099}) + -- ({\dx*-2.7400},{\dy*-1.7149}) + -- ({\dx*-2.7200},{\dy*-1.7255}) + -- ({\dx*-2.7000},{\dy*-1.7401}) + -- ({\dx*-2.6800},{\dy*-1.7575}) + -- ({\dx*-2.6600},{\dy*-1.7768}) + -- ({\dx*-2.6400},{\dy*-1.7972}) + -- ({\dx*-2.6200},{\dy*-1.8183}) + -- ({\dx*-2.6000},{\dy*-1.8397}) + -- ({\dx*-2.5800},{\dy*-1.8612}) + -- ({\dx*-2.5600},{\dy*-1.8825}) + -- ({\dx*-2.5400},{\dy*-1.9036}) + -- ({\dx*-2.5200},{\dy*-1.9245}) + -- ({\dx*-2.5000},{\dy*-1.9453}) + -- ({\dx*-2.4800},{\dy*-1.9663}) + -- ({\dx*-2.4600},{\dy*-1.9877}) + -- ({\dx*-2.4400},{\dy*-2.0099}) + -- ({\dx*-2.4200},{\dy*-2.0335}) + -- ({\dx*-2.4000},{\dy*-2.0591}) + -- ({\dx*-2.3800},{\dy*-2.0876}) + -- ({\dx*-2.3600},{\dy*-2.1200}) + -- ({\dx*-2.3400},{\dy*-2.1578}) + -- ({\dx*-2.3200},{\dy*-2.2024}) + -- ({\dx*-2.3000},{\dy*-2.2561}) + -- ({\dx*-2.2800},{\dy*-2.3216}) + -- ({\dx*-2.2600},{\dy*-2.4024}) + -- ({\dx*-2.2400},{\dy*-2.5032}) + -- ({\dx*-2.2200},{\dy*-2.6303}) + -- ({\dx*-2.2000},{\dy*-2.7928}) + -- ({\dx*-2.1800},{\dy*-3.0032}) + -- ({\dx*-2.1600},{\dy*-3.2809}) + -- ({\dx*-2.1400},{\dy*-3.6556}) + -- ({\dx*-2.1200},{\dy*-4.1772}) + -- ({\dx*-2.1000},{\dy*-4.9351}) + -- ({\dx*-2.0800},{\dy*-6.1082}) + -- ({\dx*-2.0600},{\dy*-8.1132}) + -- ({\dx*-2.0400},{\dy*-12.2003}) + -- ({\dx*-2.0200},{\dy*-24.6199}) + -- ({\dx*-2.0190},{\dy*-25.9316})} +\def\gammasinfour{({\dx*-3.9950},{\dy*8.4124}) + -- ({\dx*-3.9800},{\dy*2.2112}) + -- ({\dx*-3.9600},{\dy*1.2344}) + -- ({\dx*-3.9400},{\dy*0.9517}) + -- ({\dx*-3.9200},{\dy*0.8420}) + -- ({\dx*-3.9000},{\dy*0.8009}) + -- ({\dx*-3.8800},{\dy*0.7935}) + -- ({\dx*-3.8600},{\dy*0.8045}) + -- ({\dx*-3.8400},{\dy*0.8265}) + -- ({\dx*-3.8200},{\dy*0.8550}) + -- ({\dx*-3.8000},{\dy*0.8874}) + -- ({\dx*-3.7800},{\dy*0.9219}) + -- ({\dx*-3.7600},{\dy*0.9573}) + -- ({\dx*-3.7400},{\dy*0.9926}) + -- ({\dx*-3.7200},{\dy*1.0272}) + -- ({\dx*-3.7000},{\dy*1.0607}) + -- ({\dx*-3.6800},{\dy*1.0925}) + -- ({\dx*-3.6600},{\dy*1.1223}) + -- ({\dx*-3.6400},{\dy*1.1500}) + -- ({\dx*-3.6200},{\dy*1.1752}) + -- ({\dx*-3.6000},{\dy*1.1979}) + -- ({\dx*-3.5800},{\dy*1.2179}) + -- ({\dx*-3.5600},{\dy*1.2351}) + -- ({\dx*-3.5400},{\dy*1.2496}) + -- ({\dx*-3.5200},{\dy*1.2612}) + -- ({\dx*-3.5000},{\dy*1.2701}) + -- ({\dx*-3.4800},{\dy*1.2763}) + -- ({\dx*-3.4600},{\dy*1.2798}) + -- ({\dx*-3.4400},{\dy*1.2810}) + -- ({\dx*-3.4200},{\dy*1.2800}) + -- ({\dx*-3.4000},{\dy*1.2769}) + -- ({\dx*-3.3800},{\dy*1.2723}) + -- ({\dx*-3.3600},{\dy*1.2665}) + -- ({\dx*-3.3400},{\dy*1.2600}) + -- ({\dx*-3.3200},{\dy*1.2534}) + -- ({\dx*-3.3000},{\dy*1.2475}) + -- ({\dx*-3.2800},{\dy*1.2434}) + -- ({\dx*-3.2600},{\dy*1.2423}) + -- ({\dx*-3.2400},{\dy*1.2458}) + -- ({\dx*-3.2200},{\dy*1.2563}) + -- ({\dx*-3.2000},{\dy*1.2768}) + -- ({\dx*-3.1800},{\dy*1.3117}) + -- ({\dx*-3.1600},{\dy*1.3676}) + -- ({\dx*-3.1400},{\dy*1.4544}) + -- ({\dx*-3.1200},{\dy*1.5890}) + -- ({\dx*-3.1000},{\dy*1.8013}) + -- ({\dx*-3.0800},{\dy*2.1511}) + -- ({\dx*-3.0600},{\dy*2.7775}) + -- ({\dx*-3.0400},{\dy*4.0974}) + -- ({\dx*-3.0200},{\dy*8.1943}) + -- ({\dx*-3.0050},{\dy*33.1416})} +\def\gammasinfive{({\dx*-4.9990},{\dy*-8.3507}) + -- ({\dx*-4.9800},{\dy*-0.4942}) + -- ({\dx*-4.9600},{\dy*-0.3489}) + -- ({\dx*-4.9400},{\dy*-0.3421}) + -- ({\dx*-4.9200},{\dy*-0.3693}) + -- ({\dx*-4.9000},{\dy*-0.4094}) + -- ({\dx*-4.8800},{\dy*-0.4553}) + -- ({\dx*-4.8600},{\dy*-0.5037}) + -- ({\dx*-4.8400},{\dy*-0.5530}) + -- ({\dx*-4.8200},{\dy*-0.6021}) + -- ({\dx*-4.8000},{\dy*-0.6502}) + -- ({\dx*-4.7800},{\dy*-0.6969}) + -- ({\dx*-4.7600},{\dy*-0.7418}) + -- ({\dx*-4.7400},{\dy*-0.7846}) + -- ({\dx*-4.7200},{\dy*-0.8249}) + -- ({\dx*-4.7000},{\dy*-0.8626}) + -- ({\dx*-4.6800},{\dy*-0.8973}) + -- ({\dx*-4.6600},{\dy*-0.9291}) + -- ({\dx*-4.6400},{\dy*-0.9577}) + -- ({\dx*-4.6200},{\dy*-0.9829}) + -- ({\dx*-4.6000},{\dy*-1.0047}) + -- ({\dx*-4.5800},{\dy*-1.0230}) + -- ({\dx*-4.5600},{\dy*-1.0377}) + -- ({\dx*-4.5400},{\dy*-1.0488}) + -- ({\dx*-4.5200},{\dy*-1.0563}) + -- ({\dx*-4.5000},{\dy*-1.0600}) + -- ({\dx*-4.4800},{\dy*-1.0601}) + -- ({\dx*-4.4600},{\dy*-1.0566}) + -- ({\dx*-4.4400},{\dy*-1.0496}) + -- ({\dx*-4.4200},{\dy*-1.0390}) + -- ({\dx*-4.4000},{\dy*-1.0251}) + -- ({\dx*-4.3800},{\dy*-1.0080}) + -- ({\dx*-4.3600},{\dy*-0.9878}) + -- ({\dx*-4.3400},{\dy*-0.9647}) + -- ({\dx*-4.3200},{\dy*-0.9390}) + -- ({\dx*-4.3000},{\dy*-0.9110}) + -- ({\dx*-4.2800},{\dy*-0.8810}) + -- ({\dx*-4.2600},{\dy*-0.8495}) + -- ({\dx*-4.2400},{\dy*-0.8169}) + -- ({\dx*-4.2200},{\dy*-0.7841}) + -- ({\dx*-4.2000},{\dy*-0.7518}) + -- ({\dx*-4.1800},{\dy*-0.7215}) + -- ({\dx*-4.1600},{\dy*-0.6947}) + -- ({\dx*-4.1400},{\dy*-0.6742}) + -- ({\dx*-4.1200},{\dy*-0.6644}) + -- ({\dx*-4.1000},{\dy*-0.6730}) + -- ({\dx*-4.0800},{\dy*-0.7150}) + -- ({\dx*-4.0600},{\dy*-0.8253}) + -- ({\dx*-4.0400},{\dy*-1.1085}) + -- ({\dx*-4.0200},{\dy*-2.0855}) + -- ({\dx*-4.0010},{\dy*-41.6072})} +\def\gammasinsix{({\dx*-5.9998},{\dy*6.9477}) + -- ({\dx*-5.9800},{\dy*0.1349}) + -- ({\dx*-5.9600},{\dy*0.1629}) + -- ({\dx*-5.9400},{\dy*0.2134}) + -- ({\dx*-5.9200},{\dy*0.2691}) + -- ({\dx*-5.9000},{\dy*0.3260}) + -- ({\dx*-5.8800},{\dy*0.3829}) + -- ({\dx*-5.8600},{\dy*0.4391}) + -- ({\dx*-5.8400},{\dy*0.4940}) + -- ({\dx*-5.8200},{\dy*0.5472}) + -- ({\dx*-5.8000},{\dy*0.5985}) + -- ({\dx*-5.7800},{\dy*0.6477}) + -- ({\dx*-5.7600},{\dy*0.6945}) + -- ({\dx*-5.7400},{\dy*0.7387}) + -- ({\dx*-5.7200},{\dy*0.7800}) + -- ({\dx*-5.7000},{\dy*0.8184}) + -- ({\dx*-5.6800},{\dy*0.8537}) + -- ({\dx*-5.6600},{\dy*0.8856}) + -- ({\dx*-5.6400},{\dy*0.9142}) + -- ({\dx*-5.6200},{\dy*0.9392}) + -- ({\dx*-5.6000},{\dy*0.9606}) + -- ({\dx*-5.5800},{\dy*0.9783}) + -- ({\dx*-5.5600},{\dy*0.9923}) + -- ({\dx*-5.5400},{\dy*1.0024}) + -- ({\dx*-5.5200},{\dy*1.0086}) + -- ({\dx*-5.5000},{\dy*1.0109}) + -- ({\dx*-5.4800},{\dy*1.0094}) + -- ({\dx*-5.4600},{\dy*1.0039}) + -- ({\dx*-5.4400},{\dy*0.9947}) + -- ({\dx*-5.4200},{\dy*0.9816}) + -- ({\dx*-5.4000},{\dy*0.9648}) + -- ({\dx*-5.3800},{\dy*0.9443}) + -- ({\dx*-5.3600},{\dy*0.9203}) + -- ({\dx*-5.3400},{\dy*0.8929}) + -- ({\dx*-5.3200},{\dy*0.8621}) + -- ({\dx*-5.3000},{\dy*0.8283}) + -- ({\dx*-5.2800},{\dy*0.7914}) + -- ({\dx*-5.2600},{\dy*0.7519}) + -- ({\dx*-5.2400},{\dy*0.7098}) + -- ({\dx*-5.2200},{\dy*0.6655}) + -- ({\dx*-5.2000},{\dy*0.6193}) + -- ({\dx*-5.1800},{\dy*0.5717}) + -- ({\dx*-5.1600},{\dy*0.5230}) + -- ({\dx*-5.1400},{\dy*0.4741}) + -- ({\dx*-5.1200},{\dy*0.4260}) + -- ({\dx*-5.1000},{\dy*0.3804}) + -- ({\dx*-5.0800},{\dy*0.3405}) + -- ({\dx*-5.0600},{\dy*0.3135}) + -- ({\dx*-5.0400},{\dy*0.3204}) + -- ({\dx*-5.0200},{\dy*0.4657}) + -- ({\dx*-5.0002},{\dy*41.6531})} diff --git a/buch/papers/laguerre/images/gammaplot.pdf b/buch/papers/laguerre/images/gammaplot.pdf Binary files differnew file mode 100644 index 0000000..7c195f2 --- /dev/null +++ b/buch/papers/laguerre/images/gammaplot.pdf diff --git a/buch/papers/laguerre/images/gammaplot.tex b/buch/papers/laguerre/images/gammaplot.tex new file mode 100644 index 0000000..5a68f0a --- /dev/null +++ b/buch/papers/laguerre/images/gammaplot.tex @@ -0,0 +1,73 @@ +% +% gammaplot.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\input{gammapaths.tex} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\definecolor{mainColor}{HTML}{D72864} % OST pink + +\draw[->] (-6.1,0) -- (5.3,0) coordinate[label={$z$}]; +\draw[->] (0,-5.1) -- (0,6.4) coordinate[label={right:$\Gamma(z)$}]; + +\foreach \x in {-1,-2,-3,-4,-5,-6}{ + \draw (\x,-0.1) -- (\x,0.1); + \draw[line width=0.1pt] (\x,-5) -- (\x,6.2); +} +\foreach \x in {1,2,3,4,5}{ + \draw (\x,-0.1) -- (\x,0.1); + \node at (\x,0) [below] {$\x$}; +} +\foreach \y in {-5,-4,-3,-2,-1,1,2,3,4,5,6}{ + \draw (-0.1,\y) -- (0.1,\y); +} +\foreach \y in {1,2,3,4,5,6}{ + \node at (0,\y) [left] {$\y$}; +} +\foreach \y in {-1,-2,-3,-4,-5}{ + \node at (0,\y) [right] {$\y$}; +} +\foreach \x in {-1,-3,-5}{ + \node at (\x,0) [below left] {$\x$}; +} +\foreach \x in {-2,-4,-6}{ + \node at (\x,0) [above left] {$\x$}; +} + +\def\dx{1} +\def\dy{1} + +\begin{scope} +\clip (-6.1,-5) rectangle (4.3,6.2); + +% \draw[color=darkgreen,line width=1.4pt] \gammasinplus; +% \draw[color=darkgreen,line width=1.4pt] \gammasinone; +% \draw[color=darkgreen,line width=1.4pt] \gammasintwo; +% \draw[color=darkgreen,line width=1.4pt] \gammasinthree; +% \draw[color=darkgreen,line width=1.4pt] \gammasinfour; +% \draw[color=darkgreen,line width=1.4pt] \gammasinfive; +% \draw[color=darkgreen,line width=1.4pt] \gammasinsix; + +\draw[color=mainColor,line width=1.4pt] \gammaplus; +\draw[color=mainColor,line width=1.4pt] \gammaone; +\draw[color=mainColor,line width=1.4pt] \gammatwo; +\draw[color=mainColor,line width=1.4pt] \gammathree; +\draw[color=mainColor,line width=1.4pt] \gammafour; +\draw[color=mainColor,line width=1.4pt] \gammafive; +\draw[color=mainColor,line width=1.4pt] \gammasix; + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/laguerre/images/integrand.pdf b/buch/papers/laguerre/images/integrand.pdf Binary files differnew file mode 100644 index 0000000..76be412 --- /dev/null +++ b/buch/papers/laguerre/images/integrand.pdf diff --git a/buch/papers/laguerre/images/integrand_exp.pdf b/buch/papers/laguerre/images/integrand_exp.pdf Binary files differnew file mode 100644 index 0000000..5fe7a7b --- /dev/null +++ b/buch/papers/laguerre/images/integrand_exp.pdf diff --git a/buch/papers/laguerre/images/laguerre_poly.pdf b/buch/papers/laguerre/images/laguerre_poly.pdf Binary files differnew file mode 100644 index 0000000..f31d81d --- /dev/null +++ b/buch/papers/laguerre/images/laguerre_poly.pdf diff --git a/buch/papers/laguerre/images/rel_error_complex.pdf b/buch/papers/laguerre/images/rel_error_complex.pdf Binary files differnew file mode 100644 index 0000000..c7bb37a --- /dev/null +++ b/buch/papers/laguerre/images/rel_error_complex.pdf diff --git a/buch/papers/laguerre/images/rel_error_mirror.pdf b/buch/papers/laguerre/images/rel_error_mirror.pdf Binary files differnew file mode 100644 index 0000000..8a27d41 --- /dev/null +++ b/buch/papers/laguerre/images/rel_error_mirror.pdf diff --git a/buch/papers/laguerre/images/rel_error_range.pdf b/buch/papers/laguerre/images/rel_error_range.pdf Binary files differnew file mode 100644 index 0000000..bb8a2d7 --- /dev/null +++ b/buch/papers/laguerre/images/rel_error_range.pdf diff --git a/buch/papers/laguerre/images/rel_error_shifted.pdf b/buch/papers/laguerre/images/rel_error_shifted.pdf Binary files differnew file mode 100644 index 0000000..b7e72dc --- /dev/null +++ b/buch/papers/laguerre/images/rel_error_shifted.pdf diff --git a/buch/papers/laguerre/images/rel_error_simple.pdf b/buch/papers/laguerre/images/rel_error_simple.pdf Binary files differnew file mode 100644 index 0000000..0072d28 --- /dev/null +++ b/buch/papers/laguerre/images/rel_error_simple.pdf diff --git a/buch/papers/laguerre/images/targets.pdf b/buch/papers/laguerre/images/targets.pdf Binary files differnew file mode 100644 index 0000000..dc61c88 --- /dev/null +++ b/buch/papers/laguerre/images/targets.pdf diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex index 1fe0f8b..133d686 100644 --- a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -8,13 +8,35 @@ \begin{refsection} \chapterauthor{Patrik Müller} -Hier kommt eine Einleitung. +{\parindent0pt Die} Laguerre\--Polynome, +benannt nach Edmond Laguerre (1834 -- 1886), +sind Lösungen der ebenfalls nach %Laguerre +ihm +benannten Differentialgleichung. +Laguerre entdeckte diese Polynome, als er Approximations\-methoden +für das Integral +% $\int_0^\infty \exp(-x) / x \, dx $ +\begin{align*} +\int_0^\infty \frac{e^{-x}}{x} \, dx +\end{align*} +suchte. +Darum möchten wir uns in diesem Kapitel, +ganz im Sinne des Entdeckers, +den Laguerre-Polynomen für Approximationen von Integralen mit +exponentiell abfallenden Funktionen widmen. +Namentlich werden wir versuchen, mittels Laguerre-Polynomen und +der Gauss-Quadratur eine geeignete Approximation für die Gamma-Funktion zu +finden. + +Laguerre-Polynome tauchen zudem auch in der Quantenmechanik im radialen Anteil +der Lösung für die Schrödinger-Gleichung eines Wasserstoffatoms auf. \input{papers/laguerre/definition} \input{papers/laguerre/eigenschaften} \input{papers/laguerre/quadratur} -\input{papers/laguerre/transformation} -\input{papers/laguerre/wasserstoff} +\input{papers/laguerre/gamma} +% \input{papers/laguerre/transformation} +% \input{papers/laguerre/wasserstoff} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/laguerre/packages.tex b/buch/papers/laguerre/packages.tex index ab55228..a80d091 100644 --- a/buch/papers/laguerre/packages.tex +++ b/buch/papers/laguerre/packages.tex @@ -6,5 +6,4 @@ % if your paper needs special packages, add package commands as in the % following example -\usepackage{derivative} - +\DeclareMathOperator{\real}{Re}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/presentation.tex b/buch/papers/laguerre/presentation/presentation.tex new file mode 100644 index 0000000..3db69f5 --- /dev/null +++ b/buch/papers/laguerre/presentation/presentation.tex @@ -0,0 +1,134 @@ +\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer} + +% style +\mode<presentation>{ + \usetheme{Frankfurt} +} +%packages +\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-} +\usepackage[ngerman]{babel} +\usepackage{graphicx} +\usepackage{array} + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\usepackage{ragged2e} + +\usepackage{bm} % bold math +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{multirow} % multi row in tables +\usepackage{booktabs} %toprule midrule bottomrue in tables +\usepackage{scrextend} +\usepackage{textgreek} +\usepackage[rgb]{xcolor} + +\usepackage{ marvosym } % \Lightning + +\usepackage{multimedia} % embedded videos + +\usepackage{tikz} +\usepackage{pgf} +\usepackage{pgfplots} + +\usepackage{algorithmic} + +%citations +\usepackage[style=verbose,backend=biber]{biblatex} +\addbibresource{references.bib} + + +%math font +\usefonttheme[onlymath]{serif} + +%Beamer Template modifications +%\definecolor{mainColor}{HTML}{0065A3} % HSR blue +\definecolor{mainColor}{HTML}{D72864} % OST pink +\definecolor{invColor}{HTML}{28d79b} % OST pink +\definecolor{dgreen}{HTML}{38ad36} % Dark green + +%\definecolor{mainColor}{HTML}{000000} % HSR blue +\setbeamercolor{palette primary}{bg=white,fg=mainColor} +\setbeamercolor{palette secondary}{bg=orange,fg=mainColor} +\setbeamercolor{palette tertiary}{bg=yellow,fg=red} +\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic +\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points) +\setbeamercolor{section in toc}{fg=black} % TOC sections +\setbeamertemplate{section in toc}[sections numbered] +\setbeamertemplate{subsection in toc}{% + \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par} + +\setbeamertemplate{itemize items}[circle] +\setbeamertemplate{description item}[circle] +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\beamertemplatenavigationsymbolsempty + +\setbeamercolor{footline}{fg=gray} +\setbeamertemplate{footline}{% + \hfill\usebeamertemplate***{navigation symbols} + \hspace{0.5cm} + \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm} +} + +\usepackage{caption} +\captionsetup{labelformat=empty} + +%Title Page +\title{Laguerre-Polynome} +\subtitle{Anwendung: Approximation der Gamma-Funktion} +\author{Patrik Müller} +% \institute{OST Ostschweizer Fachhochschule} +% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}} +\date{\today} + +\input{../packages.tex} + +\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}% + +\newcommand*{\HL}{\textcolor{mainColor}} +\newcommand*{\RD}{\textcolor{red}} +\newcommand*{\BL}{\textcolor{blue}} +\newcommand*{\GN}{\textcolor{dgreen}} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + + +\makeatletter +\newcount\my@repeat@count +\newcommand{\myrepeat}[2]{% + \begingroup + \my@repeat@count=\z@ + \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}% + \endgroup +} +\makeatother + +\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings} + +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + +\begin{frame}{Inhaltsverzeichnis} + \tableofcontents +\end{frame} + +\input{sections/laguerre} + +\input{sections/gaussquad} + +\input{sections/gamma} + +\input{sections/gamma_approx} + +\appendix +\begin{frame} + % \centering + % \Large + % \textbf{Vielen Dank für die Aufmerksamkeit} +\end{frame} + +\end{document} diff --git a/buch/papers/laguerre/presentation/sections/gamma.tex b/buch/papers/laguerre/presentation/sections/gamma.tex new file mode 100644 index 0000000..7dca39b --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gamma.tex @@ -0,0 +1,51 @@ +\section{Gamma-Funktion} + +\begin{frame}{Gamma-Funktion} +\begin{columns} + +\begin{column}{0.55\textwidth} +\begin{figure}[h] +\vspace{-16pt} +\centering +% \scalebox{0.51}{\input{../images/gammaplot.pdf}} +\includegraphics[width=1\textwidth]{../images/gammaplot.pdf} +% \caption{Gamma-Funktion} +\end{figure} +\end{column} + +\begin{column}{0.45\textwidth} +Verallgemeinerung der Fakultät +\begin{align*} +\Gamma(n) = (n-1)! +\end{align*} + +Integralformel +\begin{align*} +\Gamma(z) += +\int_0^\infty x^{z-1} e^{-x} \, dx +,\quad +\operatorname{Re} z > 0 +\end{align*} + +Funktionalgleichung +\begin{align*} +z \Gamma(z) += +\Gamma(z + 1) +\end{align*} + +Reflektionsformel +\begin{align*} +\Gamma(z) \Gamma(1 - z) += +\frac{\pi}{\sin \pi z} +, \quad +\text{für } +z \notin \mathbb{Z} +\end{align*} + +\end{column} +\end{columns} + +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/gamma_approx.tex b/buch/papers/laguerre/presentation/sections/gamma_approx.tex new file mode 100644 index 0000000..b5e1131 --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gamma_approx.tex @@ -0,0 +1,201 @@ +\section{Approximieren der Gamma-Funktion} + +\begin{frame}{Anwenden der Gauss-Laguerre-Quadratur auf $\Gamma(z)$} + +\begin{align*} +\Gamma(z) + & = +\int_0^\infty x^{z-1} e^{-x} \, dx +\uncover<2->{ +\approx +\sum_{i=1}^{n} f(x_i) A_i +} +\uncover<3->{ += +\sum_{i=1}^{n} x^{z-1} A_i +} +\\\\ +\uncover<4->{ + & \text{wobei } +A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} +\text{ und $x_i$ die Nullstellen von $L_n(x)$} +} +\end{align*} + +\end{frame} + +\begin{frame}{Fehlerabschätzung} +\begin{align*} +R_n(\xi) + & = +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +\\ + & = +(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z - 2n - 1} +,\quad +0 < \xi < \infty +\end{align*} + +% \textbf{Probleme:} +\begin{itemize} +\item Funktion ist unbeschränkt +\item Maximum von $R_n$ gibt oberes Limit des Fehlers an +\uncover<2->{\item[$\Rightarrow$] Schwierig ein Maximum von $R_n(\xi)$ zu finden} +\end{itemize} +\end{frame} + +\begin{frame}{Einfacher Ansatz} + +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/rel_error_simple.pgf}} +% \resizebox{!}{0.72\textheight}{\input{../images/rel_error_simple.pgf}} +\includegraphics[width=0.77\textwidth]{../images/rel_error_simple.pdf} +\caption{Relativer Fehler des einfachen Ansatzes für verschiedene reelle Werte +von $z$ und Grade $n$ der Laguerre-Polynome} +\end{figure} + +\end{frame} + +\begin{frame}{Wieso sind die Resultate so schlecht?} + +\textbf{Beobachtungen} +\begin{itemize} +\item Wenn $z \in \mathbb{Z}$ relativer Fehler $\rightarrow 0$ +\item Gewisse Periodizität zu erkennen +\item Für grosse und kleine $z$ ergibt sich ein schlechter relativer Fehler +\item Es gibt Intervalle $[a,a+1]$ mit minimalem relativem Fehler +\item $a$ ist abhängig von $n$ +\end{itemize} + +\uncover<2->{ +\textbf{Ursache?} +\begin{itemize} +\item Vermutung: Integrand ist problematisch +} +\uncover<3->{ +\item[$\Rightarrow$] Analysieren von $f(x)$ und dem Integranden +} +\end{itemize} +\end{frame} + +\begin{frame}{$f(x) = x^z$} +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/integrand.pgf}} +\includegraphics[width=0.8\textwidth]{../images/integrand.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{frame} + +\begin{frame}{Integrand $x^z e^{-x}$} +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/integrand_exp.pgf}} +\includegraphics[width=0.8\textwidth]{../images/integrand_exp.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{frame} + +\begin{frame}{Neuer Ansatz?} + +\textbf{Vermutung} +\begin{itemize} +\item Es gibt Intervalle $[a(n), a(n)+1]$ in denen der relative Fehler minimal +ist +\item $a(n) > 0$ +\end{itemize} + +\uncover<2->{ +\textbf{Idee} +\begin{itemize} +\item[$\Rightarrow$] Berechnen von $\Gamma(z)$ im geeigneten Intervall und dann +mit Funktionalgleichung zurückverschieben +\end{itemize} +} + +\uncover<3->{ +\textbf{Wie finden wir $\boldsymbol{a(n)}$?} +\begin{itemize} +\item Minimieren des Fehlerterms mit zusätzlichem Verschiebungsterm +} +\uncover<4->{$\Rightarrow$ Schwierig das Maximum des Fehlerterms zu bestimmen} +\uncover<5->{\item Empirisch $a(n)$ bestimmen} +\uncover<6->{$\Rightarrow$ Sinnvoll, +da Gauss-Quadratur nur für kleine $n$ praktischen Nutzen hat} +\end{itemize} +\end{frame} + +\begin{frame}{Verschiebungsterm} +\begin{columns} +\begin{column}{0.625\textwidth} +\begin{figure}[h] +\centering +\includegraphics[width=1\textwidth]{../images/targets.pdf} +\caption{Optimaler Verschiebungsterm $m^*$ in Abhängigkeit von $z$ und $n$} +\end{figure} +\end{column} +\begin{column}{0.375\textwidth} +\begin{align*} +\Gamma(z) +\approx +\frac{1}{(z-m)_{m}} \sum_{i=1}^{n} x_i^{z + m - 1} A_i +\end{align*} +\end{column} +\end{columns} +\end{frame} + +\begin{frame}{Schätzen von $m^*$} +\begin{columns} +\begin{column}{0.65\textwidth} +\begin{figure} +\centering +\vspace{-12pt} +% \scalebox{0.7}{\input{../images/estimates.pgf}} +\includegraphics[width=1.0\textwidth]{../images/estimates.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{column} +\begin{column}{0.34\textwidth} +\begin{align*} +\hat{m} +&= +\alpha n + \beta +\\ +&\approx +1.34154 n + 0.848786 +\\ +m^* +&= +\lceil \hat{m} - \operatorname{Re}z \rceil +\end{align*} +\end{column} +\end{columns} + +\end{frame} + +\begin{frame}{} +\begin{figure}[h] +\centering +% \scalebox{0.6}{\input{../images/rel_error_shifted.pgf}} +\includegraphics{../images/rel_error_shifted.pdf} +\caption{Relativer Fehler mit $n=8$, unterschiedlichen Verschiebungstermen $m$ und $z\in(0, 1)$} +\end{figure} +\end{frame} + +\begin{frame}{} +\begin{figure}[h] +\centering +% \scalebox{0.6}{\input{../images/rel_error_range.pgf}} +\includegraphics{../images/rel_error_range.pdf} +\caption{Relativer Fehler mit $n=8$, Verschiebungsterm $m^*$ und $z\in(-5, 5)$} +\end{figure} +\end{frame} + +\begin{frame}{Vergleich mit Lanczos-Methode} +Maximaler relativer Fehler für $n=6$ +\begin{itemize} + \item Lanczos-Methode $< 10^{-12}$ + \item Unsere Methode $\approx 10^{-6}$ +\end{itemize} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/gaussquad.tex b/buch/papers/laguerre/presentation/sections/gaussquad.tex new file mode 100644 index 0000000..4d973b8 --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gaussquad.tex @@ -0,0 +1,67 @@ +\section{Gauss-Quadratur} + +\begin{frame}{Gauss-Quadratur} +\textbf{Idee} +\begin{itemize}[<+->] +\item Polynome können viele Funktionen approximieren +\item Wenn Verfahren gut für Polynome funktioniert, +sollte es auch für andere Funktionen funktionieren +\item Integrieren eines Interpolationspolynom +\item Interpolationspolynom ist durch Funktionswerte $f(x_i)$ bestimmt +$\Rightarrow$ Integral kann durch Funktionswerte berechnet werden +\item Evaluation der Funktionswerte an geeigneten Stellen +\end{itemize} +\end{frame} + +\begin{frame}{Gauss-Quadratur} +\begin{align*} +\int_{-1}^{1} f(x) \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\end{align*} + +\begin{itemize}[<+->] +\item Exakt für Polynome mit Grad $2n-1$ +\item Interpolationspolynome müssen orthogonal sein +\item Stützstellen $x_i$ sind Nullstellen des Polynoms +\item Fehler: +\begin{align*} +E += +\frac{f^{(2n)}(\xi)}{(2n)!} \int_{-1}^{1} l(x)^2 \, dx +,\quad +\text{wobei } +l(x) = \prod_{i=1}^n (x-x_i) +\end{align*} +\end{itemize} +\end{frame} + +\begin{frame}{Gauss-Laguerre-Quadratur} +\begin{itemize}[<+->] +\item Erweiterung des Integrationsintervall von $[-1, 1]$ auf $(a, b)$ +\item Hinzufügen einer Gewichtsfunktion +\item Bei uneigentlichen Integralen muss Gewichtsfunktion schneller als jedes +Integrationspolynom gegen $0$ gehen +\item[$\Rightarrow$] Für Laguerre-Polynome haben wir den Definitionsbereich +$(0, \infty)$ und die Gewichtsfunktion $w(x) = e^{-x}$ +\begin{align*} +\int_0^\infty & f(x) e^{-x} \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\\ + & \text{wobei } +A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} +\text{ und $x_i$ die Nullstellen von $L_n(x)$} +\end{align*} +\end{itemize} +\end{frame} + +\begin{frame}{Fehler der Gauss-Laguerre-Quadratur} +\begin{align*} +R_n += +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +,\quad +0 < \xi < \infty +\end{align*} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex new file mode 100644 index 0000000..f99214e --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -0,0 +1,91 @@ +\section{Laguerre-Polynome} + +\begin{frame}{Laguerre-Differentialgleichung} + +\begin{itemize} +\item Benannt nach Edmond Nicolas Laguerre (1834-1886) +\item Aus Artikel von 1879, +in dem er $\int_0^\infty \exp(-x)/x \, dx$ analysierte +\end{itemize} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +, \quad +n \in \mathbb{N}_0 +, \quad +x \in \mathbb{R} +\end{align*} + +\end{frame} + +\begin{frame}{Lösen der Differentialgleichung} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +\\ +\end{align*} + +\uncover<2->{ +\centering +\begin{tikzpicture}[remember picture,overlay] +%% use here too +\path[draw=mainColor, very thick,->](0, 1.1) to +node[anchor=west]{Potenzreihenansatz} (0, -0.8); +\end{tikzpicture} +} + +\begin{align*} +\uncover<3->{ +L_n(x) + & = +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +} +\end{align*} +\uncover<4->{ +\begin{itemize} + \item Die Lösungen der DGL sind die Laguerre-Polynome +\end{itemize} +} +\end{frame} + +\begin{frame} +\begin{figure}[h] +\centering +% \resizebox{0.74\textwidth}{!}{\input{../images/laguerre_poly.pgf}} +\includegraphics[width=0.7\textwidth]{../images/laguerre_poly.pdf} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\end{figure} +\end{frame} + +\begin{frame}{Orthogonalität} +\begin{itemize}[<+->] +\item Beweis: Umformen in Sturm-Liouville-Problem (siehe Paper) +\begin{alignat*}{5} +((p(x) &y'(x)))' + q(x) &y(x) +&= +\lambda &w(x) &y(x) +\\ +((x e^{-x} &y'(x)))' + 0 &y(x) +&= +n &e^{-x} &y(x) +\end{alignat*} +\item Definitionsbereich $(0, \infty)$ +\item Gewichtsfunktion $w(x) = e^{-x}$ +\end{itemize} + +\uncover<4->{ +\begin{align*} +\int_0^\infty e^{-x} L_n(x) L_m(x) \, dx += +0 +,\quad +n \neq m +,\quad +n, m \in \mathbb{N} +\end{align*} +} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/quadratur.tex b/buch/papers/laguerre/quadratur.tex index 8ab1af5..0e32012 100644 --- a/buch/papers/laguerre/quadratur.tex +++ b/buch/papers/laguerre/quadratur.tex @@ -3,27 +3,214 @@ % % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % -\section{Gauss-Laguerre Quadratur -\label{laguerre:section:quadratur}} - +\section{Gauss-Quadratur% + \label{laguerre:section:quadratur}} +\rhead{Gauss-Quadratur}% +Die Gauss-Quadratur ist ein numerisches Integrationsverfahren, +welches die Eigenschaften von orthogonalen Polynomen verwendet. +Herleitungen und Analysen der Gauss-Quadratur können im +Abschnitt~\ref{buch:orthogonal:section:gauss-quadratur} gefunden werden. +Als grundlegende Idee wird die Beobachtung, +dass viele Funktionen sich gut mit Polynomen approximieren lassen, +verwendet. +Stellt man also sicher, +dass ein Verfahren gut für Polynome funktioniert, +sollte es auch für andere Funktionen angemessene Resultate liefern. +Es wird ein Interpolationspolynom verwendet, +welches an den Punkten $x_0 < x_1 < \ldots < x_n$ +die Funktionwerte~$f(x_i)$ annimmt. +Als Resultat kann das Integral via einer gewichteten Summe der Form \begin{align} - \int_a^b f(x) w(x) - \approx - \sum_{i=1}^N f(x_i) A_i - \label{laguerre:gaussquadratur} +\int_a^b f(x) w(x) \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\label{laguerre:gaussquadratur} \end{align} +berechnet werden. +Die Gauss-Quadratur ist exakt für Polynome mit Grad $2n -1$, +wenn ein Interpolationspolynom von Grad $n$ gewählt wurde. +\subsection{Gauss-Laguerre-Quadratur% +\label{laguerre:subsection:gausslag-quadratur}} +Wir möchten nun die Gauss-Quadratur auf die Berechnung +von uneigentlichen Integralen erweitern, +spezifisch auf das Intervall~$(0, \infty)$. +Mit dem vorher beschriebenen Verfahren ist dies nicht direkt möglich. +% Mit einer Transformation +% \begin{align*} +% x +% = +% % a + +% \frac{1 - t}{t} +% \end{align*} +% die das unendliche Intervall~$(0, \infty)$ +% auf das Intervall~$[0, 1]$ transformiert, +% kann dies behoben werden. +% % Für unseren Fall gilt $a = 0$. +Das Integral eines Polynomes in diesem Intervall ist immer divergent. +Es ist also nötig, +den Integranden durch Funktionen zu approximieren, +die genügend schnell gegen $0$ gehen. +Man kann Polynome beliebigen Grades verwenden, +wenn sie mit einer Funktion multipliziert werden, +die schneller gegen $0$ geht als jedes Polynom. +Damit stellen wir sicher, +dass das Integral immer noch konvergiert. +% Darum müssen wir das Polynom mit einer Funktion multiplizieren, +% die schneller als jedes Polynom gegen $0$ geht, +% damit das Integral immer noch konvergiert. +Die Laguerre-Polynome $L_n$ schaffen hier Abhilfe, +da ihre Gewichtsfunktion $w(x) = e^{-x}$ schneller +gegen $0$ konvergiert als jedes Polynom. +% In unserem Falle möchten wir die Gauss Quadratur auf die Laguerre-Polynome +% $L_n$ ausweiten. +% Diese sind orthogonal im Intervall $(0, \infty)$ bezüglich +% der Gewichtsfunktion $e^{-x}$. +Um also das Integral einer Funktion $g(x)$ im Intervall~$(0,\infty)$ zu +berechen, +formt man das Integral wie folgt um: +\begin{align*} +\int_0^\infty g(x) \, dx += +\int_0^\infty f(x) e^{-x} \, dx +\end{align*} +Wir approximieren dann $f(x)$ durch ein Interpolationspolynom +wie bei der Gauss-Quadratur. +% Die Gleichung~\eqref{laguerre:gaussquadratur} lässt sich daher wie folgt +% umformulieren: +Die Gleichung~\eqref{laguerre:gaussquadratur} wird also +für die Gauss-Laguerre-Quadratur zu \begin{align} - \int_{0}^{\infty} f(x) e^{-x} dx - \approx - \sum_{i=1}^{N} f(x_i) A_i - \label{laguerre:laguerrequadratur} +\int_{0}^{\infty} f(x) e^{-x} dx +\approx +\sum_{i=1}^{n} f(x_i) A_i +\label{laguerre:laguerrequadratur} +. \end{align} +\subsubsection{Stützstellen und Gewichte} +Nach der Definition der Gauss-Quadratur müssen als Stützstellen die Nullstellen +des Approximationspolynoms genommen werden. +Für das Laguerre-Polynom $L_n(x)$ müssen demnach dessen Nullstellen $x_i$ und +als Gewichte $A_i$ die Integrale von $l_i(x) e^{-x}$ verwendet werden. +Dabei sind +\begin{align*} +l_i(x_j) += +\delta_{ij} += +\begin{cases} +1 & i=j \\ +0 & \text{sonst} +\end{cases} +% . +\end{align*} +die Lagrangeschen Interpolationspolynome. +Laut \cite{laguerre:hildebrand2013introduction} können die Gewichte mit +\begin{align*} +A_i + & = +-\frac{C_{n+1} \gamma_n}{C_n \phi'_n(x_i) \phi_{n+1} (x_i)} +\end{align*} +berechnet werden. +$C_i$ entspricht dabei dem Koeffizienten von $x^i$ +des orthogonalen Polynoms $\phi_n(x)$, $\forall i =0,\ldots,n$ und +\begin{align*} +\gamma_n += +\int_0^\infty w(x) \phi_n^2(x)\,dx +\end{align*} +dem Normalisierungsfaktor. + +Wir setzen nun $\phi_n(x) = L_n(x)$ und +nutzen den Vorzeichenwechsel der Laguerre-Koeffizienten +(ersichtlich am Term $(-1)^k$ in \eqref{laguerre:polynom}) +aus, +damit erhalten wir +\begin{align*} +A_i + & = +-\frac{C_{n+1} \gamma_n}{C_n L'_n(x_i) L_{n+1} (x_i)} +\\ + & = \frac{C_n}{C_{n-1}} \frac{\gamma_{n-1}}{L_{n-1}(x_i) L'_n(x_i)} +. +\end{align*} +Für Laguerre-Polynome gilt +\begin{align*} +\frac{C_n}{C_{n-1}} += +-\frac{1}{n} +\quad \text{und} \quad +\gamma_n += +1 +. +\end{align*} +Daraus folgt +\begin{align} +A_i + & = +- \frac{1}{n L_{n-1}(x_i) L'_n(x_i)} +\label{laguerre:gewichte_lag_temp} +. +\end{align} +Nun kann die Rekursionseigenschaft der Laguerre-Polynome +\cite{laguerre:hildebrand2013introduction} +% (siehe \cite{laguerre:hildebrand2013introduction}) +\begin{align*} +x L'_n(x) + & = +n L_n(x) - n L_{n-1}(x) +\\ + & = (x - n - 1) L_n(x) + (n + 1) L_{n+1}(x) +\end{align*} +umgeformt werden und da $x_i$ die Nullstellen von $L_n(x)$ sind, +vereinfacht sich die Gleichung zu +\begin{align*} +x_i L'_n(x_i) + & = +- n L_{n-1}(x_i) +\\ + & = +(n + 1) L_{n+1}(x_i) +. +\end{align*} +Setzen wir diese Beziehung nun in \eqref{laguerre:gewichte_lag_temp} ein, +ergibt sich \begin{align} - A_i - = - \frac{x_i}{(n + 1)^2 \left[ L_{n + 1}(x_i)\right]^2} - \label{laguerre:quadratur_gewichte} +\nonumber +A_i + & = +\frac{1}{x_i \left[ L'_n(x_i) \right]^2} +\\ + & = +\frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} +. +\label{laguerre:quadratur_gewichte} \end{align} +\subsubsection{Fehlerterm} +Die Gauss-Laguerre-Quadratur mit $n$ Stützstellen berechnet Integrale +von Polynomen bis zum Grad $2n - 1$ exakt. +Für beliebige Funktionen kann eine Fehlerabschätzung angegeben werden. +Der Fehlerterm $R_n$ folgt direkt aus der Approximation +\begin{align*} +\int_0^{\infty} f(x) e^{-x} \, dx += +\sum_{i=1}^n f(x_i) A_i + R_n +\end{align*} +und \cite{laguerre:abramowitz+stegun} gibt ihn als +\begin{align} +R_n + & = +\frac{f^{(2n)}(\xi)}{(2n)!} \int_0^\infty l(x)^2 e^{-x}\,dx +\\ + & = +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +,\quad +0 < \xi < \infty +\label{laguerre:lag_error} +\end{align} +an. +Der Fehler ist also abhängig von der $2n$-ten Ableitung +der zu integrierenden Funktion. diff --git a/buch/papers/laguerre/references.bib b/buch/papers/laguerre/references.bib index caf270f..1a4a903 100644 --- a/buch/papers/laguerre/references.bib +++ b/buch/papers/laguerre/references.bib @@ -3,33 +3,46 @@ % % (c) 2020 Autor, Hochschule Rapperswil % - -@online{laguerre:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@book{laguerre:hildebrand2013introduction, + title={Introduction to Numerical Analysis: Second Edition}, + author={Hildebrand, F.B.}, + isbn={9780486318554}, + series={Dover Books on Mathematics}, + year={2013}, + publisher={Dover Publications}, + pages = {389-392} } -@book{laguerre:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@book{laguerre:abramowitz+stegun, + added-at = {2008-06-25T06:25:58.000+0200}, + address = {New York}, + author = {Abramowitz, Milton and Stegun, Irene A.}, + edition = {ninth Dover printing, tenth GPO printing}, + interhash = {d4914a420f489f7c5129ed01ec3cf80c}, + intrahash = {23ec744709b3a776a1af0a3fd65cd09f}, + keywords = {Handbook}, + publisher = {Dover}, + pages = {890}, + timestamp = {2008-06-25T06:25:58.000+0200}, + title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, + year = 1972 } -@article{laguerre:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@article{laguerre:Cassity1965AbcissasCA, + title={Abcissas, coefficients, and error term for the generalized Gauss-Laguerre quadrature formula using the zero ordinate}, + author={C. Ronald Cassity}, + journal={Mathematics of Computation}, + year={1965}, + volume={19}, + pages={287-296} } +@online{laguerre:lanczos, + title = {Lanczos Approximation}, + author={Eric W. Weisstein}, + url = {https://mathworld.wolfram.com/LanczosApproximation.html}, + date = {2022-07-18}, + year = {2022}, + month = {7}, + day = {18} +}
\ No newline at end of file diff --git a/buch/papers/laguerre/scripts/estimates.py b/buch/papers/laguerre/scripts/estimates.py new file mode 100644 index 0000000..1acd7f7 --- /dev/null +++ b/buch/papers/laguerre/scripts/estimates.py @@ -0,0 +1,49 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + import gamma_approx as ga + import targets + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 200 + ns = np.arange(1, 13) + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + + bests = targets.find_best_loc(N, ns=ns) + mean_m = np.mean(bests, -1) + + intercept, bias = np.polyfit(ns, mean_m, 1) + fig, axs = plt.subplots( + 2, num=1, sharex=True, clear=True, constrained_layout=True, figsize=(4.5, 3.6) + ) + xl = np.array([ns[0] - 0.5, ns[-1] + 0.5]) + axs[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$") + axs[0].plot(ns, mean_m, "x", label=r"$\overline{m}$") + axs[1].plot( + ns, ((intercept * ns + bias) - mean_m), "-x", label=r"$\hat{m} - \overline{m}$" + ) + axs[0].set_xlim(*xl) + axs[0].set_xticks(ns) + axs[0].set_yticks(np.arange(np.floor(mean_m[0]), np.ceil(mean_m[-1]) + 0.1, 2)) + # axs[0].set_title("Schätzung von Mittelwert") + # axs[1].set_title("Fehler") + axs[-1].set_xlabel(r"$n$") + for ax in axs: + ax.grid(1) + ax.legend() + # fig.savefig(f"{ga.img_path}/estimates.pgf") + fig.savefig(f"{ga.img_path}/estimates.pdf") + + print(f"Intercept={intercept:.6g}, Bias={bias:.6g}") + predicts = np.ceil(intercept * ns[:, None] + bias - np.real(x)) + print(f"Error: {np.mean(np.abs(bests - predicts))}") diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py new file mode 100644 index 0000000..5b09e59 --- /dev/null +++ b/buch/papers/laguerre/scripts/gamma_approx.py @@ -0,0 +1,116 @@ +from pathlib import Path + +import numpy as np +import scipy.special + +EPSILON = 1e-7 +root = str(Path(__file__).parent) +img_path = f"{root}/../images" +fontsize = "medium" +cmap = "plasma" + + +def _prep_zeros_and_weights(x, w, n): + if x is None or w is None: + return np.polynomial.laguerre.laggauss(n) + return x, w + + +def drop_imag(z): + if abs(z.imag) <= EPSILON: + z = z.real + return z + + +def pochhammer(z, n): + return np.prod(z + np.arange(n)) + + +def find_optimal_shift(z, n): + mhat = 1.34093 * n + 0.854093 + steps = int(np.floor(mhat - np.real(z))) + return steps + + +def get_shifting_factor(z, steps): + factor = 1.0 + if steps > 0: + factor = 1 / pochhammer(z, steps) + elif steps < 0: + factor = pochhammer(z + steps, -steps) + return factor + + +def laguerre_gamma_shifted(z, x=None, w=None, n=8, target=11): + x, w = _prep_zeros_and_weights(x, w, n) + n = len(x) + + z += 0j + steps = int(np.floor(target - np.real(z))) + z_shifted = z + steps + correction_factor = get_shifting_factor(z, steps) + + res = np.sum(x ** (z_shifted - 1) * w) + res *= correction_factor + res = drop_imag(res) + return res + + +def laguerre_gamma_opt_shifted(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + n = len(x) + + z += 0j + opt_shift = find_optimal_shift(z, n) + correction_factor = get_shifting_factor(z, opt_shift) + z_shifted = z + opt_shift + + res = np.sum(x ** (z_shifted - 1) * w) + res *= correction_factor + res = drop_imag(res) + return res + + +def laguerre_gamma_simple(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + z += 0j + res = np.sum(x ** (z - 1) * w) + res = drop_imag(res) + return res + + +def laguerre_gamma_mirror(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + z += 0j + if z.real < 1e-3: + return np.pi / ( + np.sin(np.pi * z) * laguerre_gamma_simple(1 - z, x, w) + ) # Reflection formula + return laguerre_gamma_simple(z, x, w) + + +def eval_laguerre_gamma(z, x=None, w=None, n=8, func="simple", **kwargs): + x, w = _prep_zeros_and_weights(x, w, n) + if func == "simple": + f = laguerre_gamma_simple + elif func == "mirror": + f = laguerre_gamma_mirror + elif func == "optimal_shifted": + f = laguerre_gamma_opt_shifted + else: + f = laguerre_gamma_shifted + return np.array([f(zi, x, w, n, **kwargs) for zi in z]) + + +def calc_rel_error(x, y): + mask = np.abs(x) != np.infty + rel_error = np.zeros_like(y) + rel_error[mask] = (y[mask] - x[mask]) / x[mask] + rel_error[~mask] = 0.0 + return rel_error diff --git a/buch/papers/laguerre/scripts/integrand.py b/buch/papers/laguerre/scripts/integrand.py new file mode 100644 index 0000000..e970721 --- /dev/null +++ b/buch/papers/laguerre/scripts/integrand.py @@ -0,0 +1,42 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Plot for integrand of gamma function with shifting terms.""" + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + EPSILON = 1e-12 + xlims = np.array([-3, 3]) + + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + t = np.logspace(*xlims, 1001)[:, None] + + z = np.array([-4.5, -2, -1, -0.5, 0.0, 0.5, 1, 2, 4.5]) + r = t ** z + + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(4, 2.4)) + ax.semilogx(t, r) + ax.set_xlim(*(10.0 ** xlims)) + ax.set_ylim(1e-3, 40) + ax.set_xlabel(r"$x$") + # ax.set_ylabel(r"$x^z$") + ax.grid(1, "both") + labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)] + ax.legend(labels, ncol=2, loc="upper left", fontsize="small") + fig.savefig(f"{img_path}/integrand.pdf") diff --git a/buch/papers/laguerre/scripts/integrand_exp.py b/buch/papers/laguerre/scripts/integrand_exp.py new file mode 100644 index 0000000..e649b26 --- /dev/null +++ b/buch/papers/laguerre/scripts/integrand_exp.py @@ -0,0 +1,46 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Plot for integrand of gamma function with shifting terms.""" + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + EPSILON = 1e-12 + xlims = np.array([-3, 3]) + + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + t = np.logspace(*xlims, 1001)[:, None] + + z = np.array([-1, -0.5, 0.0, 0.5, 1, 2, 3, 4, 4.5]) + e = np.exp(-t) + r = t ** z * e + + fig, ax = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(4, 2.4)) + ax.semilogx(t, r) + # ax.plot(t,np.exp(-t)) + ax.set_xlim(10 ** (-2), 20) + ax.set_ylim(1e-3, 10) + ax.set_xlabel(r"$x$") + # ax.set_ylabel(r"$x^z e^{-x}$") + ax.grid(1, "both") + labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)] + ax.legend(labels, ncol=2, loc="upper left", fontsize="small") + # fig.savefig(f"{img_path}/integrand_exp.pgf") + fig.savefig(f"{img_path}/integrand_exp.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/laguerre_poly.py b/buch/papers/laguerre/scripts/laguerre_poly.py new file mode 100644 index 0000000..05db5d3 --- /dev/null +++ b/buch/papers/laguerre/scripts/laguerre_poly.py @@ -0,0 +1,108 @@ +import numpy as np + + +def get_ticks(start, end, step=1): + ticks = np.arange(start, end, step) + return ticks[ticks != 0] + + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import scipy.special as ss + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 1000 + step = 5 + t = np.linspace(-1.05, 10.5, N)[:, None] + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + # fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7)) + # ax = fig.add_subplot(axes_class=AxesZero) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) + for n in np.arange(0, 8): + k = np.arange(0, n + 1)[None] + L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) + ax.plot(t, L, label=f"$n={n}$") + + ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True) + ax.set_xticks(get_ticks(0, t[-1], step)) + ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0])) + ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large") + + ylim = 13 + ax.set_yticks(get_ticks(-ylim, ylim), minor=True) + ax.set_yticks(get_ticks(-step * (ylim // step), ylim, step)) + ax.set_ylim(-ylim, ylim) + ax.set_ylabel(r"$y$", y=0.95, labelpad=-14, rotation=0, fontsize="large") + + ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large") + + # set the x-spine + ax.spines[["left", "bottom"]].set_position("zero") + ax.spines[["right", "top"]].set_visible(False) + ax.xaxis.set_ticks_position("bottom") + hlx = 0.4 + dx = t[-1, 0] - t[0, 0] + dy = 2 * ylim + hly = dy / dx * hlx + dps = fig.dpi_scale_trans.inverted() + bbox = ax.get_window_extent().transformed(dps) + width, height = bbox.width, bbox.height + + # manual arrowhead width and length + hw = 1.0 / 60.0 * dy + hl = 1.0 / 30.0 * dx + lw = 0.5 # axis line width + ohg = 0.0 # arrow overhang + + # compute matching arrowhead length and width + yhw = hw / dy * dx * height / width + yhl = hl / dx * dy * width / height + + # draw x and y axis + ax.arrow( + t[-1, 0] - hl, + 0, + hl, + 0.0, + fc="k", + ec="k", + lw=lw, + head_width=hw, + head_length=hl, + overhang=ohg, + length_includes_head=True, + clip_on=False, + ) + + ax.arrow( + 0, + ylim - yhl, + 0.0, + yhl, + fc="k", + ec="k", + lw=lw, + head_width=yhw, + head_length=yhl, + overhang=ohg, + length_includes_head=True, + clip_on=False, + ) + + # fig.savefig(f"{img_path}/laguerre_poly.pgf") + fig.savefig(f"{img_path}/laguerre_poly.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_complex.py b/buch/papers/laguerre/scripts/rel_error_complex.py new file mode 100644 index 0000000..4a714fa --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_complex.py @@ -0,0 +1,43 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + xmax = 4 + vals = np.linspace(-xmax + ga.EPSILON, xmax, 100) + x, y = np.meshgrid(vals, vals) + mesh = x + 1j * y + input = mesh.flatten() + + lanczos = scipy.special.gamma(mesh) + lag = ga.eval_laguerre_gamma(input, n=8, func="optimal_shifted").reshape(mesh.shape) + rel_error = np.abs(ga.calc_rel_error(lanczos, lag)) + + fig, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(3.5, 2.1)) + _c = ax.pcolormesh( + x, y, rel_error, shading="gouraud", cmap=ga.cmap, norm=mpl.colors.LogNorm() + ) + cbr = fig.colorbar(_c, ax=ax) + cbr.minorticks_off() + # ax.set_title("Relative Error") + ax.set_xlabel("Re($z$)") + ax.set_ylabel("Im($z$)") + minor_ticks = np.arange(-xmax, xmax + ga.EPSILON) + ticks = np.arange(-xmax, xmax + ga.EPSILON, 2) + ax.set_xticks(ticks) + ax.set_xticks(minor_ticks, minor=True) + ax.set_yticks(ticks) + ax.set_yticks(minor_ticks, minor=True) + fig.savefig(f"{ga.img_path}/rel_error_complex.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_mirror.py b/buch/papers/laguerre/scripts/rel_error_mirror.py new file mode 100644 index 0000000..7348d5e --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_mirror.py @@ -0,0 +1,38 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + xmin = -15 + xmax = 15 + ns = np.arange(2, 12, 2) + ylim = np.array([-11, 1]) + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, 400) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.5)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n, func="mirror") + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 5)) + ax.set_xticks(np.arange(xmin, xmax), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_mirror.pgf") + fig.savefig(f"{ga.img_path}/rel_error_mirror.pdf") diff --git a/buch/papers/laguerre/scripts/rel_error_range.py b/buch/papers/laguerre/scripts/rel_error_range.py new file mode 100644 index 0000000..ece3b6d --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_range.py @@ -0,0 +1,41 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + N = 1201 + xmax = 6 + xmin = -xmax + ns = np.arange(2, 12, 2) + ylim = np.array([-11, -1.2]) + + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, N) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted") + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 2)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_yticks(10.0 ** np.arange(*ylim, 1), "", minor=True) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_range.pgf") + fig.savefig(f"{ga.img_path}/rel_error_range.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_shifted.py b/buch/papers/laguerre/scripts/rel_error_shifted.py new file mode 100644 index 0000000..f53c89b --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_shifted.py @@ -0,0 +1,40 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + n = 8 # order of Laguerre polynomial + N = 200 # number of points in interval + + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + targets = np.arange(10, 14) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.1)) + for target in targets: + gamma_lag = ga.eval_laguerre_gamma(x, target=target, n=n, func="shifted") + rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lag)) + ax.semilogy(x, rel_error, label=f"$m={target}$", linewidth=3) + gamma_lgo = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted") + rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lgo)) + ax.semilogy(x, rel_error, "m", linestyle=":", label="$m^*$", linewidth=3) + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(5e-9, 5e-8) + ax.set_xlabel(r"$z$") + ax.set_xticks(np.linspace(0, 1, 6)) + ax.set_xticks(np.linspace(0, 1, 11), minor=True) + ax.grid(1, "both") + ax.legend(ncol=1, fontsize=ga.fontsize) + # fig.savefig(f"{ga.img_path}/rel_error_shifted.pgf") + fig.savefig(f"{ga.img_path}/rel_error_shifted.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_simple.py b/buch/papers/laguerre/scripts/rel_error_simple.py new file mode 100644 index 0000000..e1ea36a --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_simple.py @@ -0,0 +1,41 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + # mpl.rc("text", usetex=True) + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + # mpl.rcParams.update({"font.family": "serif", "font.serif": "TeX Gyre Termes"}) + + # Simple / naive + xmin = -5 + xmax = 25 + ns = np.arange(2, 12, 2) + ylim = np.array([-11, 6]) + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, 400) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.5)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n) + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 5)) + ax.set_xticks(np.arange(xmin, xmax), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=3, fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_simple.pgf") + fig.savefig(f"{ga.img_path}/rel_error_simple.pdf") diff --git a/buch/papers/laguerre/scripts/targets.py b/buch/papers/laguerre/scripts/targets.py new file mode 100644 index 0000000..69f94ba --- /dev/null +++ b/buch/papers/laguerre/scripts/targets.py @@ -0,0 +1,58 @@ +import numpy as np +import scipy.special + +import gamma_approx as ga + + +def find_best_loc(N=200, a=1.375, b=0.5, ns=None): + if ns is None: + ns = np.arange(2, 13) + bests = [] + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + gamma = scipy.special.gamma(x) + for n in ns: + zeros, weights = np.polynomial.laguerre.laggauss(n) + est = np.ceil(b + a * n) + targets = np.arange(max(est - 2, 0), est + 3) + rel_error = [] + for target in targets: + gamma_lag = ga.eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted") + rel_error.append(np.abs(ga.calc_rel_error(gamma, gamma_lag))) + rel_error = np.stack(rel_error, -1) + best = np.argmin(rel_error, -1) + targets[0] + bests.append(best) + return np.stack(bests, 0) + + +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 200 + ns = np.arange(1, 13) + + bests = find_best_loc(N, ns=ns) + + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(3.5, 2.1)) + v = ax.imshow(bests, cmap=ga.cmap, aspect="auto", interpolation="nearest") + plt.colorbar(v, ax=ax, label=r"$m^*$") + ticks = np.arange(0, N + 1, N // 5) + ax.set_xlim(0, 1) + ax.set_xticks(ticks) + ax.set_xticklabels([f"{v:.2f}" for v in ticks / N]) + ax.set_xticks(np.arange(0, N + 1, N // 20), minor=True) + ax.set_yticks(np.arange(len(ns))) + ax.set_yticklabels(ns) + ax.set_xlabel(r"$z$") + ax.set_ylabel(r"$n$") + # fig.savefig(f"{ga.img_path}/targets.pgf") + fig.savefig(f"{ga.img_path}/targets.pdf") diff --git a/buch/papers/laguerre/transformation.tex b/buch/papers/laguerre/transformation.tex deleted file mode 100644 index 4de86b6..0000000 --- a/buch/papers/laguerre/transformation.tex +++ /dev/null @@ -1,31 +0,0 @@ -% -% transformation.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Laguerre Transformation -\label{laguerre:section:transformation}} -\begin{align} - L \left\{ f(x) \right\} - = - \tilde{f}_\alpha(n) - = - \int_0^\infty e^{-x} x^\alpha L_n^\alpha(x) f(x) dx - \label{laguerre:transformation} -\end{align} - -\begin{align} - L^{-1} \left\{ \tilde{f}_\alpha(n) \right\} - = - f(x) - = - \sum_{n=0}^{\infty} - \begin{pmatrix} - n + \alpha \\ - n - \end{pmatrix}^{-1} - \frac{1}{\Gamma(\alpha + 1)} - \tilde{f}_\alpha(n) - L_n^\alpha(x) - \label{laguerre:inverse_transformation} -\end{align}
\ No newline at end of file diff --git a/buch/papers/laguerre/wasserstoff.tex b/buch/papers/laguerre/wasserstoff.tex deleted file mode 100644 index caaa6af..0000000 --- a/buch/papers/laguerre/wasserstoff.tex +++ /dev/null @@ -1,29 +0,0 @@ -% -% wasserstoff.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Radialer Schwingungsanteil eines Wasserstoffatoms -\label{laguerre:section:radial_h_atom}} - -\begin{align} - \nonumber - - \frac{\hbar^2}{2m} - & - \left( - \frac{1}{r^2} \pdv{}{r} - \left( r^2 \pdv{}{r} \right) - + - \frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} - \left( \sin \vartheta \pdv{}{\vartheta} \right) - + - \frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} - \right) - u(r, \vartheta, \varphi) - \\ - & - - \frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) - = - E u(r, \vartheta, \varphi) - \label{laguerre:pdg_h_atom} -\end{align} diff --git a/buch/papers/nav/Makefile.inc b/buch/papers/nav/Makefile.inc index b30377e..5e86543 100644 --- a/buch/papers/nav/Makefile.inc +++ b/buch/papers/nav/Makefile.inc @@ -6,9 +6,10 @@ dependencies-nav = \ papers/nav/packages.tex \ papers/nav/main.tex \ - papers/nav/references.bib \ - papers/nav/teil0.tex \ - papers/nav/teil1.tex \ - papers/nav/teil2.tex \ - papers/nav/teil3.tex + papers/nav/einleitung.tex \ + papers/nav/flatearth.tex \ + papers/nav/nautischesdreieck.tex \ + papers/nav/sincos.tex \ + papers/nav/trigo.tex \ + papers/nav/references.bib diff --git a/buch/papers/nav/beispiel.txt b/buch/papers/nav/beispiel.txt new file mode 100644 index 0000000..b8716fc --- /dev/null +++ b/buch/papers/nav/beispiel.txt @@ -0,0 +1,24 @@ +Datum: 28. 5. 2022 +Zeit: 15:29:49 UTC +Sternzeit: 7h 54m 26.593s 7.90738694h + +Deneb + +RA 20h 42m 12.14s 20.703372h +DEC 45 21' 40.3" 45.361194 + +H 50g 15' 17.1" 50.254750 +Azi 59g 36' 02.0" 59.600555 + +Spica + +RA 13h 26m 23.44s 13.439844h +DEC -11g 16' 46.8" 11.279666 + +H 18g 27' 30.0" 18.458333 +Azi 240g 23' 52.5" 240.397916 + +Position: + +l = 140 14' 00.01" E 140.233336 E +b = 35 43' 00.02" N 35.716672 N diff --git a/buch/papers/nav/bilder/beispiele1.pdf b/buch/papers/nav/bilder/beispiele1.pdf Binary files differnew file mode 100644 index 0000000..1f91809 --- /dev/null +++ b/buch/papers/nav/bilder/beispiele1.pdf diff --git a/buch/papers/nav/bilder/beispiele2.pdf b/buch/papers/nav/bilder/beispiele2.pdf Binary files differnew file mode 100644 index 0000000..4b28f2f --- /dev/null +++ b/buch/papers/nav/bilder/beispiele2.pdf diff --git a/buch/papers/nav/bilder/dreieck.pdf b/buch/papers/nav/bilder/dreieck.pdf Binary files differnew file mode 100644 index 0000000..9d630aa --- /dev/null +++ b/buch/papers/nav/bilder/dreieck.pdf diff --git a/buch/papers/nav/bilder/dreieck.png b/buch/papers/nav/bilder/dreieck.png Binary files differnew file mode 100644 index 0000000..2b02105 --- /dev/null +++ b/buch/papers/nav/bilder/dreieck.png diff --git a/buch/papers/nav/bilder/ephe.png b/buch/papers/nav/bilder/ephe.png Binary files differnew file mode 100644 index 0000000..3f99a36 --- /dev/null +++ b/buch/papers/nav/bilder/ephe.png diff --git a/buch/papers/nav/bilder/kugel1.png b/buch/papers/nav/bilder/kugel1.png Binary files differnew file mode 100644 index 0000000..b3188b7 --- /dev/null +++ b/buch/papers/nav/bilder/kugel1.png diff --git a/buch/papers/nav/bilder/kugel2.png b/buch/papers/nav/bilder/kugel2.png Binary files differnew file mode 100644 index 0000000..057740f --- /dev/null +++ b/buch/papers/nav/bilder/kugel2.png diff --git a/buch/papers/nav/bilder/kugel3.png b/buch/papers/nav/bilder/kugel3.png Binary files differnew file mode 100644 index 0000000..97066a2 --- /dev/null +++ b/buch/papers/nav/bilder/kugel3.png diff --git a/buch/papers/nav/bilder/position1.pdf b/buch/papers/nav/bilder/position1.pdf Binary files differnew file mode 100644 index 0000000..ba7755f --- /dev/null +++ b/buch/papers/nav/bilder/position1.pdf diff --git a/buch/papers/nav/bilder/position2.pdf b/buch/papers/nav/bilder/position2.pdf Binary files differnew file mode 100644 index 0000000..3333dd4 --- /dev/null +++ b/buch/papers/nav/bilder/position2.pdf diff --git a/buch/papers/nav/bilder/position3.pdf b/buch/papers/nav/bilder/position3.pdf Binary files differnew file mode 100644 index 0000000..fae0b85 --- /dev/null +++ b/buch/papers/nav/bilder/position3.pdf diff --git a/buch/papers/nav/bilder/position4.pdf b/buch/papers/nav/bilder/position4.pdf Binary files differnew file mode 100644 index 0000000..ac80c46 --- /dev/null +++ b/buch/papers/nav/bilder/position4.pdf diff --git a/buch/papers/nav/bilder/position5.pdf b/buch/papers/nav/bilder/position5.pdf Binary files differnew file mode 100644 index 0000000..afe120e --- /dev/null +++ b/buch/papers/nav/bilder/position5.pdf diff --git a/buch/papers/nav/bilder/projektion.png b/buch/papers/nav/bilder/projektion.png Binary files differnew file mode 100644 index 0000000..5dcc0c8 --- /dev/null +++ b/buch/papers/nav/bilder/projektion.png diff --git a/buch/papers/nav/bilder/recht.jpg b/buch/papers/nav/bilder/recht.jpg Binary files differnew file mode 100644 index 0000000..3f60370 --- /dev/null +++ b/buch/papers/nav/bilder/recht.jpg diff --git a/buch/papers/nav/bilder/sextant.jpg b/buch/papers/nav/bilder/sextant.jpg Binary files differnew file mode 100644 index 0000000..472e61f --- /dev/null +++ b/buch/papers/nav/bilder/sextant.jpg diff --git a/buch/papers/nav/bsp.tex b/buch/papers/nav/bsp.tex new file mode 100644 index 0000000..ff01828 --- /dev/null +++ b/buch/papers/nav/bsp.tex @@ -0,0 +1,182 @@ +\section{Beispielrechnung} +\rhead{Beispielrechnung} + +\subsection{Einführung} +In diesem Abschnitt wird die Theorie vom Abschnitt 21.6 in einem Praxisbeispiel angewendet. +Wir haben die Deklination, Rektaszension, Höhe der beiden Planeten Deneb und Arktur und die Sternzeit von Greenwich als Ausgangslage. +Die Deklinationen und Rektaszensionen sind von einem vergangenen Datum und die Höhe der Gestirne und die Sternzeit wurden von unserem Dozenten digital in einer Stadt in Japan mit den Koordinaten 35.716672 N, 140.233336 E bestimmt. +Wir werden rechnerisch beweisen, dass wir mit diesen Ergebnissen genau auf diese Koordinaten kommen. +\subsection{Vorgehen} + +\begin{center} + \begin{tabular}{l l l} + 1. & Koordinaten der Bildpunkte der Gestirne bestimmen \\ + 2. & Dreiecke aufzeichnen und richtig beschriften\\ + 3. & Dreieck ABC bestimmmen\\ + 4. & Dreieck BPC bestimmen \\ + 5. & Dreieck ABP bestimmen \\ + 6. & Geographische Breite bestimmen \\ + 7. & Geographische Länge bestimmen \\ + \end{tabular} +\end{center} + +\subsection{Ausgangslage} +\begin{wrapfigure}{R}{5.6cm} + \includegraphics{papers/nav/bilder/position1.pdf} + \caption{Ausgangslage} +\end{wrapfigure} +Die Rektaszension und die Sternzeit sind in der Regeln in Stunden angegeben. +Für die Umrechnung in Grad kann folgender Zusammenhang verwendet werden: +\[ Stunden \cdot 15 = Grad\]. +Dies wurde hier bereits gemacht. +\begin{center} + \begin{tabular}{l l l} + Sternzeit $s$ & $118.610804^\circ$ \\ + Deneb&\\ + & Rektaszension $RA_{Deneb}$& $310.55058^\circ$ \\ + & Deklination $DEC_{Deneb}$& $45.361194^\circ$ \\ + & Höhe $h_c$ & $50.256027^\circ$ \\ + Arktur &\\ + & Rektaszension $RA_{Arktur}$& $214.17558^\circ$ \\ + & Deklination $DEC_{Arktur}$& $19.063222^\circ$ \\ + & Höhe $h_b$ & $47.427444^\circ$ \\ + \end{tabular} +\end{center} +\subsection{Koordinaten der Bildpunkte} +Als erstes benötigen wir die Koordinaten der Bildpunkte von Arktur und Deneb. +$\delta$ ist die Breite, $\lambda$ die Länge. +\begin{align} +\delta_{Deneb}&=DEC_{Deneb} = \underline{\underline{45.361194^\circ}} \nonumber \\ +\lambda_{Deneb}&=RA_{Deneb} - s = 310.55058^\circ -118.610804^\circ =\underline{\underline{191.939776^\circ}} \nonumber \\ +\delta_{Arktur}&=DEC_{Arktur} = \underline{\underline{19.063222^\circ}} \nonumber \\ +\lambda_{Arktur}&=RA_{Arktur} - s = 214.17558^\circ -118.610804^\circ = \underline{\underline{5.5647759^\circ}} \nonumber +\end{align} + + +\subsection{Dreiecke definieren} +\begin{figure} + \begin{center} + \includegraphics[width=6cm]{papers/nav/bilder/beispiele1.pdf} + \includegraphics[width=6cm]{papers/nav/bilder/beispiele2.pdf} + \caption{Arktur-Deneb; Spica-Altiar} +\end{center} +\end{figure} +Das Festlegen der Dreiecke ist essenziell für die korrekten Berechnungen. +Ein Problem, welches in der Theorie nicht berücksichtigt wurde ist, dass der Punkt $P$ nicht zwingend unterhalb der Seite $a$ sein muss. +Wenn man das nicht berücksichtigt, erhält man falsche oder keine Ergebnisse. +In der Realität weiss man jedoch ungefähr auf welchem Breitengrad man ist, so kann man relativ einfach entscheiden, ob der eigene Standort über $a$ ist oder nicht. +Beim unserem genutzten Paar Arktur-Deneb ist dies kein Problem, da der Punkt unterhalb der Seite $a$ liegt. +Würde man aber das Paar Altair-Spica nehmen, liegt $P$ über $a$ (vgl. Abbildung 21.11) und man müsste trigonometrisch anders vorgehen. + +\subsection{Dreieck $ABC$} +\begin{wrapfigure}{R}{5.6cm} + \includegraphics{papers/nav/bilder/position2.pdf} + \caption{Dreieck ABC} +\end{wrapfigure} +Nun berechnen wir alle Seitenlängen $a$, $b$, $c$ und die Innnenwinkel $\alpha$, $\beta$ und $\gamma$ +Wir können $b$ und $c$ mit den geltenten Zusammenhängen des nautischen Dreiecks wie folgt bestimmen: +\begin{align} + b=90^\circ-DEC_{Deneb} = 90^\circ - 45.361194^\circ = \underline{\underline{44.638806^\circ}}\nonumber \\ + c=90^\circ-DEC_{Arktur} = 90^\circ - 19.063222^\circ = \underline{\underline{70.936778^\circ}} \nonumber +\end{align} +Um $a$ zu bestimmen, benötigen wir zuerst den Winkel \[\alpha= RA_{Deneb} - RA_{Arktur} = 310.55058^\circ -214.17558^\circ = \underline{\underline{96.375^\circ}}.\] +Danach nutzen wir den sphärischen Winkelkosinussatz, um $a$ zu berechnen: +\begin{align} + a &= \cos^{-1}(\cos(b) \cdot \cos(c) + \sin(b) \cdot \sin(c) \cdot \cos(\alpha)) \nonumber \\ + &= \cos^{-1}(\cos(44.638806) \cdot \cos(70.936778) + \sin(44.638806) \cdot \sin(70.936778) \cdot \cos(96.375)) \nonumber \\ + &= \underline{\underline{80.8707801^\circ}} \nonumber +\end{align} +Für $\beta$ und $\gamma$ nutzen wir den sphärischen Seitenkosinussatz: +\begin{align} + \beta &= \cos^{-1} \bigg[\frac{\cos(b)-\cos(a) \cdot \cos(c)}{\sin(a) \cdot \sin(c)}\bigg] \nonumber \\ + &= \cos^{-1} \bigg[\frac{\cos(44.638806)-\cos(80.8707801) \cdot \cos(70.936778)}{\sin(80.8707801) \cdot \sin(70.936778)}\bigg] \nonumber \\ + &= \underline{\underline{45.0115314^\circ}} \nonumber +\end{align} + + \begin{align} + \gamma &= \cos^{-1} \bigg[\frac{\cos(c)-\cos(b) \cdot \cos(a)}{\sin(a) \cdot \sin(b)}\bigg] \nonumber \\ + &= \cos^{-1} \bigg[\frac{\cos(70.936778)-\cos(44.638806) \cdot \cos(80.8707801)}{\sin(80.8707801) \cdot \sin(44.638806)}\bigg] \nonumber \\ + &=\underline{\underline{72.0573328^\circ}} \nonumber +\end{align} +\subsection{Dreieck $BPC$} +\begin{wrapfigure}{R}{5.6cm} + \includegraphics{papers/nav/bilder/position3.pdf} + \caption{Dreieck BPC} +\end{wrapfigure} +Als nächstes berechnen wir die Seiten $h_b$, $h_c$ und die Innenwinkel $\beta_1$ und $\gamma_1$. +\begin{align} + h_b&=90^\circ - h_b \nonumber \\ + &= 90^\circ - 47.42744^\circ \nonumber \\ + &= \underline{\underline{42.572556^\circ}} \nonumber +\end{align} +\begin{align} + h_c &= 90^\circ - h_c \nonumber \\ + &= 90^\circ - 50.256027^\circ \nonumber \\ + &= \underline{\underline{39.743973^\circ}} \nonumber +\end{align} +\begin{align} + \beta_1 &= \cos^{-1} \bigg[\frac{\cos(h_c)-\cos(a) \cdot \cos(h_b)}{\sin(a) \cdot \sin(h_b)}\bigg] \nonumber \\ + &= \cos^{-1} \bigg[\frac{\cos(39.743973)-\cos(80.8707801) \cdot \cos(42.572556)}{\sin(80.8707801) \cdot \sin(42.572556)}\bigg] \nonumber \\ + &=\underline{\underline{12.5211127^\circ}} \nonumber +\end{align} +\begin{align} + \gamma_1 &= \cos^{-1} \bigg[\frac{\cos(h_b)-\cos(a) \cdot \cos(h_c)}{\sin(a) \cdot \sin(h_c)}\bigg] \nonumber \\ + &= \cos^{-1} \bigg[\frac{\cos(42.572556)-\cos(80.8707801) \cdot \cos(39.743973)}{\sin(80.8707801) \cdot \sin(39.743973)}\bigg] \nonumber \\ + &=\underline{\underline{13.2618475^\circ}} \nonumber +\end{align} + +\subsection{Dreieck $ABP$} +\begin{wrapfigure}{R}{5.6cm} + \includegraphics{papers/nav/bilder/position4.pdf} + \caption{Dreieck ABP} +\end{wrapfigure} +Als erster müssen wir den Winkel $\beta_2$ berechnen: +\begin{align} + \beta_2 &= \beta + \beta_1 = 45.011513^\circ + 12.5211127^\circ \nonumber \\ + &=\underline{\underline{44.6687451^\circ}} \nonumber +\end{align} +Danach können wir mithilfe von $\beta_2$, $c$ und $h_b$ die Seite $l$ berechnen: +\begin{align} + l &= \cos^{-1}(\cos(c) \cdot \cos(h_b) + \sin(c) \cdot \sin(h_b) \cdot \cos(\beta_2)) \nonumber \\ + &= \cos^{-1}(\cos(70.936778) \cdot \cos(42.572556) + \sin(70.936778) \cdot \sin(42.572556) \cdot \cos(57.5326442)) \nonumber \\ + &= \underline{\underline{54.2833404^\circ}} \nonumber +\end{align} +Damit wir gleich den Längengrad berechnen können, benötigen wir noch den Winkel $\omega$: +\begin{align} + \omega &= \cos^{-1} \bigg[\frac{\cos(h_b)-\cos(c) \cdot \cos(l)}{\sin(c) \cdot \sin(l)}\bigg] \nonumber \\ + &=\cos^{-1} \bigg[\frac{\cos(42.572556)-\cos(70.936778) \cdot \cos(54.2833404)}{\sin(70.936778) \cdot \sin(54.2833404)}\bigg] \nonumber \\ + &= \underline{\underline{44.6687451^\circ}} \nonumber +\end{align} + +\subsection{Längengrad und Breitengrad bestimmen} + +\begin{align} + \delta &= 90^\circ - l \nonumber \\ + &= 90^\circ - 54.2833404 \nonumber \\ + &= \underline{\underline{35.7166596^\circ}} \nonumber +\end{align} +\begin{align} + \lambda &= \lambda_{Arktur} + \omega \nonumber \\ + &= 95.5647759^\circ + 44.6687451^\circ \nonumber \\ + &= \underline{\underline{140.233521^\circ}} \nonumber +\end{align} +Wie wir sehen, stimmen die berechneten Koordinaten mit den Koordinaten des Punktes, an welchem gemessen wurde überein. + +\subsection{Fazit} +Die theoretische Anleitung im Abschnitt 21.6 scheint grundsätzlich zu funktionieren. +Allerdings gab es zwei interessante Probleme. + +Einerseits das Problem, ob der Punkt P sich oberhalb oder unterhalb von $a$ befindet. +Da wir eigentlich wussten, wo der gesuchte Punkt P ist, konnten wir das Dreieck anhand der Koordinaten der Bildpunkte richtig aufstellen. +In der Praxis muss man aber schon wissen, auf welchem Breitengrad man ungefähr ist. +Dies weis man in der Regeln aber, da die eigene Breite die Höhe des Polarsterns ist. +Diese Höhe wird mit dem Sextant gemessen. + +Andererseits ist da noch ein Problem mit dem Sinussatz. +Beim Sinussatz gibt es immer zwei Lösungen, weil \[ \sin(\pi-a)=\sin(a).\] +Da kann es sein (und war in unserem Fall auch so), dass man das falsche Ergebnis erwischt. +Durch diese Erkenntnis haben wir nur Kosinussätze verwendet und dies ebenfalls im Abschnitt 21.6 abgeändert, da es für den Leser auch relevant sein kann, wenn er es Probieren möchte. + + + + diff --git a/buch/papers/nav/bsp2.tex b/buch/papers/nav/bsp2.tex new file mode 100644 index 0000000..8d9083b --- /dev/null +++ b/buch/papers/nav/bsp2.tex @@ -0,0 +1,236 @@ +\section{Beispielrechnung} +\rhead{Beispielrechnung} + +\subsection{Einführung} +In diesem Abschnitt wird die Theorie vom Abschnitt \ref{sta} in einem Praxisbeispiel angewendet. +Wir haben die Deklination, Rektaszension, Höhe der beiden Planeten Deneb und Arktur und die Sternzeit von Greenwich als Ausgangslage. +Die Deklinationen und Rektaszensionen sind von einem vergangenen Datum und die Höhe der Gestirne und die Sternzeit wurden digital in einer Stadt in Japan mit den Koordinaten 35.716672 N, 140.233336 E bestimmt. +Wir werden nachrechnen, dass wir mit unserer Methode genau auf diese Koordinaten kommen. +\subsection{Vorgehen} +Unser Vorgehen erschliesst sich aus unserer Methode, die wir im Abschnitt \ref{p} theoretisch erklärt haben. +\begin{compactenum} +\item +Koordinaten der Bildpunkte der Gestirne bestimmen +\item +Dreiecke aufzeichnen und richtig beschriften +\item +Dreieck ABC bestimmmen +\item +Dreieck BPC bestimmen +\item +Dreieck ABP bestimmen +\item +Geographische Breite bestimmen +\item +Geographische Länge bestimmen +\end{compactenum} + +\subsection{Ausgangslage} +\hbox to\textwidth{% +\begin{minipage}{8.4cm} +Die Rektaszension und die Sternzeit sind in der Regel in Stunden angegeben. +Für die Umrechnung in Grad kann folgender Zusammenhang verwendet werden: +\[ +\text{Stunden} \cdot 15 = \text{Grad}. +\] +Dies wurde hier bereits gemacht. +\begin{center} +\begin{tabular}{l l >{$}l<{$}} +Sternzeit $s$ & $118.610804^\circ$ \\ +Deneb &\\ + & Rektaszension $RA_{\text{Deneb}}$ & 310.55058^\circ\\ + & Deklination $DEC_{\text{Deneb}}$ & \phantom{0}45.361194^\circ \\ + & Höhe $h_c$ & \phantom{0}50.256027^\circ \\ +Arktur &\\ + & Rektaszension $RA_{\text{Arktur}}$& 214.17558^\circ \\ + & Deklination $DEC_{\text{Arktur}}$ & \phantom{0}19.063222^\circ \\ + & Höhe $h_b$ & \phantom{0}47.427444^\circ \\ +\end{tabular} +\end{center} +\end{minipage}% +\hfill% +\raisebox{-2cm}{\includegraphics{papers/nav/bilder/position1.pdf}}% +} +\medskip + +\subsection{Koordinaten der Bildpunkte} +Als erstes benötigen wir die Koordinaten der Bildpunkte von Arktur und Deneb. +$\delta$ ist die Breite, $\lambda$ die Länge. +\begin{align} +\delta_{\text{Deneb}}&=DEC_{\text{Deneb}} = \underline{\underline{45.361194^\circ}} \nonumber \\ +\lambda_{\text{Deneb}}&=RA_{\text{Deneb}} - s = 310.55058^\circ -118.610804^\circ =\underline{\underline{191.939776^\circ}} \nonumber \\ +\delta_{\text{Arktur}}&=DEC_{\text{Arktur}} = \underline{\underline{19.063222^\circ}} \nonumber \\ +\lambda_{\text{Arktur}}&=RA_{\text{Arktur}} - s = 214.17558^\circ -118.610804^\circ = \underline{\underline{5.5647759^\circ}} \nonumber +\end{align} + + +\subsection{Dreiecke definieren} +\begin{figure} +\hbox{% +\includegraphics{papers/nav/bilder/beispiele1.pdf}% +\hfill% +\includegraphics{papers/nav/bilder/beispiele2.pdf}} +\caption{Arktur-Deneb; Spica-Altiar +\label{nav:beispiele}} +\end{figure} +Das Festlegen der Dreiecke ist essenziell für die korrekten Berechnungen. +Ein Problem, welches in der Theorie nicht berücksichtigt wurde ist, dass der Punkt $P$ nicht zwingend unterhalb der Seite $a$ sein muss. +Wenn man das nicht berücksichtigt, erhält man falsche oder keine Ergebnisse. +In der Realität weiss man jedoch ungefähr auf welchem Breitengrad man ist, so kann man relativ einfach entscheiden, ob der eigene Standort über $a$ ist oder nicht. +Beim unserem genutzten Paar Arktur-Deneb ist dies kein Problem, da der Punkt unterhalb der Seite $a$ liegt. +Würde man aber das Paar Altair-Spica nehmen, liegt $P$ über $a$ +(vgl. Abbildung\ref{nav:beispiele}) und man müsste trigonometrisch +anders vorgehen. + +\subsection{Dreieck $ABC$} +\vspace*{-3mm} +\hbox to\textwidth{% +\begin{minipage}{8.4cm}% +Nun berechnen wir alle Seitenlängen $a$, $b$, $c$ und die +Innnenwinkel $\alpha$, $\beta$ und $\gamma$. +Wir können $b$ und $c$ mit den geltenten Zusammenhängen des nautischen Dreiecks wie folgt bestimmen: +\begin{align*} +b +&= +90^\circ-DEC_{\text{Deneb}} += +90^\circ - 45.361194^\circ +\\ +&= +\underline{\underline{44.638806^\circ}} +\\ +c +&= +90^\circ-DEC_{\text{Arktur}} += +90^\circ - 19.063222^\circ +\\ +&= +\underline{\underline{70.936778^\circ}} +\end{align*} +\end{minipage}% +\hfill% +\raisebox{-2.4cm}{\includegraphics{papers/nav/bilder/position2.pdf}}% +} +Um $a$ zu bestimmen, benötigen wir zuerst den Winkel +\begin{align*} +\alpha +&= +RA_{\text{Deneb}} - RA_{\text{Arktur}} += +310.55058^\circ -214.17558^\circ +\\ +&= +\underline{\underline{96.375^\circ}}. +\end{align*} +Danach nutzen wir den sphärischen Winkelkosinussatz, um $a$ zu berechnen: +\begin{align*} + a &= \cos^{-1}(\cos(b) \cdot \cos(c) + \sin(b) \cdot \sin(c) \cdot \cos(\alpha)) \\ + &= \cos^{-1}(\cos(44.638806^\circ) \cdot \cos(70.936778^\circ) + \sin(44.638806^\circ) \cdot \sin(70.936778^\circ) \cdot \cos(96.375^\circ)) \\ + &= \underline{\underline{80.8707801^\circ}} +\end{align*} +Für $\beta$ und $\gamma$ nutzen wir den sphärischen Seitenkosinussatz: +\begin{align*} + \beta &= \cos^{-1} \bigg[\frac{\cos(b)-\cos(a) \cdot \cos(c)}{\sin(a) \cdot \sin(c)}\bigg] \\ + &= \cos^{-1} \bigg[\frac{\cos(44.638806^\circ)-\cos(80.8707801^\circ) \cdot \cos(70.936778^\circ)}{\sin(80.8707801^\circ) \cdot \sin(70.936778^\circ)}\bigg] \\ + &= \underline{\underline{45.0115314^\circ}} +\\ +\gamma &= \cos^{-1} \bigg[\frac{\cos(c)-\cos(b) \cdot \cos(a)}{\sin(a) \cdot \sin(b)}\bigg] \\ + &= \cos^{-1} \bigg[\frac{\cos(70.936778^\circ)-\cos(44.638806^\circ) \cdot \cos(80.8707801^\circ)}{\sin(80.8707801^\circ) \cdot \sin(44.638806^\circ)}\bigg] \\ + &=\underline{\underline{72.0573328^\circ}} +\end{align*} + + + +\subsection{Dreieck $BPC$} +\vspace*{-4mm} +\hbox to\textwidth{% +\begin{minipage}{8.4cm}% +Als nächstes berechnen wir die Seiten $h_B$, $h_B$ und die Innenwinkel $\beta_1$ und $\gamma_1$. +\begin{align*} +h_B&=90^\circ - pbb + = 90^\circ - 47.42744^\circ \\ + &= \underline{\underline{42.572556^\circ}} +\\ + h_C &= 90^\circ - pc + = 90^\circ - 50.256027^\circ \\ + &= \underline{\underline{39.743973^\circ}} +\end{align*} +\end{minipage}% +\hfill% +\raisebox{-2.8cm}{\includegraphics{papers/nav/bilder/position3.pdf}}% +} +\begin{align*} +\beta_1 &= \cos^{-1} \bigg[\frac{\cos(h_c)-\cos(a) \cdot \cos(h_B)}{\sin(a) \cdot \sin(h_B)}\bigg] \\ + &= \cos^{-1} \bigg[\frac{\cos(39.743973^\circ)-\cos(80.8707801^\circ) \cdot \cos(42.572556^\circ)}{\sin(80.8707801^\circ) \cdot \sin(42.572556^\circ)}\bigg] \\ + &=\underline{\underline{12.5211127^\circ}} +\\ +\gamma_1 &= \cos^{-1} \bigg[\frac{\cos(h_b)-\cos(a) \cdot \cos(h_C)}{\sin(a) \cdot \sin(h_C)}\bigg] \\ + &= \cos^{-1} \bigg[\frac{\cos(42.572556^\circ)-\cos(80.8707801^\circ) \cdot \cos(39.743973^\circ)}{\sin(80.8707801^\circ) \cdot \sin(39.743973^\circ)}\bigg] \\ + &=\underline{\underline{13.2618475^\circ}} +\end{align*} + +\subsection{Dreieck $ABP$} +\vspace*{-2mm} +\hbox to\textwidth{% +\begin{minipage}{8.4cm}% +Als erstes müssen wir den Winkel $\beta_2$ berechnen: +\begin{align*} + \beta_2 &= \beta + \beta_1 = 45.011513^\circ + 12.5211127^\circ \\ + &=\underline{\underline{44.6687451^\circ}} +\end{align*} +Danach können wir mithilfe von $\beta_2$, $c$ und $h_B$ die Seite $l$ berechnen: +\begin{align*} +l +&= +\cos^{-1}(\cos(c) \cdot \cos(h_B) + + \sin(c) \cdot \sin(h_B) \cdot \cos(\beta_2)) \\ +&= +\cos^{-1}(\cos(70.936778^\circ) \cdot \cos(42.572556^\circ)\\ +&\qquad + \sin(70.936778^\circ) \cdot \sin(42.572556^\circ) \cdot \cos(57.5326442^\circ)) \\ +&= \underline{\underline{54.2833404^\circ}} +\end{align*} +\end{minipage}% +\hfill% +\raisebox{-2.0cm}{\includegraphics{papers/nav/bilder/position4.pdf}}% +} + +\medskip + +Damit wir gleich den Längengrad berechnen können, benötigen wir noch den Winkel $\omega$: +\begin{align*} + \omega &= \cos^{-1} \bigg[\frac{\cos(h_B)-\cos(c) \cdot \cos(l)}{\sin(c) \cdot \sin(l)}\bigg] \\ + &=\cos^{-1} \bigg[\frac{\cos(42.572556^\circ)-\cos(70.936778^\circ) \cdot \cos(54.2833404^\circ)}{\sin(70.936778^\circ) \cdot \sin(54.2833404^\circ)}\bigg] \\ + &= \underline{\underline{44.6687451^\circ}} +\end{align*} + +\subsection{Längengrad und Breitengrad bestimmen} + +\begin{align*} +\delta &= 90^\circ - l & + \lambda &= \lambda_{\text{Arktur}} + \omega \\ +&= 90^\circ - 54.2833404 & + &= 95.5647759^\circ + 44.6687451^\circ \\ +&= \underline{\underline{35.7166596^\circ}} & + &= \underline{\underline{140.233521^\circ}} +\end{align*} +Wie wir sehen, stimmen die berechneten Koordinaten mit den Koordinaten des Punktes, an welchem gemessen wurde überein. + +\subsection{Fazit} +Die theoretische Anleitung im Abschnitt \ref{sta} scheint grundsätzlich zu funktionieren. +Allerdings gab es zwei interessante Probleme. + +Einerseits das Problem, ob der Punkt $P$ sich oberhalb oder unterhalb von $a$ befindet. +Da wir eigentlich wussten, wo der gesuchte Punkt $P$ ist, konnten wir das Dreieck anhand der Koordinaten der Bildpunkte richtig aufstellen. +In der Praxis muss man aber schon wissen, auf welchem Breitengrad man ungefähr ist. +Dies weis man in der Regeln aber, da die eigene Breite die Höhe des Polarsterns ist. +Diese Höhe wird mit dem Sextant gemessen. + +Andererseits ist da noch ein Problem mit dem Sinussatz. +Beim Sinussatz gibt es immer zwei Lösungen, weil \[ \sin(\pi-a)=\sin(a).\] +Da kann es sein (und war in unserem Fall auch so), dass man das falsche Ergebnis erwischt. +Wegen dieser Erkenntnis haben wir nur Kosinussätze verwendet und dies ebenfalls im Abschnitt \ref{sta} abgeändert, da es für den Leser auch relevant sein kann, wenn er es Probieren möchte. + + + + diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex new file mode 100644 index 0000000..c778d5c --- /dev/null +++ b/buch/papers/nav/einleitung.tex @@ -0,0 +1,10 @@ + + +\section{Einleitung} +\rhead{Einleitung} +Heutzutage ist die Navigation ein Teil des Lebens. +Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. +Dies wird durch Technologien wie Funknavigation, welches ein auf Laufzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist, oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. +Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? +In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet.
\ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex new file mode 100644 index 0000000..9745cdc --- /dev/null +++ b/buch/papers/nav/flatearth.tex @@ -0,0 +1,28 @@ + + +\section{Warum ist die Erde nicht flach?} +\rhead{Warum ist die Erde nicht flach?} +\begin{figure} + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/projektion.png} + \caption[Mercator Projektion]{Mercator Projektion} + \label{merc} + \end{center} +\end{figure} + +Es gibt heutzutage viele Beweise dafür, dass die Erde eine Kugel ist. +Die Fotos von unserem Planeten oder die Berichte der Astronauten. +Aber schon vor ca. 2300 Jahren hat Aristoteles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist. +Auch der Erdschatten bei einer Mondfinsternis ist immer rund. +Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. +Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. +Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. + +Der Kartograph Gerhard Mercator projizierte die Erdkugel wie in Abbildung \ref{merc} dargestellt auf ein Papier und erstellte so eine winkeltreue Karte. +Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. +Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. +Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. +Grönland ist auf der Weltkarte so gross wie Afrika. +In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. +Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. + diff --git a/buch/papers/nav/images/2k_earth_daymap.png b/buch/papers/nav/images/2k_earth_daymap.png Binary files differnew file mode 100644 index 0000000..4d55da8 --- /dev/null +++ b/buch/papers/nav/images/2k_earth_daymap.png diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile new file mode 100644 index 0000000..39bfbcf --- /dev/null +++ b/buch/papers/nav/images/Makefile @@ -0,0 +1,130 @@ +# +# Makefile to build images +# +# (c) 2022 +# +all: dreiecke3d + +dreieck.pdf: dreieck.tex dreieckdata.tex macros.tex + pdflatex dreieck.tex + +dreieckdata.tex: pk.m + octave pk.m + +DREIECKE = \ + dreieck1.pdf \ + dreieck2.pdf \ + dreieck3.pdf \ + dreieck4.pdf \ + dreieck5.pdf \ + dreieck6.pdf \ + dreieck7.pdf + +dreiecke: $(DREIECKE) + +dreieck1.pdf: dreieck1.tex dreieckdata.tex macros.tex + pdflatex dreieck1.tex + +dreieck2.pdf: dreieck2.tex dreieckdata.tex macros.tex + pdflatex dreieck2.tex + +dreieck3.pdf: dreieck3.tex dreieckdata.tex macros.tex + pdflatex dreieck3.tex + +dreieck4.pdf: dreieck4.tex dreieckdata.tex macros.tex + pdflatex dreieck4.tex + +dreieck5.pdf: dreieck5.tex dreieckdata.tex macros.tex + pdflatex dreieck5.tex + +dreieck6.pdf: dreieck6.tex dreieckdata.tex macros.tex + pdflatex dreieck6.tex + +dreieck7.pdf: dreieck7.tex dreieckdata.tex macros.tex + pdflatex dreieck7.tex + +DREIECKE3D = \ + dreieck3d1.pdf \ + dreieck3d2.pdf \ + dreieck3d3.pdf \ + dreieck3d4.pdf \ + dreieck3d5.pdf \ + dreieck3d6.pdf \ + dreieck3d7.pdf \ + dreieck3d8.pdf + +dreiecke3d: $(DREIECKE3D) + +POVRAYOPTIONS = -W1080 -H1080 +#POVRAYOPTIONS = -W480 -H480 + +dreieck3d1.png: dreieck3d1.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d1.png dreieck3d1.pov +dreieck3d1.jpg: dreieck3d1.png + convert dreieck3d1.png -density 300 -units PixelsPerInch dreieck3d1.jpg +dreieck3d1.pdf: dreieck3d1.tex dreieck3d1.jpg + pdflatex dreieck3d1.tex + +dreieck3d2.png: dreieck3d2.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d2.png dreieck3d2.pov +dreieck3d2.jpg: dreieck3d2.png + convert dreieck3d2.png -density 300 -units PixelsPerInch dreieck3d2.jpg +dreieck3d2.pdf: dreieck3d2.tex dreieck3d2.jpg + pdflatex dreieck3d2.tex + +dreieck3d3.png: dreieck3d3.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d3.png dreieck3d3.pov +dreieck3d3.jpg: dreieck3d3.png + convert dreieck3d3.png -density 300 -units PixelsPerInch dreieck3d3.jpg +dreieck3d3.pdf: dreieck3d3.tex dreieck3d3.jpg + pdflatex dreieck3d3.tex + +dreieck3d4.png: dreieck3d4.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d4.png dreieck3d4.pov +dreieck3d4.jpg: dreieck3d4.png + convert dreieck3d4.png -density 300 -units PixelsPerInch dreieck3d4.jpg +dreieck3d4.pdf: dreieck3d4.tex dreieck3d4.jpg + pdflatex dreieck3d4.tex + +dreieck3d5.png: dreieck3d5.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d5.png dreieck3d5.pov +dreieck3d5.jpg: dreieck3d5.png + convert dreieck3d5.png -density 300 -units PixelsPerInch dreieck3d5.jpg +dreieck3d5.pdf: dreieck3d5.tex dreieck3d5.jpg + pdflatex dreieck3d5.tex + +dreieck3d6.png: dreieck3d6.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d6.png dreieck3d6.pov +dreieck3d6.jpg: dreieck3d6.png + convert dreieck3d6.png -density 300 -units PixelsPerInch dreieck3d6.jpg +dreieck3d6.pdf: dreieck3d6.tex dreieck3d6.jpg + pdflatex dreieck3d6.tex + +dreieck3d7.png: dreieck3d7.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d7.png dreieck3d7.pov +dreieck3d7.jpg: dreieck3d7.png + convert dreieck3d7.png -density 300 -units PixelsPerInch dreieck3d7.jpg +dreieck3d7.pdf: dreieck3d7.tex dreieck3d7.jpg + pdflatex dreieck3d7.tex + +dreieck3d8.png: dreieck3d8.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d8.png dreieck3d8.pov +dreieck3d8.jpg: dreieck3d8.png + convert dreieck3d8.png -density 300 -units PixelsPerInch dreieck3d8.jpg +dreieck3d8.pdf: dreieck3d8.tex dreieck3d8.jpg + pdflatex dreieck3d8.tex + +dreieck3d9.png: dreieck3d9.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d9.png dreieck3d9.pov +dreieck3d9.jpg: dreieck3d9.png + convert dreieck3d9.png -density 300 -units PixelsPerInch dreieck3d9.jpg +dreieck3d9.pdf: dreieck3d9.tex dreieck3d9.jpg + pdflatex dreieck3d9.tex + +dreieck3d10.png: dreieck3d10.pov common.inc macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d10.png dreieck3d10.pov +dreieck3d10.jpg: dreieck3d10.png + convert dreieck3d10.png -density 300 -units PixelsPerInch dreieck3d10.jpg +dreieck3d10.pdf: dreieck3d10.tex dreieck3d10.jpg macros.inc + pdflatex dreieck3d10.tex + diff --git a/buch/papers/nav/images/beispiele/2k_earth_daymap.png b/buch/papers/nav/images/beispiele/2k_earth_daymap.png Binary files differnew file mode 100644 index 0000000..4d55da8 --- /dev/null +++ b/buch/papers/nav/images/beispiele/2k_earth_daymap.png diff --git a/buch/papers/nav/images/beispiele/Makefile b/buch/papers/nav/images/beispiele/Makefile new file mode 100644 index 0000000..9546c8e --- /dev/null +++ b/buch/papers/nav/images/beispiele/Makefile @@ -0,0 +1,38 @@ +# +# Makefile to build images +# +# (c) 2022 +# +all: beispiele + +POSITION = \ + beispiele1.pdf \ + beispiele2.pdf \ + beispiele3.pdf + +beispiele: $(POSITION) + +POVRAYOPTIONS = -W1080 -H1080 +#POVRAYOPTIONS = -W480 -H480 + +beispiele1.png: beispiele1.pov common.inc geometrie.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Obeispiele1.png beispiele1.pov +beispiele1.jpg: beispiele1.png + convert beispiele1.png -density 300 -units PixelsPerInch beispiele1.jpg +beispiele1.pdf: beispiele1.tex common.tex beispiele1.jpg + pdflatex beispiele1.tex + +beispiele2.png: beispiele2.pov common.inc geometrie.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Obeispiele2.png beispiele2.pov +beispiele2.jpg: beispiele2.png + convert beispiele2.png -density 300 -units PixelsPerInch beispiele2.jpg +beispiele2.pdf: beispiele2.tex common.tex beispiele2.jpg + pdflatex beispiele2.tex + +beispiele3.png: beispiele3.pov common.inc geometrie.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Obeispiele3.png beispiele3.pov +beispiele3.jpg: beispiele3.png + convert beispiele3.png -density 300 -units PixelsPerInch beispiele3.jpg +beispiele3.pdf: beispiele3.tex common.tex beispiele3.jpg + pdflatex beispiele3.tex + diff --git a/buch/papers/nav/images/beispiele/beispiele1.pdf b/buch/papers/nav/images/beispiele/beispiele1.pdf Binary files differnew file mode 100644 index 0000000..1f91809 --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele1.pdf diff --git a/buch/papers/nav/images/beispiele/beispiele1.pov b/buch/papers/nav/images/beispiele/beispiele1.pov new file mode 100644 index 0000000..7fb3de2 --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele1.pov @@ -0,0 +1,12 @@ +// +// beispiele1.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +#declare Stern1 = Deneb; +#declare Stern2 = Arktur; + +#include "geometrie.inc" + diff --git a/buch/papers/nav/images/beispiele/beispiele1.tex b/buch/papers/nav/images/beispiele/beispiele1.tex new file mode 100644 index 0000000..0dfae2f --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele1.tex @@ -0,0 +1,49 @@ +% +% beispiele1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math,calc} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.8125] + +% Povray Bild +\node at (0,0) {\includegraphics[width=6.5cm]{beispiele1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelP +\labelDeneb +\labelArktur +\labelhDeneb +\labelhArktur +\labellone +\labeldDeneb +\labeldArktur + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/beispiele/beispiele2.pdf b/buch/papers/nav/images/beispiele/beispiele2.pdf Binary files differnew file mode 100644 index 0000000..4b28f2f --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele2.pdf diff --git a/buch/papers/nav/images/beispiele/beispiele2.pov b/buch/papers/nav/images/beispiele/beispiele2.pov new file mode 100644 index 0000000..b69f0f9 --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele2.pov @@ -0,0 +1,12 @@ +// +// beispiele1.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +#declare Stern1 = Altair; +#declare Stern2 = Spica; + +#include "geometrie.inc" + diff --git a/buch/papers/nav/images/beispiele/beispiele2.tex b/buch/papers/nav/images/beispiele/beispiele2.tex new file mode 100644 index 0000000..04c1e4d --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele2.tex @@ -0,0 +1,50 @@ +% +% beispiele2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math,calc} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.8125] + +% Povray Bild +\node at (0,0) {\includegraphics[width=6.5cm]{beispiele2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelP +\labelAltair +\labelSpica +\labelhAltair +\labelhSpica +\labelltwo +\labeldAltair +\labeldSpica + + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/beispiele/beispiele3.pdf b/buch/papers/nav/images/beispiele/beispiele3.pdf Binary files differnew file mode 100644 index 0000000..049ccdf --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele3.pdf diff --git a/buch/papers/nav/images/beispiele/beispiele3.pov b/buch/papers/nav/images/beispiele/beispiele3.pov new file mode 100644 index 0000000..af9a468 --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele3.pov @@ -0,0 +1,12 @@ +// +// beispiele1.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +#declare Stern1 = Deneb; +#declare Stern2 = Altair; + +#include "geometrie.inc" + diff --git a/buch/papers/nav/images/beispiele/beispiele3.tex b/buch/papers/nav/images/beispiele/beispiele3.tex new file mode 100644 index 0000000..2573199 --- /dev/null +++ b/buch/papers/nav/images/beispiele/beispiele3.tex @@ -0,0 +1,49 @@ +% +% beispiele3.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math,calc} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{beispiele3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelP +\labelDeneb +\labelAltair +\labelhDeneb +\labelhAltair +\labellone +%\labeldDeneb +%\labeldAltair + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/beispiele/common.inc b/buch/papers/nav/images/beispiele/common.inc new file mode 100644 index 0000000..51fbd1f --- /dev/null +++ b/buch/papers/nav/images/beispiele/common.inc @@ -0,0 +1,50 @@ +// +// common.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "../macros.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.034; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + +light_source { + <30, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +erde(0) +achse(fein, White) +koordinatennetz(gitterfarbe, 9, 0.001) + +union { + punkt(Sakura, fett) + pigment { + color rot + } + finish { + metallic + specular 0.9 + } +} + diff --git a/buch/papers/nav/images/beispiele/common.tex b/buch/papers/nav/images/beispiele/common.tex new file mode 100644 index 0000000..81dc037 --- /dev/null +++ b/buch/papers/nav/images/beispiele/common.tex @@ -0,0 +1,79 @@ +% +% common.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\def\labelA{\node at (0.7,3.8) {$A$};} + +\def\labelSpica{ + \node at (-3.6,-2.8) {Spica}; +} +\def\labelAltair{ + \node at (3.0,-2.3) {Altair}; +} +\def\labelArktur{ + \node at (-3.3,-0.7) {Arktur}; +} +\def\labelDeneb{ + \node at (3.4,0.9) {Deneb}; +} + +\def\labelP{\node at (0,-0.2) {$P$};} + +\def\labellone{\node at (0.1,1.9) {$l$};} +\def\labelltwo{\node at (0.1,2.0) {$l$};} + +\def\labelhSpica{ + \coordinate (Spica) at (-1.8,-0.3); + \node at (Spica) {$h_{\text{Spica}}\mathstrut$}; +} +\def\labelhAltair{ + \coordinate (Altair) at (1.1,-1.0); + \node at (Altair) {$h_{\text{Altair}}\mathstrut$}; +} +\def\labelhArktur{ + \coordinate (Arktur) at (-1.3,-0.3); + \node at (Arktur) {$h_{\text{Arktur}}\mathstrut$}; +} +\def\labelhDeneb{ + \coordinate (Deneb) at (1.6,0.45); + \node at (Deneb) {$h_{\text{Deneb}}\mathstrut$}; +} + +\def\labeldSpica{ + \coordinate (dSpica) at (-1.5,2.6); + \fill[color=white,opacity=0.5] + ($(dSpica)+(-1.8,0.13)$) + rectangle + ($(dSpica)+(-0.06,0.60)$); + \node at (dSpica) [above left] + {$90^\circ-\delta_{\text{Spica}}\mathstrut$}; +} +\def\labeldAltair{ + \coordinate (dAltair) at (2.0,2.1); + \fill[color=white,opacity=0.5] + ($(dAltair)+(0.10,0.10)$) + rectangle + ($(dAltair)+(2.0,0.60)$); + \node at (dAltair) [above right] + {$90^\circ-\delta_{\text{Altair}}\mathstrut$}; +} +\def\labeldArktur{ + \coordinate (dArktur) at (-1.2,2.5); + \fill[color=white,opacity=0.5] + ($(dArktur)+(-1.8,0.10)$) + rectangle + ($(dArktur)+(-0.06,0.55)$); + \node at (dArktur) [above left] + {$90^\circ-\delta_{\text{Arktur}}\mathstrut$}; +} +\def\labeldDeneb{ + \coordinate (dDeneb) at (2.0,2.8); + \fill[color=white,opacity=0.5] + ($(dDeneb)+(0.05,0.60)$) + rectangle + ($(dDeneb)+(1.87,0.10)$); + \node at (dDeneb) [above right] + {$90^\circ-\delta_{\text{Deneb}}\mathstrut$}; +} diff --git a/buch/papers/nav/images/beispiele/geometrie.inc b/buch/papers/nav/images/beispiele/geometrie.inc new file mode 100644 index 0000000..2f6084e --- /dev/null +++ b/buch/papers/nav/images/beispiele/geometrie.inc @@ -0,0 +1,41 @@ +union { + punkt(A, fett) + punkt(Stern1, fein) + punkt(Stern2, fein) + seite(Stern1, Stern2, fein) + pigment { + color kugelfarbe + } + finish { + metallic + specular 0.9 + } +} + +union { + seite(A, Stern1, fein) + seite(A, Stern2, fein) + seite(Stern1, Sakura, fein) + seite(Stern2, Sakura, fein) + winkel(A, Stern1, Stern2, 0.5*fein, gross) + pigment { + color bekannt + } + finish { + metallic + specular 0.9 + } +} + +union { + seite(A, Sakura, fein) + winkel(A, Sakura, Stern1, 0.5*fett, klein) + pigment { + color unbekannt + } + finish { + metallic + specular 0.9 + } +} + diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc new file mode 100644 index 0000000..7b861de --- /dev/null +++ b/buch/papers/nav/images/common.inc @@ -0,0 +1,35 @@ +// +// common.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "macros.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.034; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + +light_source { + <30, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + diff --git a/buch/papers/nav/images/dreieck.tex b/buch/papers/nav/images/dreieck.tex new file mode 100644 index 0000000..55f6a81 --- /dev/null +++ b/buch/papers/nav/images/dreieck.tex @@ -0,0 +1,68 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\def\punktbeschriftung{ + \node at (A) [above] {$A$}; + \node at (B) [left] {$B$}; + \node at (C) [right] {$C$}; + \node at (P) [below] {$P$}; +} + +\winkelKappa{gray} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\winkelOmega{gray} +\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\punktbeschriftung + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck1.pdf b/buch/papers/nav/images/dreieck1.pdf Binary files differnew file mode 100644 index 0000000..5bdf23d --- /dev/null +++ b/buch/papers/nav/images/dreieck1.pdf diff --git a/buch/papers/nav/images/dreieck1.tex b/buch/papers/nav/images/dreieck1.tex new file mode 100644 index 0000000..436314c --- /dev/null +++ b/buch/papers/nav/images/dreieck1.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck2.pdf b/buch/papers/nav/images/dreieck2.pdf Binary files differnew file mode 100644 index 0000000..a872b25 --- /dev/null +++ b/buch/papers/nav/images/dreieck2.pdf diff --git a/buch/papers/nav/images/dreieck2.tex b/buch/papers/nav/images/dreieck2.tex new file mode 100644 index 0000000..99aabb7 --- /dev/null +++ b/buch/papers/nav/images/dreieck2.tex @@ -0,0 +1,59 @@ +% +% dreieck2.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck3.pdf b/buch/papers/nav/images/dreieck3.pdf Binary files differnew file mode 100644 index 0000000..65070c6 --- /dev/null +++ b/buch/papers/nav/images/dreieck3.pdf diff --git a/buch/papers/nav/images/dreieck3.tex b/buch/papers/nav/images/dreieck3.tex new file mode 100644 index 0000000..0cf5363 --- /dev/null +++ b/buch/papers/nav/images/dreieck3.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d1.pdf b/buch/papers/nav/images/dreieck3d1.pdf Binary files differnew file mode 100644 index 0000000..fecaece --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.pdf diff --git a/buch/papers/nav/images/dreieck3d1.pov b/buch/papers/nav/images/dreieck3d1.pov new file mode 100644 index 0000000..336161c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.pov @@ -0,0 +1,61 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, gross) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, A, fein, gross) + pigment { + color gruen + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, A, B, fein, gross) + pigment { + color blau + } + finish { + specular 0.95 + metallic + } +} diff --git a/buch/papers/nav/images/dreieck3d1.tex b/buch/papers/nav/images/dreieck3d1.tex new file mode 100644 index 0000000..799b21a --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.tex @@ -0,0 +1,53 @@ +% +% dreieck3d1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +\node at (-2.5,-0.5) {$\beta$}; +\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d10.pov b/buch/papers/nav/images/dreieck3d10.pov new file mode 100644 index 0000000..2dd7c79 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d10.pov @@ -0,0 +1,46 @@ +// +// dreiecke3d10.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +erde() + +#declare Stern1 = Deneb; +#declare Stern2 = Spica; + +koordinatennetz(gitterfarbe, 9, 0.001) + +union { + seite(A, Stern1, 0.5*fein) + seite(A, Stern2, 0.5*fein) + seite(A, Sakura, 0.5*fein) + seite(Stern1, Sakura, 0.5*fein) + seite(Stern2, Sakura, 0.5*fein) + seite(Stern1, Stern2, 0.5*fein) + + punkt(A, fein) + punkt(Sakura, fett) + punkt(Deneb, fein) + punkt(Spica, fein) + punkt(Altair, fein) + punkt(Arktur, fein) + pigment { + color Red + } +} + +//arrow(<-1.3,0,0>, <1.3,0,0>, fein, White) +arrow(<0,-1.3,0>, <0,1.3,0>, fein, White) +//arrow(<0,0,-1.3>, <0,0,1.3>, fein, White) + +#declare imagescale = 0.044; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + diff --git a/buch/papers/nav/images/dreieck3d2.pdf b/buch/papers/nav/images/dreieck3d2.pdf Binary files differnew file mode 100644 index 0000000..28af5fe --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.pdf diff --git a/buch/papers/nav/images/dreieck3d2.pov b/buch/papers/nav/images/dreieck3d2.pov new file mode 100644 index 0000000..9e57d22 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.pov @@ -0,0 +1,28 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d2.tex b/buch/papers/nav/images/dreieck3d2.tex new file mode 100644 index 0000000..0f6e10c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.tex @@ -0,0 +1,53 @@ +% +% dreieck3d2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d3.pdf b/buch/papers/nav/images/dreieck3d3.pdf Binary files differnew file mode 100644 index 0000000..4fc4fc1 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.pdf diff --git a/buch/papers/nav/images/dreieck3d3.pov b/buch/papers/nav/images/dreieck3d3.pov new file mode 100644 index 0000000..bde780b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.pov @@ -0,0 +1,39 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, gross) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d3.tex b/buch/papers/nav/images/dreieck3d3.tex new file mode 100644 index 0000000..a047b1b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.tex @@ -0,0 +1,53 @@ +% +% dreieck3d3.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d4.pdf b/buch/papers/nav/images/dreieck3d4.pdf Binary files differnew file mode 100644 index 0000000..0d57fc2 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.pdf diff --git a/buch/papers/nav/images/dreieck3d4.pov b/buch/papers/nav/images/dreieck3d4.pov new file mode 100644 index 0000000..08f266b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.pov @@ -0,0 +1,39 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugelfarbe) + +union { + seite(A, B, fein) + seite(A, C, fein) + punkt(A, fein) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, P, fein, gross) + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d4.tex b/buch/papers/nav/images/dreieck3d4.tex new file mode 100644 index 0000000..d49fb66 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.tex @@ -0,0 +1,54 @@ +% +% dreieck3d4.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d4.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.3,-1.5) {$\beta_1$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d5.pdf b/buch/papers/nav/images/dreieck3d5.pdf Binary files differnew file mode 100644 index 0000000..a5dd0ae --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.pdf diff --git a/buch/papers/nav/images/dreieck3d5.pov b/buch/papers/nav/images/dreieck3d5.pov new file mode 100644 index 0000000..1aac0dc --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.pov @@ -0,0 +1,28 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, B, fein) + seite(A, C, fein) + punkt(A, fein) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d5.tex b/buch/papers/nav/images/dreieck3d5.tex new file mode 100644 index 0000000..8011b37 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.tex @@ -0,0 +1,53 @@ +% +% dreieck3d5.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d5.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d6.pov b/buch/papers/nav/images/dreieck3d6.pov new file mode 100644 index 0000000..6bbd1a9 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.pov @@ -0,0 +1,39 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, B, fett) + seite(A, C, fett) + seite(B, P, fett) + seite(C, P, fett) + seite(A, P, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, A, P, fein, gross) + pigment { + color rgb<0.6,0.2,0.6> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d6.tex b/buch/papers/nav/images/dreieck3d6.tex new file mode 100644 index 0000000..bbca2ca --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.tex @@ -0,0 +1,55 @@ +% +% dreieck3d6.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d6.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.4,-0.6) {$\kappa$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d7.pov b/buch/papers/nav/images/dreieck3d7.pov new file mode 100644 index 0000000..45dc5d6 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.pov @@ -0,0 +1,41 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +kugel(kugeldunkel) + +union { + seite(A, C, fett) + seite(A, P, fett) + seite(C, P, fett) + + seite(A, B, fein) + seite(B, C, fein) + seite(B, P, fein) + punkt(A, fett) + punkt(C, fett) + punkt(P, fett) + punkt(B, fein) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fein, gross) + pigment { + color rgb<0.4,0.4,1> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d7.tex b/buch/papers/nav/images/dreieck3d7.tex new file mode 100644 index 0000000..4027a8b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.tex @@ -0,0 +1,55 @@ +% +% dreieck3d7.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d7.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (0.8,3.1) {$\omega$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d8.jpg b/buch/papers/nav/images/dreieck3d8.jpg Binary files differnew file mode 100644 index 0000000..f24ea33 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.jpg diff --git a/buch/papers/nav/images/dreieck3d8.pdf b/buch/papers/nav/images/dreieck3d8.pdf Binary files differnew file mode 100644 index 0000000..da3b110 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.pdf diff --git a/buch/papers/nav/images/dreieck3d8.pov b/buch/papers/nav/images/dreieck3d8.pov new file mode 100644 index 0000000..dae7f67 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.pov @@ -0,0 +1,97 @@ +// +// dreiecke3d8.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + seite(A, P, fein) + seite(B, P, fett) + seite(C, P, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, klein) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, A, fein, klein) + pigment { + color gruen + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, A, B, fein, gross) + pigment { + color blau + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fein/2, gross) + pigment { + color rgb<0.8,0,0.8> + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, P, C, fein, klein) + pigment { + color rgb<1,0.8,0> + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, P, A, fein/2, gross) + pigment { + color rgb<0.4,0.6,0.8> + } + finish { + specular 0.95 + metallic + } +} + +dreieck(A, B, C, White) + +kugel(kugeldunkel) + diff --git a/buch/papers/nav/images/dreieck3d8.tex b/buch/papers/nav/images/dreieck3d8.tex new file mode 100644 index 0000000..c59c7b0 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.tex @@ -0,0 +1,57 @@ +% +% dreieck3d8.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d8.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.8,0) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3.3) {$\alpha$}; +\node at (0.8,2.85) {$\omega$}; +\node at (-2.6,-0.6) {$\beta$}; +\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.6,-1.3) {$\beta_1$}; +\node at (-2.1,-0.8) {$\kappa$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d9.pov b/buch/papers/nav/images/dreieck3d9.pov new file mode 100644 index 0000000..24d3843 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d9.pov @@ -0,0 +1,66 @@ +// +// dreiecke3d8.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +//union { +// seite(A, B, fein) +// seite(B, C, fein) +// seite(A, C, fein) +// seite(A, P, fein) +// seite(B, P, fett) +// seite(C, P, fett) +// punkt(A, fein) +// punkt(B, fett) +// punkt(C, fett) +// punkt(P, fett) +// pigment { +// color dreieckfarbe +// } +// finish { +// specular 0.95 +// metallic +// } +//} + +//dreieck(A, B, C, White) + +kugel(kugeltransparent) + +ebenerwinkel(O, C, P, 0.01, 1.001, rot) +ebenerwinkel(P, C, P, 0.01, 0.3, rot) +komplement(P, C, P, 0.01, 0.3, Yellow) + +ebenerwinkel(O, B, P, 0.01, 1.001, blau) +ebenerwinkel(P, B, P, 0.01, 0.3, blau) +komplement(P, B, P, 0.01, 0.3, Green) + +arrow(B, 1.5 * B, 0.015, White) +arrow(C, 1.5 * C, 0.015, White) +arrow(P, 1.5 * P, 0.015, White) + +union { + cylinder { O, P, 0.7 * fein } + + cylinder { P, P + 3 * B, 0.7 * fein } + cylinder { O, B + 3 * B, 0.7 * fein } + + cylinder { P, P + 3 * C, 0.7 * fein } + cylinder { O, C + 3 * C, 0.7 * fein } + + pigment { + color White + } +} + +#declare imagescale = 0.044; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + diff --git a/buch/papers/nav/images/dreieck4.pdf b/buch/papers/nav/images/dreieck4.pdf Binary files differnew file mode 100644 index 0000000..4871a1e --- /dev/null +++ b/buch/papers/nav/images/dreieck4.pdf diff --git a/buch/papers/nav/images/dreieck4.tex b/buch/papers/nav/images/dreieck4.tex new file mode 100644 index 0000000..19a7d12 --- /dev/null +++ b/buch/papers/nav/images/dreieck4.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck5.pdf b/buch/papers/nav/images/dreieck5.pdf Binary files differnew file mode 100644 index 0000000..cf686e0 --- /dev/null +++ b/buch/papers/nav/images/dreieck5.pdf diff --git a/buch/papers/nav/images/dreieck5.tex b/buch/papers/nav/images/dreieck5.tex new file mode 100644 index 0000000..d1117d1 --- /dev/null +++ b/buch/papers/nav/images/dreieck5.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +%\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck6.pdf b/buch/papers/nav/images/dreieck6.pdf Binary files differnew file mode 100644 index 0000000..7efd673 --- /dev/null +++ b/buch/papers/nav/images/dreieck6.pdf diff --git a/buch/papers/nav/images/dreieck6.tex b/buch/papers/nav/images/dreieck6.tex new file mode 100644 index 0000000..87db1c2 --- /dev/null +++ b/buch/papers/nav/images/dreieck6.tex @@ -0,0 +1,64 @@ +% +% dreieck6.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +%\seiteA{gray} + +\seiteL{black} +\seitePB{black} +\seitePC{black} + +\draw[color=black,line width=1.4pt] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +%\draw[color=gray] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck7.pdf b/buch/papers/nav/images/dreieck7.pdf Binary files differnew file mode 100644 index 0000000..aa83e28 --- /dev/null +++ b/buch/papers/nav/images/dreieck7.pdf diff --git a/buch/papers/nav/images/dreieck7.tex b/buch/papers/nav/images/dreieck7.tex new file mode 100644 index 0000000..f084708 --- /dev/null +++ b/buch/papers/nav/images/dreieck7.tex @@ -0,0 +1,64 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{gray} + +\seiteL{black} +\seitePB{gray} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +\draw[color=gray] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{gray}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node[color=gray] at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieckdata.tex b/buch/papers/nav/images/dreieckdata.tex new file mode 100644 index 0000000..c0fb720 --- /dev/null +++ b/buch/papers/nav/images/dreieckdata.tex @@ -0,0 +1,16 @@ +\coordinate (P) at (0.0000,0.0000); +\coordinate (A) at (1.0000,8.0000); +\coordinate (B) at (-3.0000,3.0000); +\coordinate (C) at (4.0000,4.0000); +\def\kanteAB{(1.0000,8.0000) arc (114.77514:167.90524:7.1589)} +\def\kanteBA{(-3.0000,3.0000) arc (167.90524:114.77514:7.1589)} +\def\kanteAC{(1.0000,8.0000) arc (63.43495:10.30485:5.5902)} +\def\kanteCA{(4.0000,4.0000) arc (10.30485:63.43495:5.5902)} +\def\kanteAP{(1.0000,8.0000) arc (146.30993:199.44003:9.0139)} +\def\kantePA{(0.0000,0.0000) arc (199.44003:146.30993:9.0139)} +\def\kanteBC{(-3.0000,3.0000) arc (-95.90614:-67.83365:14.5774)} +\def\kanteCB{(4.0000,4.0000) arc (-67.83365:-95.90614:14.5774)} +\def\kanteBP{(-3.0000,3.0000) arc (-161.56505:-108.43495:4.7434)} +\def\kantePB{(0.0000,0.0000) arc (-108.43495:-161.56505:4.7434)} +\def\kanteCP{(4.0000,4.0000) arc (-30.96376:-59.03624:11.6619)} +\def\kantePC{(0.0000,0.0000) arc (-59.03624:-30.96376:11.6619)} diff --git a/buch/papers/nav/images/macros.inc b/buch/papers/nav/images/macros.inc new file mode 100644 index 0000000..20cb2ff --- /dev/null +++ b/buch/papers/nav/images/macros.inc @@ -0,0 +1,345 @@ +// +// macros.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +// +// Dimensions +// +#declare fett = 0.015; +#declare fein = 0.010; + +#declare klein = 0.3; +#declare gross = 0.4; + +// +// colors +// +#declare dreieckfarbe = rgb<0.6,0.6,0.6>; +#declare rot = rgb<0.8,0.2,0.2>; +#declare gruen = rgb<0,0.6,0>; +#declare blau = rgb<0.2,0.2,0.8>; + +#declare bekannt = rgb<0.2,0.6,1>; +#declare unbekannt = rgb<1.0,0.6,0.8>; + +#declare kugelfarbe = rgb<0.8,0.8,0.8>; +#declare kugeldunkel = rgb<0.4,0.4,0.4>; +#declare kugeltransparent = rgbt<0.8,0.8,0.8,0.5>; + +#declare gitterfarbe = rgb<0.2,0.6,1>; +#declare gitterfarbe = rgb<1.0,0.8,0>; + +// +// Points Points +// +#declare O = <0, 0, 0>; +#declare Nordpol = vnormalize(< 0, 1, 0>); +#declare A = vnormalize(< 0, 1, 0>); +#declare B = vnormalize(< 1, 2, -8>); +#declare C = vnormalize(< 5, 1, 0>); +#declare P = vnormalize(< 5, -1, -7>); + +// +// \brief convert spherical coordinates to recctangular coordinates +// +// \param phi +// \param theta +// +#macro kugelpunkt(phi, theta) + < sin(theta) * cos(phi - pi), cos(theta), sin(theta) * sin(phi - pi) > +#end + +#declare Sakura = kugelpunkt(radians(140.2325498), radians(90 - 35.71548014)); +#declare Deneb = kugelpunkt(radians(191.9397759), radians(90 - 45.361194)); +#declare Spica = kugelpunkt(radians(82.9868559), radians(90 - (-11.279666))); +#declare Altair = kugelpunkt(radians(179.3616609), radians(90 - 8.928416)); +#declare Arktur = kugelpunkt(radians(95.5647759), radians(90 - 19.063222)); + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#declare ntsteps = 100; + +// +// \brief Draw a circle +// +// \param b1 basis vector for a coordinate system of the plane containing +// the circle +// \param b2 the other basis vector +// \param o center of the circle +// \param thick diameter of the circular tube +// +#macro kreis(b1, b2, o, thick, maxwinkel) + #declare tpstep = pi / ntsteps; + #declare tp = tpstep; + #declare p1 = b1 + o; + sphere { p1, thick } + #declare tpstep = pi/ntsteps; + #while (tp < (maxwinkel - tpstep/2)) + #declare p2 = cos(tp) * b1 + sin(tp) * b2 + o; + cylinder { p1, p2, thick } + sphere { p2, thick } + #declare p1 = p2; + #declare tp = tp + tpstep; + #end + #if ((tp - tpstep) < maxwinkel) + #declare p2 = cos(maxwinkel) * b1 + sin(maxwinkel) * b2 + o; + cylinder { p1, p2, thick } + sphere { p2, thick } + #end +#end + +// +// \brief Draw a great circle +// +// \param normale the normal of the plane containing the great circle +// \param thick diameter +// +#macro grosskreis(normale, thick) + #declare other = < normale.y, -normale.x, normale.z >; + #declare b1 = vnormalize(vcross(other, normale)); + #declare b2 = vnormalize(vcross(normale, b1)); + kreis(b1, b2, <0,0,0>, thick, 2*pi) +#end + +// +// \brief Draw a circle of latitude +// +// \param theta latitude +// \param thick diameter +// +#macro breitenkreis(theta, thick) + #declare b1 = sin(theta) * kugelpunkt(0, pi/2); + #declare b2 = sin(theta) * kugelpunkt(pi/2, pi/2); + #declare o = < 0, cos(theta), 0 >; + kreis(b1, b2, o, thick, 2*pi) +#end + +// +// \brief Draw the great circle connecting the two points +// +// \param p first point +// \param q second point +// \param staerke diameter +// + +#macro seite(p, q, staerke) + #declare s1 = vnormalize(p); + #declare s2 = vnormalize(q); + #declare w = acos(vdot(s1, s2)); + #declare n = vnormalize(vcross(p, q)); + #declare s2 = vnormalize(vcross(n, s1)); + kreis(s1, s2, O, staerke, w) +#end + +// +// \brief Draw an angle +// +// \param w the edge where the angle is located +// \param p point on the first leg +// \param q point on the second leg +// \param r diameter of the angle +// +#macro winkel(w, p, q, staerke, r) + #declare n = vnormalize(w); + #declare pp = vnormalize(p - vdot(n, p) * n); + #declare qq = vnormalize(q - vdot(n, q) * n); + intersection { + sphere { O, 1 + staerke } + cone { O, 0, 1.2 * vnormalize(w), r } + plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 } + plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 } + } +#end + +// +// \brief Draw a point on the sphere as a circle +// +// \param p the point +// \param staerke the diameter of the point +// +#macro punkt(p, staerke) + sphere { p, 1.5 * staerke } +#end + +// +// \brief Draw a circle as a part of the differently colored cutout from +// the sphere +// +// \param p first point of the triangle +// \param q second point of the triangle +// \param r third point of the triangle +// \param farbe color +// +#macro dreieck(p, q, r, farbe) + #declare n1 = vnormalize(vcross(p, q)); + #declare n2 = vnormalize(vcross(q, r)); + #declare n3 = vnormalize(vcross(r, p)); + intersection { + plane { n1, 0 } + plane { n2, 0 } + plane { n3, 0 } + sphere { <0, 0, 0>, 1 + 0.001 } + pigment { + color farbe + } + finish { + metallic + specular 0.4 + } + } +#end + +// +// \brief +// +// \param a axis of the angle +// \param p first leg +// \param q second leg +// \param s thickness of the angle disk +// \param r radius of the angle disk +// \param farbe color +// +#macro ebenerwinkel(a, p, q, s, r, farbe) + #declare n = vnormalize(-vcross(p, q)); + #declare np = vnormalize(-vcross(p, n)); + #declare nq = -vnormalize(-vcross(q, n)); +// arrow(a, a + n, 0.02, White) +// arrow(a, a + np, 0.01, Red) +// arrow(a, a + nq, 0.01, Blue) + intersection { + cylinder { a - (s/2) * n, a + (s/2) * n, r } + plane { np, vdot(np, a) } + plane { nq, vdot(nq, a) } + pigment { + farbe + } + finish { + metallic + specular 0.5 + } + } +#end + +// +// \brief Show the complement angle +// +// +#macro komplement(a, p, q, s, r, farbe) + #declare n = vnormalize(-vcross(p, q)); +// arrow(a, a + n, 0.015, Orange) + #declare m = vnormalize(-vcross(q, n)); +// arrow(a, a + m, 0.015, Pink) + ebenerwinkel(a, p, m, s, r, farbe) +#end + +// +// \brief Show a coordinate grid on the sphere +// +// \param farbe the color of the grid +// \param thick the line thickness +// +#macro koordinatennetz(farbe, netzschritte, thick) +union { + // circles of latitude + #declare theta = pi/(2*netzschritte); + #declare thetastep = pi/(2*netzschritte); + #while (theta < pi - thetastep/2) + breitenkreis(theta, thick) + #declare theta = theta + thetastep; + #end + // cirles of longitude + #declare phi = 0; + #declare phistep = pi/(2*netzschritte); + #while (phi < pi-phistep/2) + grosskreis(kugelpunkt(phi, pi/2), thick) + #declare phi = phi + phistep; + #end + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } +} +#end + +// +// \brief Display a color of given color +// +// \param farbe the color +// +#macro kugel(farbe) +sphere { + <0, 0, 0>, 1 + pigment { + color farbe + } +} +#end + +// +// \brief Display the earth +// +#macro erde(winkel) +sphere { + <0, 0, 0>, 1 + pigment { + image_map { + png "2k_earth_daymap.png" gamma 1.0 + map_type 1 + } + } + rotate <0,winkel,0> +} +#end + +// +// achse +// +#macro achse(durchmesser, farbe) + cylinder { + < 0, -1.2, 0 >, <0, 1.2, 0 >, durchmesser + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } + } +#end diff --git a/buch/papers/nav/images/macros.tex b/buch/papers/nav/images/macros.tex new file mode 100644 index 0000000..69a620d --- /dev/null +++ b/buch/papers/nav/images/macros.tex @@ -0,0 +1,54 @@ +\def\winkelAlpha#1{ + \begin{scope} + \clip (A) circle[radius=1.1]; + \fill[color=#1!20] \kanteAB -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(222:0.82)$) {$\alpha$}; +} + +\def\winkelOmega#1{ + \begin{scope} + \clip (A) circle[radius=0.7]; + \fill[color=#1!20] \kanteAP -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(285:0.50)$) {$\omega$}; +} + +\def\winkelGamma#1{ + \begin{scope} + \clip (C) circle[radius=1.0]; + \fill[color=#1!20] \kanteCA -- \kanteBC -- cycle; + \end{scope} + \node[color=#1] at ($(C)+(155:0.60)$) {$\gamma$}; +} + +\def\winkelKappa#1{ + \begin{scope} + \clip (B) circle[radius=1.2]; + \fill[color=#1!20] \kanteBP -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(15:1.00)$) {$\kappa$}; +} + +\def\winkelBeta#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBC -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(35:0.40)$) {$\beta$}; +} + +\def\winkelBetaEins#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBP -- \kanteCB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(330:0.60)$) {$\beta_1$}; +} + +\def\seiteC#1{ \node[color=#1] at (-1.9,5.9) {$c$}; } +\def\seiteB#1{ \node[color=#1] at (3.2,6.5) {$b$}; } +\def\seiteL#1{ \node[color=#1] at (-0.2,4.5) {$l$}; } +\def\seiteA#1{ \node[color=#1] at (2,3) {$a$}; } +\def\seitePB#1{ \node[color=#1] at (-2.1,1) {$p_b$}; } +\def\seitePC#1{ \node[color=#1] at (2.5,1.5) {$p_c$}; } diff --git a/buch/papers/nav/images/pk.m b/buch/papers/nav/images/pk.m new file mode 100644 index 0000000..6e89e9a --- /dev/null +++ b/buch/papers/nav/images/pk.m @@ -0,0 +1,55 @@ +# +# pk.m -- Punkte und Kanten für sphärisches Dreieck +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +A = [ 1, 8 ]; +B = [ -3, 3 ]; +C = [ 4, 4 ]; +P = [ 0, 0 ]; + +global fn; +fn = fopen("dreieckdata.tex", "w"); + +fprintf(fn, "\\coordinate (P) at (%.4f,%.4f);\n", P(1,1), P(1,2)); +fprintf(fn, "\\coordinate (A) at (%.4f,%.4f);\n", A(1,1), A(1,2)); +fprintf(fn, "\\coordinate (B) at (%.4f,%.4f);\n", B(1,1), B(1,2)); +fprintf(fn, "\\coordinate (C) at (%.4f,%.4f);\n", C(1,1), C(1,2)); + +function retval = seite(A, B, l, nameA, nameB) + global fn; + d = fliplr(B - A); + d(1, 2) = -d(1, 2); + # Zentrum + C = 0.5 * (A + B) + l * d; + # Radius: + r = hypot(C(1,1)-A(1,1), C(1,2)-A(1,2)) + # Winkel von + winkelvon = atan2(A(1,2)-C(1,2),A(1,1)-C(1,1)); + # Winkel bis + winkelbis = atan2(B(1,2)-C(1,2),B(1,1)-C(1,1)); + if (abs(winkelvon - winkelbis) > pi) + if (winkelbis < winkelvon) + winkelbis = winkelbis + 2 * pi + else + winkelvon = winkelvon + 2 * pi + end + end + # Kurve + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameA, nameB, + A(1,1), A(1,2), winkelvon * 180 / pi, winkelbis * 180 / pi, r); + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameB, nameA, + B(1,1), B(1,2), winkelbis * 180 / pi, winkelvon * 180 / pi, r); +endfunction + +seite(A, B, -1, "A", "B"); +seite(A, C, 1, "A", "C"); +seite(A, P, -1, "A", "P"); +seite(B, C, -2, "B", "C"); +seite(B, P, -1, "B", "P"); +seite(C, P, 2, "C", "P"); + +fclose(fn); diff --git a/buch/papers/nav/images/position/2k_earth_daymap.png b/buch/papers/nav/images/position/2k_earth_daymap.png Binary files differnew file mode 100644 index 0000000..4d55da8 --- /dev/null +++ b/buch/papers/nav/images/position/2k_earth_daymap.png diff --git a/buch/papers/nav/images/position/Makefile b/buch/papers/nav/images/position/Makefile new file mode 100644 index 0000000..eed2e56 --- /dev/null +++ b/buch/papers/nav/images/position/Makefile @@ -0,0 +1,69 @@ +# +# Makefile to build images +# +# (c) 2022 +# +all: position + +POSITION = \ + position1.pdf position1-small.pdf \ + position2.pdf position2-small.pdf \ + position3.pdf position3-small.pdf \ + position4.pdf position4-small.pdf \ + position5.pdf position5-small.pdf + +position: $(POSITION) + +POVRAYOPTIONS = -W1080 -H1080 +#POVRAYOPTIONS = -W480 -H480 + +position1.png: position1.pov common.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Oposition1.png position1.pov +position1.jpg: position1.png + convert position1.png -density 300 -units PixelsPerInch position1.jpg +position1.pdf: position1.tex common.tex position1.jpg + pdflatex position1.tex + +position2.png: position2.pov common.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Oposition2.png position2.pov +position2.jpg: position2.png + convert position2.png -density 300 -units PixelsPerInch position2.jpg +position2.pdf: position2.tex common.tex position2.jpg + pdflatex position2.tex + +position3.png: position3.pov common.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Oposition3.png position3.pov +position3.jpg: position3.png + convert position3.png -density 300 -units PixelsPerInch position3.jpg +position3.pdf: position3.tex common.tex position3.jpg + pdflatex position3.tex + +position4.png: position4.pov common.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Oposition4.png position4.pov +position4.jpg: position4.png + convert position4.png -density 300 -units PixelsPerInch position4.jpg +position4.pdf: position4.tex common.tex position4.jpg + pdflatex position4.tex + +position5.png: position5.pov common.inc ../macros.inc + povray +A0.1 $(POVRAYOPTIONS) -Oposition5.png position5.pov +position5.jpg: position5.png + convert position5.png -density 300 -units PixelsPerInch position5.jpg +position5.pdf: position5.tex common.tex position5.jpg + pdflatex position5.tex + +position1-small.pdf: position1-small.tex common.tex position1.jpg + pdflatex position1-small.tex +position2-small.pdf: position2-small.tex common.tex position2.jpg + pdflatex position2-small.tex +position3-small.pdf: position3-small.tex common.tex position3.jpg + pdflatex position3-small.tex +position4-small.pdf: position4-small.tex common.tex position4.jpg + pdflatex position4-small.tex +position5-small.pdf: position5-small.tex common.tex position5.jpg + pdflatex position5-small.tex + +test: test.pdf + +test.pdf: test.tex $(POSITION) + pdflatex test.tex diff --git a/buch/papers/nav/images/position/common-small.tex b/buch/papers/nav/images/position/common-small.tex new file mode 100644 index 0000000..9430608 --- /dev/null +++ b/buch/papers/nav/images/position/common-small.tex @@ -0,0 +1,32 @@ +% +% common.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\def\labelA{\node at (0.7,3.8) {$A$};} +\def\labelB{\node at (-3.4,-0.8) {$B$};} +\def\labelC{\node at (3.3,-2.1) {$C$};} +\def\labelP{\node at (-1.4,-3.5) {$P$};} + +\def\labelc{\node at (-1.9,2.1) {$c$};} +\def\labela{\node at (-0.2,-1.2) {$a$};} +\def\labelb{\node at (2.6,1.5) {$b$};} + +\def\labelhb{\node at (-2.6,-2.2) {$h_B$};} +\def\labelhc{\node at (1,-2.9) {$h_C$};} +\def\labell{\node at (-0.7,0.3) {$l$};} + +\def\labelalpha{\node at (0.6,2.85) {$\alpha$};} +\def\labelbeta{\node at (-2.5,-0.5) {$\beta$};} +\def\labelgamma{\node at (2.3,-1.2) {$\gamma$};} +\def\labelomega{\node at (0.85,3.3) {$\omega$};} + +\def\labelgammaone{\node at (2.1,-2.0) {$\gamma_1$};} +\def\labelgammatwo{\node at (2.3,-1.3) {$\gamma_2$};} +\def\labelbetaone{\node at (-2.4,-1.4) {$\beta_1$};} +\def\labelbetatwo{\node at (-2.5,-0.8) {$\beta_2$};} + +\def\labelomegalinks{\node at (0.25,3.25) {$\omega$};} +\def\labelomegarechts{\node at (0.85,3.1) {$\omega$};} + diff --git a/buch/papers/nav/images/position/common.inc b/buch/papers/nav/images/position/common.inc new file mode 100644 index 0000000..56e2836 --- /dev/null +++ b/buch/papers/nav/images/position/common.inc @@ -0,0 +1,39 @@ +// +// common.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "../macros.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.034; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + +light_source { + <30, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +//kugel(kugeldunkel) +erde(-100) +koordinatennetz(gitterfarbe, 9, 0.001) +achse(fein, White) diff --git a/buch/papers/nav/images/position/common.tex b/buch/papers/nav/images/position/common.tex new file mode 100644 index 0000000..9430608 --- /dev/null +++ b/buch/papers/nav/images/position/common.tex @@ -0,0 +1,32 @@ +% +% common.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\def\labelA{\node at (0.7,3.8) {$A$};} +\def\labelB{\node at (-3.4,-0.8) {$B$};} +\def\labelC{\node at (3.3,-2.1) {$C$};} +\def\labelP{\node at (-1.4,-3.5) {$P$};} + +\def\labelc{\node at (-1.9,2.1) {$c$};} +\def\labela{\node at (-0.2,-1.2) {$a$};} +\def\labelb{\node at (2.6,1.5) {$b$};} + +\def\labelhb{\node at (-2.6,-2.2) {$h_B$};} +\def\labelhc{\node at (1,-2.9) {$h_C$};} +\def\labell{\node at (-0.7,0.3) {$l$};} + +\def\labelalpha{\node at (0.6,2.85) {$\alpha$};} +\def\labelbeta{\node at (-2.5,-0.5) {$\beta$};} +\def\labelgamma{\node at (2.3,-1.2) {$\gamma$};} +\def\labelomega{\node at (0.85,3.3) {$\omega$};} + +\def\labelgammaone{\node at (2.1,-2.0) {$\gamma_1$};} +\def\labelgammatwo{\node at (2.3,-1.3) {$\gamma_2$};} +\def\labelbetaone{\node at (-2.4,-1.4) {$\beta_1$};} +\def\labelbetatwo{\node at (-2.5,-0.8) {$\beta_2$};} + +\def\labelomegalinks{\node at (0.25,3.25) {$\omega$};} +\def\labelomegarechts{\node at (0.85,3.1) {$\omega$};} + diff --git a/buch/papers/nav/images/position/position1-small.pdf b/buch/papers/nav/images/position/position1-small.pdf Binary files differnew file mode 100644 index 0000000..ba7755f --- /dev/null +++ b/buch/papers/nav/images/position/position1-small.pdf diff --git a/buch/papers/nav/images/position/position1-small.tex b/buch/papers/nav/images/position/position1-small.tex new file mode 100644 index 0000000..05fad44 --- /dev/null +++ b/buch/papers/nav/images/position/position1-small.tex @@ -0,0 +1,55 @@ +% +% position1-small.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common-small.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.625] + +% Povray Bild +\node at (0,0) {\includegraphics[width=5cm]{position1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelC +\labelP + +\labelc +\labela +\labelb +\labell + +\labelhb +\labelhc + +\labelalpha +\labelomega + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position1.pdf b/buch/papers/nav/images/position/position1.pdf Binary files differnew file mode 100644 index 0000000..fc4f760 --- /dev/null +++ b/buch/papers/nav/images/position/position1.pdf diff --git a/buch/papers/nav/images/position/position1.pov b/buch/papers/nav/images/position/position1.pov new file mode 100644 index 0000000..a79a9f1 --- /dev/null +++ b/buch/papers/nav/images/position/position1.pov @@ -0,0 +1,71 @@ +// +// position1.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +union { + seite(B, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(A, P, fett) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + + +union { + seite(A, B, fett) + seite(A, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, gross) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fett, klein) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/position/position1.tex b/buch/papers/nav/images/position/position1.tex new file mode 100644 index 0000000..d6c21c3 --- /dev/null +++ b/buch/papers/nav/images/position/position1.tex @@ -0,0 +1,55 @@ +% +% dreieck3d1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{position1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelC +\labelP + +\labelc +\labela +\labelb +\labell + +\labelhb +\labelhc + +\labelalpha +\labelomega + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position2-small.pdf b/buch/papers/nav/images/position/position2-small.pdf Binary files differnew file mode 100644 index 0000000..3333dd4 --- /dev/null +++ b/buch/papers/nav/images/position/position2-small.pdf diff --git a/buch/papers/nav/images/position/position2-small.tex b/buch/papers/nav/images/position/position2-small.tex new file mode 100644 index 0000000..e5c33cf --- /dev/null +++ b/buch/papers/nav/images/position/position2-small.tex @@ -0,0 +1,53 @@ +% +% position2-small.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common-small.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.625] + +% Povray Bild +\node at (0,0) {\includegraphics[width=5cm]{position2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelC + +\labelc +\labela +\labelb + +\begin{scope}[yshift=0.3cm,xshift=0.1cm] +\labelalpha +\end{scope} +\labelbeta +\labelgamma + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position2.pdf b/buch/papers/nav/images/position/position2.pdf Binary files differnew file mode 100644 index 0000000..dbd2ea9 --- /dev/null +++ b/buch/papers/nav/images/position/position2.pdf diff --git a/buch/papers/nav/images/position/position2.pov b/buch/papers/nav/images/position/position2.pov new file mode 100644 index 0000000..2abcd94 --- /dev/null +++ b/buch/papers/nav/images/position/position2.pov @@ -0,0 +1,70 @@ +// +// position3.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +dreieck(A, B, C, kugelfarbe) + +union { + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(A, B, fett) + seite(A, C, fett) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(B, C, fett) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, gross) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +union { + winkel(B, C, A, fein, gross) + winkel(C, A, B, fein, gross) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + + diff --git a/buch/papers/nav/images/position/position2.tex b/buch/papers/nav/images/position/position2.tex new file mode 100644 index 0000000..339592c --- /dev/null +++ b/buch/papers/nav/images/position/position2.tex @@ -0,0 +1,53 @@ +% +% position2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{position2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelC + +\labelc +\labela +\labelb + +\begin{scope}[yshift=0.3cm,xshift=0.1cm] +\labelalpha +\end{scope} +\labelbeta +\labelgamma + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position3-small.pdf b/buch/papers/nav/images/position/position3-small.pdf Binary files differnew file mode 100644 index 0000000..fae0b85 --- /dev/null +++ b/buch/papers/nav/images/position/position3-small.pdf diff --git a/buch/papers/nav/images/position/position3-small.tex b/buch/papers/nav/images/position/position3-small.tex new file mode 100644 index 0000000..4f7b0e9 --- /dev/null +++ b/buch/papers/nav/images/position/position3-small.tex @@ -0,0 +1,51 @@ +% +% position3-small.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common-small.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.625] + +% Povray Bild +\node at (0,0) {\includegraphics[width=5cm]{position3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelB +\labelC +\labelP + +\labela + +\labelhb +\labelhc + +\labelbetaone +\labelgammaone + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position3.pdf b/buch/papers/nav/images/position/position3.pdf Binary files differnew file mode 100644 index 0000000..2c940d2 --- /dev/null +++ b/buch/papers/nav/images/position/position3.pdf diff --git a/buch/papers/nav/images/position/position3.pov b/buch/papers/nav/images/position/position3.pov new file mode 100644 index 0000000..f6823eb --- /dev/null +++ b/buch/papers/nav/images/position/position3.pov @@ -0,0 +1,48 @@ +// +// position3.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +dreieck(B, P, C, kugelfarbe) + +union { + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +union { + winkel(B, P, C, fein, gross) + winkel(C, B, P, fein, gross) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/position/position3.tex b/buch/papers/nav/images/position/position3.tex new file mode 100644 index 0000000..d5480da --- /dev/null +++ b/buch/papers/nav/images/position/position3.tex @@ -0,0 +1,51 @@ +% +% dreieck3d1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{position3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelB +\labelC +\labelP + +\labela + +\labelhb +\labelhc + +\labelbetaone +\labelgammaone + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position4-small.pdf b/buch/papers/nav/images/position/position4-small.pdf Binary files differnew file mode 100644 index 0000000..ac80c46 --- /dev/null +++ b/buch/papers/nav/images/position/position4-small.pdf diff --git a/buch/papers/nav/images/position/position4-small.tex b/buch/papers/nav/images/position/position4-small.tex new file mode 100644 index 0000000..e06523b --- /dev/null +++ b/buch/papers/nav/images/position/position4-small.tex @@ -0,0 +1,50 @@ +% +% position4-small.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common-small.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.625] + +% Povray Bild +\node at (0,0) {\includegraphics[width=5cm]{position4.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelP + +\labelc +\labell +\labelhb + +\labelomegalinks +\labelbetatwo + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position4.pdf b/buch/papers/nav/images/position/position4.pdf Binary files differnew file mode 100644 index 0000000..8eeeaac --- /dev/null +++ b/buch/papers/nav/images/position/position4.pdf diff --git a/buch/papers/nav/images/position/position4.pov b/buch/papers/nav/images/position/position4.pov new file mode 100644 index 0000000..80628f9 --- /dev/null +++ b/buch/papers/nav/images/position/position4.pov @@ -0,0 +1,69 @@ +// +// position4.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +dreieck(A, B, P, kugelfarbe) + +union { + punkt(A, fett) + punkt(B, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(A, P, fett) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + + +union { + seite(A, B, fett) + seite(B, P, fett) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, P, A, fein, gross) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, P, fein, gross) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/position/position4.tex b/buch/papers/nav/images/position/position4.tex new file mode 100644 index 0000000..27c1757 --- /dev/null +++ b/buch/papers/nav/images/position/position4.tex @@ -0,0 +1,50 @@ +% +% position4.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{position4.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelB +\labelP + +\labelc +\labell +\labelhb + +\labelomegalinks +\labelbetatwo + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position5-small.pdf b/buch/papers/nav/images/position/position5-small.pdf Binary files differnew file mode 100644 index 0000000..afe120e --- /dev/null +++ b/buch/papers/nav/images/position/position5-small.pdf diff --git a/buch/papers/nav/images/position/position5-small.tex b/buch/papers/nav/images/position/position5-small.tex new file mode 100644 index 0000000..0a0e229 --- /dev/null +++ b/buch/papers/nav/images/position/position5-small.tex @@ -0,0 +1,50 @@ +% +% position5-small.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common-small.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick,scale=0.625] + +% Povray Bild +\node at (0,0) {\includegraphics[width=5cm]{position5.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelC +\labelP + +\labelb +\labell +\labelhc + +\labelomegarechts +\labelgammatwo + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/position5.pdf b/buch/papers/nav/images/position/position5.pdf Binary files differnew file mode 100644 index 0000000..05a64cb --- /dev/null +++ b/buch/papers/nav/images/position/position5.pdf diff --git a/buch/papers/nav/images/position/position5.pov b/buch/papers/nav/images/position/position5.pov new file mode 100644 index 0000000..7ed33c5 --- /dev/null +++ b/buch/papers/nav/images/position/position5.pov @@ -0,0 +1,69 @@ +// +// position5.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "common.inc" + +dreieck(A, P, C, kugelfarbe) + +union { + punkt(A, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +union { + seite(A, P, fett) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + + +union { + seite(A, C, fett) + seite(C, P, fett) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, P, A, fein, gross) + pigment { + color bekannt + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, C, P, fein, gross) + pigment { + color unbekannt + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/position/position5.tex b/buch/papers/nav/images/position/position5.tex new file mode 100644 index 0000000..b234429 --- /dev/null +++ b/buch/papers/nav/images/position/position5.tex @@ -0,0 +1,50 @@ +% +% position5.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\input{common.tex} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{position5.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\labelA +\labelC +\labelP + +\labelb +\labell +\labelhc + +\labelomegarechts +\labelgammatwo + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/position/test.tex b/buch/papers/nav/images/position/test.tex new file mode 100644 index 0000000..3247ed1 --- /dev/null +++ b/buch/papers/nav/images/position/test.tex @@ -0,0 +1,135 @@ +% +% test.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[12pt]{article} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{etex} +\usepackage[ngerman]{babel} +\usepackage{times} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{graphicx} +\usepackage{wrapfig} +\begin{document} + +\begin{wrapfigure}{R}{5.6cm} +\includegraphics{position1-small.pdf} +\end{wrapfigure} +Lorem ipsum dolor sit amet, consectetuer adipiscing elit. +Aenean +commodo ligula eget dolor. +Aenean massa. +Cum sociis natoque penatibus +et magnis dis parturient montes, nascetur ridiculus mus. +Donec quam +felis, ultricies nec, pellentesque eu, pretium quis, sem. +Nulla +consequat massa quis enim. +Donec pede justo, fringilla vel, aliquet +nec, vulputate eget, arcu. +In enim justo, rhoncus ut, imperdiet a, +venenatis vitae, justo. +Nullam dictum felis eu pede mollis pretium. +Integer tincidunt. +Cras dapibus. +Vivamus elementum semper nisi. +Aenean vulputate eleifend tellus. +Aenean leo ligula, porttitor eu, +consequat vitae, eleifend ac, enim. +Aliquam lorem ante, dapibus in, +viverra quis, feugiat a, tellus. + +\begin{wrapfigure}{R}{5.2cm} +\includegraphics{position2-small.pdf} +\end{wrapfigure} +Maecenas tempus, tellus eget condimentum rhoncus, sem quam semper +libero, sit amet adipiscing sem neque sed ipsum. Nam quam nunc, +blandit vel, luctus pulvinar, hendrerit id, lorem. Maecenas nec +odio et ante tincidunt tempus. Donec vitae sapien ut libero venenatis +faucibus. Nullam quis ante. Etiam sit amet orci eget eros faucibus +tincidunt. Duis leo. Sed fringilla mauris sit amet nibh. Donec +sodales sagittis magna. Sed consequat, leo eget bibendum sodales, +augue velit cursus nunc, quis gravida magna mi a libero. Fusce +vulputate eleifend sapien. Vestibulum purus quam, scelerisque ut, +mollis sed, nonummy id, metus. Nullam accumsan lorem in dui. Cras +ultricies mi eu turpis hendrerit fringilla. Vestibulum ante ipsum +primis in faucibus orci luctus et ultrices posuere cubilia Curae; + +\pagebreak + +\begin{wrapfigure}{R}{5.2cm} +\includegraphics{position3-small.pdf} +\end{wrapfigure} +Integer ante arcu, accumsan a, consectetuer eget, posuere ut, mauris. +Praesent adipiscing. Phasellus ullamcorper ipsum rutrum nunc. Nunc +nonummy metus. Vestibulum volutpat pretium libero. Cras id dui. +Aenean ut eros et nisl sagittis vestibulum. Nullam nulla eros, +ultricies sit amet, nonummy id, imperdiet feugiat, pede. Sed lectus. +Donec mollis hendrerit risus. Phasellus nec sem in justo pellentesque +facilisis. Etiam imperdiet imperdiet orci. Nunc nec neque. Phasellus +leo dolor, tempus non, auctor et, hendrerit quis, nisi. Curabitur +ligula sapien, tincidunt non, euismod vitae, posuere imperdiet, +leo. Maecenas malesuada. Praesent congue erat at massa. Sed cursus +turpis vitae tortor. Donec posuere vulputate arcu. Phasellus accumsan +cursus velit. Vestibulum ante ipsum primis in faucibus orci luctus +et ultrices posuere cubilia Curae; Sed aliquam, nisi quis porttitor +congue, elit erat euismod orci, ac placerat dolor lectus quis orci. +Phasellus consectetuer vestibulum elit. + +\begin{wrapfigure}{R}{5.2cm} +\includegraphics{position4-small.pdf} +\end{wrapfigure} +Aenean tellus metus, bibendum sed, posuere ac, mattis non, nunc. +Vestibulum fringilla pede sit amet augue. In turpis. Pellentesque +posuere. Praesent turpis. Aenean posuere, tortor sed cursus feugiat, +nunc augue blandit nunc, eu sollicitudin urna dolor sagittis lacus. +Donec elit libero, sodales nec, volutpat a, suscipit non, turpis. +Nullam sagittis. Suspendisse pulvinar, augue ac venenatis condimentum, +sem libero volutpat nibh, nec pellentesque velit pede quis nunc. +Vestibulum ante ipsum primis in faucibus orci luctus et ultrices +posuere cubilia Curae; Fusce id purus. Ut varius tincidunt libero. +Phasellus dolor. Maecenas vestibulum mollis diam. Pellentesque ut +neque. Pellentesque habitant morbi tristique senectus et netus et +malesuada fames ac turpis egestas. In dui magna, posuere eget, +vestibulum et, tempor auctor, justo. In ac felis quis tortor malesuada +pretium. Pellentesque auctor neque nec urna. + +\pagebreak + +\begin{wrapfigure}{R}{5.2cm} +\includegraphics{position5-small.pdf} +\end{wrapfigure} +Proin sapien ipsum, porta a, auctor quis, euismod ut, mi. Aenean +viverra rhoncus pede. Pellentesque habitant morbi tristique senectus +et netus et malesuada fames ac turpis egestas. Ut non enim eleifend +felis pretium feugiat. Vivamus quis mi. Phasellus a est. Phasellus +magna. In hac habitasse platea dictumst. Curabitur at lacus ac velit +ornare lobortis. Curabitur a felis in nunc fringilla tristique. +Morbi mattis ullamcorper velit. Phasellus gravida semper nisi. +Nullam vel sem. Pellentesque libero tortor, tincidunt et, tincidunt +eget, semper nec, quam. Sed hendrerit. Morbi ac felis. Nunc egestas, +augue at pellentesque laoreet, felis eros vehicula leo, at malesuada +velit leo quis pede. Donec interdum, metus et hendrerit aliquet, +dolor diam sagittis ligula, eget egestas libero turpis vel mi. Nunc +nulla. Fusce risus nisl, viverra et, tempor et, pretium in, sapien. +Donec venenatis vulputate lorem. Morbi nec metus. Phasellus blandit +leo ut odio. Maecenas ullamcorper, dui et placerat feugiat, eros +pede varius nisi, condimentum viverra felis nunc et lorem. Sed magna +purus, fermentum eu, tincidunt eu, varius ut, felis. In auctor +lobortis lacus. Quisque libero metus, condimentum nec, tempor a, +commodo mollis, magna. Vestibulum ullamcorper mauris at ligula. +Fusce fermentum. Nullam cursus lacinia erat. Praesent blandit laoreet +nibh. Fusce convallis metus id felis luctus adipiscing. Pellentesque +egestas, neque sit amet convallis pulvinar, justo nulla eleifend +augue, ac auctor orci leo non est. Quisque id mi. Ut tincidunt +tincidunt erat. Etiam feugiat lorem non metus. Vestibulum dapibus +nunc ac augue. Curabitur vestibulum aliquam leo. Praesent egestas +neque eu enim. In hac habitasse platea dictumst. Fusce a quam. Etiam +ut purus mattis mauris + +\end{document} diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index e11e2c0..f993559 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -3,34 +3,21 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:nav}} -\lhead{Thema} +\chapter{Sphärische Navigation\label{chapter:nav}} +\lhead{Sphärische Navigation} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Enez Erdem und Marc Kühne} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{papers/nav/teil0.tex} -\input{papers/nav/teil1.tex} -\input{papers/nav/teil2.tex} -\input{papers/nav/teil3.tex} + +\input{papers/nav/einleitung.tex} +\input{papers/nav/flatearth.tex} +\input{papers/nav/sincos.tex} +\input{papers/nav/trigo.tex} +\input{papers/nav/nautischesdreieck.tex} +\input{papers/nav/bsp2.tex} + \printbibliography[heading=subbibliography] \end{refsection} + diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex new file mode 100644 index 0000000..32d1b8b --- /dev/null +++ b/buch/papers/nav/nautischesdreieck.tex @@ -0,0 +1,172 @@ +\section{Das Nautische Dreieck} +\rhead{Das nautische Dreieck} +\subsection{Definition des Nautischen Dreiecks} +Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient. +Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. +Als Gestirne kommen Sterne und Planeten in Frage, zu welchen in diversen Jahrbüchern die für die Navigation nötigen Daten publiziert sind. +Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. +Das nautische Dreieck hat die Ecken Zenit, Gestirn und Himmelspol, wie man in der Abbildung \ref{naut} sehen kann. + +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. + +\subsection{Das Bilddreieck} +\begin{figure} + \begin{center} + \includegraphics[width=8cm]{papers/nav/bilder/kugel3.png} + \caption[Nautisches Dreieck]{Nautisches Dreieck} + \label{naut} + \end{center} +\end{figure} + Man kann das nautische Dreieck auf die Erdkugel projizieren. +Dieses Dreieck nennt man dann Bilddreieck. +Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. +Die Projektion des nautischen Dreiecks auf die Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. + +\section{Standortbestimmung ohne elektronische Hilfsmittel} +\label{sta} +Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. +Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. +Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt \ref{ephe} erklärt. +\begin{figure} + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} + \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} + \label{d1} + \end{center} +\end{figure} + + + + +\subsection{Ecke $P$ und $A$} +Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. +Der Vorteil an der Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so einfach. + +\subsection{Ecke $B$ und $C$ - Bildpunkt von $X$ und $Y$} +Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. +Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. +Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mond oder die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn. + +Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung \ref{d1}. +\subsection{Ephemeriden} +\label{ephe} +Zu all diesen Gestirnen gibt es Ephemeridentabellen. +Diese Tabellen enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Zeit. + +\begin{figure} + \begin{center} + \includegraphics[width=\textwidth]{papers/nav/bilder/ephe.png} + \caption[Nautical Almanac Mai 2002]{Nautical Almanac Mai 2002} + \end{center} +\end{figure} + +\subsubsection{Deklination} +Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns. + +\subsubsection{Rektaszension und Sternzeit} +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt, welcher der Nullpunkt auf dem Himmelsäquator ist, steht und geht vom Koordinatensystem der Himmelskugel aus. + +Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. +Die Lösung ist die Sternzeit $\theta$. +Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen. +Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet und $\theta=0$ ist. +Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. +Für die Sternzeit von Greenwich $\theta$ braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht nachschlagen lässt. +Im Anschluss berechnet man die Sternzeit von Greenwich + +\[\theta = 6^h 41^m 50^s.54841 + 8640184^s.812866 \cdot T + 0^s.093104 \cdot T^2 - 0^s.0000062 \cdot T^3.\] + +Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ bestimmen, wobei $\alpha$ die Rektaszension und $\theta$ die Sternzeit von Greenwich ist. +Dies gilt analog auch für das zweite Gestirn. +\subsubsection{Sextant} +Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann. Es wird vor allem der Winkelabstand vom Horizont zum Gestirn gemessen. +Man benutzt ihn vor allem für die astronomische Navigation auf See. + +\begin{figure} + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/sextant.jpg} + \caption[Sextant]{Sextant} + \end{center} +\end{figure} +\subsection{Bestimmung des eigenen Standortes $P$} \label{p} +Wir nehmen die Abbildung \ref{d2} zur Hilfe. +Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. +Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. +Auf diese Dreiecke können wir die einfachen Sätze der sphärischen Trigonometrie anwenden und benötigen lediglich ein Ephemeride zu den Gestirnen und einen Sextant. + +\begin{figure} + \begin{center} + \includegraphics[width=8cm]{papers/nav/bilder/dreieck.pdf} + \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} + \label{d2} + \end{center} +\end{figure} + +\subsubsection{Dreieck $ABC$} + +\begin{center} + \begin{tabular}{ l l l } + Ecke && Name \\ + \hline + $A$ && Nordpol \\ + $B$ && Bildpunkt des Gestirns $X$ \\ + $C$&& Bildpunkt des Gestirns $Y$ + \end{tabular} +\end{center} + +Mit unserem erlangten Wissen können wir nun alle Seiten des Dreiecks $ABC$ berechnen. +Dazu sind die folgenden vorbereiteten Berechnungen nötigt: + +\begin{enumerate} + \item Die Seite vom Nordpol zum Bildpunkt $X$ sei $c$, dann ist $c = \frac{\pi}{2} - \delta_1$. + \item Die Seite vom Nordpol zum Bildpunkt $Y$ sei $b$, dann ist $b = \frac{\pi}{2} - \delta_2$. + \item Der Innenwinkel bei der Ecke, wo der Nordpol ist sei $\alpha$, dann ist $ \alpha = |\lambda_1 - \lambda_2|$. +\end{enumerate} + +mit +\begin{center} + \begin{tabular}{ l l l } + Ecke && Name \\ + \hline + $\delta_1$ && Deklination vom Bildpunkt $X$ \\ + $\delta_2$ && Deklination vom Bildpunk $Y$ \\ + $\lambda_1 $&& Längengrad vom Bildpunkt $X$\\ + $\lambda_2$ && Längengrad vom Bildpunkt $Y$ + \end{tabular} +\end{center} + +Nun haben wir die beiden Seiten $c$ und $b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. +Mithilfe des Seiten-Kosinussatzes +$\cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ +können wir nun die dritte Seitenlänge bestimmen. +Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. + +Jetzt fehlen noch die beiden anderen Innenwinkel $\beta$ und\ $\gamma$. +Diese bestimmen wir mithilfe des Kosinussatzes: \[\beta=\cos^{-1} \bigg[\frac{\cos(b)-\cos(a) \cdot \cos(c)}{\sin(a) \cdot \sin(c)}\bigg]\] und \[\gamma = \cos^{-1} \bigg[\frac{\cos(c)-\cos(b) \cdot \cos(a)}{\sin(a) \cdot \sin(b)}\bigg]\]. + +Schlussendlich haben wir die Seiten $a$, $b$ und $c$, die Ecken $A$,$B$ und $C$ und die Winkel $\alpha$, $\beta$ und $\gamma$ bestimmt und somit das ganze Kugeldreieck $ABC$ berechnet. + +\subsubsection{Dreieck $BPC$} +Wir bilden nun ein zweites Dreieck, welches die Ecken $B$ und $C$ des ersten Dreiecks besitzt. +Die dritte Ecke ist der eigene Standort $P$. +Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. + +Die Seite von $P$ zu $B$ sei $pb$ und die Seite von $P$ zu $C$ sei $pc$. +Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ +mit $h_B=$ Höhe von Gestirn in $B$ und $h_C=$ Höhe von Gestirn in $C$ mit Sextant gemessen. + +Zum Schluss müssen wir noch den Winkel $\beta_1$ mithilfe des Seiten-Kosinussatzes \[\cos(pb)=\cos(pc)\cdot\cos(a)+\sin(pc)\cdot\sin(a)\cdot\cos(\beta_1)\] mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen. +\subsubsection{Dreieck $ABP$} +Nun muss man eine Verbindungslinie des Standorts zwischen $P$ und $A$ ziehen. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c$ und $pb$ und des Seiten-Kosinussatzes berechnen. +Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta_1$. +Somit ist \[\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)\] +und +\[ +\delta =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)]. +\] + +Für die geographische Länge $\lambda$ des eigenen Standortes nutzt man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet. +Mithilfe des Kosinussatzes können wir \[\omega = \cos^{-1} \bigg[\frac{\cos(pb)-\cos(c) \cdot \cos(l)}{\sin(c) \cdot \sin(l)}\bigg]\] berechnen und bekommen schlussendlich die geographische Länge +\[\lambda=\lambda_1 - \omega,\] +wobei $\lambda_1$ die Länge des Bildpunktes $X$ von $C$ ist. diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 9faa48d..bedaccd 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,3 +8,5 @@ % following example %\usepackage{packagename} +\usepackage{amsmath} +\usepackage{cancel} diff --git a/buch/papers/nav/references.bib b/buch/papers/nav/references.bib index 236323b..c67aaac 100644 --- a/buch/papers/nav/references.bib +++ b/buch/papers/nav/references.bib @@ -32,4 +32,10 @@ pages = {607--627}, url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@online{nav:winkel, + editor={Unbekannt}, + title = {Sphärische Trigonometrie}, + year={2022}, + url = {https://de.wikipedia.org/wiki/Sphärische_Trigonometrie} +} diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex new file mode 100644 index 0000000..b64d100 --- /dev/null +++ b/buch/papers/nav/sincos.tex @@ -0,0 +1,24 @@ + + + +\section{Sphärische Navigation und Winkelfunktionen} +\rhead{Sphärische Navigation und Winkelfunktionen} +Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren mit Problemen der sphärischen Trigonometrie beschäftigt haben, um den Lauf von Gestirnen zu berechnen. +Jedoch konnten sie dieses Problem nicht lösen. +Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 BCE dachten die Griechen über Kugelgeometrie nach, sie wurde damit zu einer Hilfswissenschaft der Astronomen. + +Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom namens Hipparchos. +Dieser entwickelte unter anderem die Chordentafeln, welche die Chordfunktionen, auch Chord genannt, beinhalten. +Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie. +In dieser Zeit wurden auch die ersten Sternenkarten angefertigt. Damals kannte man die Sinusfunktionen noch nicht. + +Die Definition der trigonometrischen Funktionen aus Griechenland ermöglicht nur, rechtwinklige Dreiecke zu berechnen. +Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. +Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. +Die Beziehung zwischen Seiten und Winkeln sind komplizierter und als Sinus- und Kosinussätze bekannt. +Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. +Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. +Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. + +Durch weitere mathematische Entwicklungen wie den Logarithmus wurden im Laufe des nächsten Jahrhunderts viele neue Methoden und kartographische Anwendungen der Kugelgeometrie entdeckt. +Im 19. und 20. Jahrhundert wurden weitere nicht-euklidische Geometrien entwickelt und die sphärische Trigonometrie fand auch ihre Anwendung in der Relativitätstheorie.
\ No newline at end of file diff --git a/buch/papers/nav/teil0.tex b/buch/papers/nav/teil0.tex deleted file mode 100644 index f3323a9..0000000 --- a/buch/papers/nav/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{nav:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{nav:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/nav/teil1.tex b/buch/papers/nav/teil1.tex deleted file mode 100644 index 996202f..0000000 --- a/buch/papers/nav/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{nav:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{nav:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{nav:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{nav:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/nav/teil2.tex b/buch/papers/nav/teil2.tex deleted file mode 100644 index 5a52e03..0000000 --- a/buch/papers/nav/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{nav:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/nav/teil3.tex b/buch/papers/nav/teil3.tex deleted file mode 100644 index 2b5d2d5..0000000 --- a/buch/papers/nav/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{nav:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex new file mode 100644 index 0000000..483b612 --- /dev/null +++ b/buch/papers/nav/trigo.tex @@ -0,0 +1,140 @@ + +\section{Sphärische Trigonometrie} +\rhead{Sphärische Trigonometrie} + +\subsection{Das Kugeldreieck} +Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie Grosskreisebene und Grosskreisbögen verstehen. +Ein Grosskreis ist ein grösstmöglicher Kreis auf einer Kugeloberfläche. +Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. +Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. +Grosskreisbögen sind die kürzesten Verbindungslinien zwischen zwei Punkten auf der Kugel. + +Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. +Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ die Erdmitte ist. + +Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. +Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. + +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden werden, so entsteht ein Kugeldreieck $ABC$. +Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $3\pi$ aber grösser als 0 ist. +$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung \ref{kugel}). + +\begin{figure} + \begin{center} + \includegraphics[width=3.5cm]{papers/nav/bilder/kugel1.png} + \caption[Das Kugeldreieck]{Das Kugeldreieck} + \label{kugel} + \end{center} + +\end{figure} + +\subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck} +In der sphärischen Trigonometrie gibt es eine Symmetrie zwischen Seiten und Winkeln, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. + +Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. +Ein rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss, wie man in der Abbildung \ref{recht} sehen kann. + +\begin{figure} + + \begin{center} + \includegraphics[width=5cm]{papers/nav/bilder/recht.jpg} + \caption[Rechtseitiges und rechtwinkliges Kugeldreieck]{Rechtseitiges und rechtwinkliges Kugeldreieck} + \label{recht} + \end{center} +\end{figure} + +\subsection{Winkelsumme und Flächeninhalt} +\label{trigo} +%\begin{figure} ----- Brauche das Bild eigentlich nicht! + +% \begin{center} +% \includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} +% \caption[Winkelangabe im Kugeldreieck]{Winkelangabe im Kugeldreieck} +% \end{center} +%\end{figure} + + +Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. +Für die Summe der Innenwinkel gilt +\begin{align} + \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \quad \text{und} \quad \alpha+\beta+\gamma > \pi, \nonumber +\end{align} +wobei $F$ der Flächeninhalt des Kugeldreiecks ist. +\subsubsection{Sphärischer Exzess} +Der sphärische Exzess +\begin{align} + \epsilon = \alpha+\beta+\gamma - \pi \nonumber +\end{align} +beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. + +\subsubsection{Flächeninnhalt} +Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt +\[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon = 2 \cdot r^2 \cdot \epsilon.\] + +In diesem Kapitel sind keine Begründungen für die erhaltenen Resultate im Abschnitt \ref{trigo} zu erwarten und können in der Referenz \cite{nav:winkel} nachgeschlagen werden. +\subsection{Seiten und Winkelberechnung} +Es gibt in der sphärischen Trigonometrie eigentlich gar keinen Satz des Pythagoras, wie man ihn aus der zweidimensionalen Geometrie kennt. +Es gibt aber einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. Dieser Satz gilt jedoch nicht für das rechtseitige Kugeldreieck. +Die Approximation im nächsten Abschnitt wird erklären, warum man dies als eine Form des Satzes des Pythagoras sehen kann. +Es gilt nämlich: +\begin{align} + \cos c = \cos a \cdot \cos b \quad \text{wenn} \nonumber & + \quad \alpha = \frac{\pi}{2}. \nonumber +\end{align} + +\subsubsection{Approximation von kleinen Dreiecken} +Die Sätze in der ebenen Trigonometrie sind eigentlich Approximationen der sphärischen Trigonometrie. +So ist der Sinussatz in der Ebene nur eine Annäherung des sphärischen Sinussatzes. Das Gleiche gilt für den Kosinussatz und dem Satz des Pythagoras. +So kann mit dem Taylorpolynom 2. Grades den Sinus und den Kosinus vom Sphärischen in die Ebene approximieren: +\begin{align} + \sin(a) &\approx a \nonumber \intertext{und} + \cos(a)&\approx 1-\frac{a^2}{2}.\nonumber +\end{align} +Es gibt ebenfalls folgende Approximierung der Seiten von der Sphäre in die Ebene: +\begin{align} + a &\approx \sin(a) \nonumber \intertext{und} + \frac{a^2}{2} &\approx 1-\cos(a). \nonumber +\end{align} +Die Korrespondenzen zwischen der ebenen und sphärischen Trigonometrie werden in den kommenden Abschnitten erläutert. + +\subsubsection{Sphärischer Satz des Pythagoras} +Die Korrespondenz \[ a^2 \approx 1- \cos(a)\] liefert unter anderem einen entsprechenden Satz des Pythagoras, nämlich + +\begin{align*} + \cos(a)\cdot \cos(b) &= \cos(c), \\ + \bigg[1-\frac{a^2}{2}\bigg] \cdot \bigg[1-\frac{b^2}{2}\bigg] &= 1-\frac{c^2}{2}. + \intertext{Höhere Potenzen vernachlässigen:} + \xcancel{1}- \frac{a^2}{2} - \frac{b^2}{2} + \xcancel{\frac{a^2b^2}{4}}&= \xcancel{1}- \frac{c^2}{2} \\ + -a^2-b^2 &=-c^2\\ + a^2+b^2&=c^2. +\end{align*} +Dies ist der wohlbekannte ebene Satz des Pythagoras. + +\subsubsection{Sphärischer Sinussatz} +Den sphärischen Sinussatz +\begin{align} + \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber +\end{align} +kann man ebenfalls mit der Korrespondenz \[a \approx \sin(a) \] zum entsprechenden ebenen Sinussatz \[\frac{a}{\sin (\alpha)} =\frac{b}{\sin (\beta)} = \frac{c}{\sin (\gamma)}\] approximieren. + + +\subsubsection{Sphärische Kosinussätze} +In der sphärischen Trigonometrie gibt es den Seitenkosinussatz +\begin{align} + \cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber +\end{align} %Seitenkosinussatz +und den Winkelkosinussatz +\begin{align} + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c, \nonumber +\end{align} der nur in der sphärischen Trigonometrie vorhanden ist. + +Analog gibt es auch beim Seitenkosinussatz eine Korrespondenz zu \[ a^2 \leftrightarrow 1-\cos(a),\] die den ebenen Kosinussatz herleiten lässt, nämlich +\begin{align} + \cos(a)&= \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha) \\ + 1-\frac{a^2}{2} &= \bigg[1-\frac{b^2}{2}\bigg]\bigg[1-\frac{c^2}{2}\bigg]+bc\cdot\cos(\alpha). \intertext{Höhere Potenzen vernachlässigen:} + \xcancel{1}-\frac{a^2}{2} &= \xcancel{1}-\frac{b^2}{2}-\frac{c^2}{2} \xcancel{+\frac{b^2c^2}{4}}+bc \cdot \cos(\alpha)\\ + a^2&=b^2+c^2-2bc \cdot \cos(\alpha). +\end{align} + + +
\ No newline at end of file diff --git a/buch/papers/parzyl/img/koordinaten.png b/buch/papers/parzyl/img/koordinaten.png Binary files differnew file mode 100644 index 0000000..3ee582d --- /dev/null +++ b/buch/papers/parzyl/img/koordinaten.png diff --git a/buch/papers/parzyl/main.tex b/buch/papers/parzyl/main.tex index ff21c9f..528a2e2 100644 --- a/buch/papers/parzyl/main.tex +++ b/buch/papers/parzyl/main.tex @@ -8,29 +8,11 @@ \begin{refsection} \chapterauthor{Thierry Schwaller, Alain Keller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} + \input{papers/parzyl/teil0.tex} \input{papers/parzyl/teil1.tex} \input{papers/parzyl/teil2.tex} -\input{papers/parzyl/teil3.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex index 09b4024..4b251db 100644 --- a/buch/papers/parzyl/teil0.tex +++ b/buch/papers/parzyl/teil0.tex @@ -3,20 +3,239 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{parzyl:section:teil0}} +\section{Einleitung\label{parzyl:section:teil0}} \rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{parzyl:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +Die Laplace-Gleichung ist eine wichtige Gleichung in der Physik. +Mit ihr lässt sich zum Beispiel das elektrische Feld in einem ladungsfreien Raum bestimmen. +In diesem Kapitel wird die Lösung der Laplace-Gleichung im +parabolischen Zylinderkoordinatensystem genauer untersucht. +\subsection{Laplace Gleichung} +Die partielle Differentialgleichung +\begin{equation} + \Delta f = 0 +\end{equation} +ist als Laplace-Gleichung bekannt. +Sie ist eine spezielle Form der Poisson-Gleichung +\begin{equation} + \Delta f = g +\end{equation} +mit g als beliebige Funktion. +In der Physik hat die Laplace-Gleichung in verschieden Gebieten +verwendet, zum Beispiel im Elektromagnetismus. +Das Gaussche Gesetz in den Maxwellgleichungen +\begin{equation} + \nabla \cdot E = \frac{\varrho}{\epsilon_0} +\label{parzyl:eq:max1} +\end{equation} +besagt das die Divergenz eines Elektrischen Feldes an einem +Punkt gleich der Ladung an diesem Punkt ist. +Das elektrische Feld ist hierbei der Gradient des elektrischen +Potentials +\begin{equation} + \nabla \phi = E. +\end{equation} +Eingesetzt in \eqref{parzyl:eq:max1} resultiert +\begin{equation} + \nabla \cdot \nabla \phi = \Delta \phi = \frac{\varrho}{\epsilon_0}, +\end{equation} +was eine Possion-Gleichung ist. +An Ladungsfreien Stellen, ist der rechte Teil der Gleichung $0$. +\subsection{Parabolische Zylinderkoordinaten +\label{parzyl:subsection:finibus}} +Im parabolischen Zylinderkoordinatensystem bilden parabolische Zylinder die Koordinatenflächen. +Die Koordinate $(\sigma, \tau, z)$ sind in kartesischen Koordinaten ausgedrückt mit +\begin{align} + x & = \sigma \tau \\ + \label{parzyl:coordRelationsa} + y & = \frac{1}{2}\left(\tau^2 - \sigma^2\right) \\ + z & = z. + \label{parzyl:coordRelationse} +\end{align} +Wird $\tau$ oder $\sigma$ konstant gesetzt resultieren die Parabeln +\begin{equation} + y = \frac{1}{2} \left( \frac{x^2}{\sigma^2} - \sigma^2 \right) +\end{equation} +und +\begin{equation} + y = \frac{1}{2} \left( -\frac{x^2}{\tau^2} + \tau^2 \right). +\end{equation} + +\begin{figure} + \centering + \includegraphics[scale=0.4]{papers/parzyl/img/koordinaten.png} + \caption{Das parabolische Koordinatensystem. Die roten Parabeln haben ein + konstantes $\sigma$ und die grünen ein konstantes $\tau$.} + \label{parzyl:fig:cordinates} +\end{figure} + +Abbildung \ref{parzyl:fig:cordinates} zeigt das Parabolische Koordinatensystem. +Das parabolische Zylinderkoordinatensystem entsteht wenn die Parabeln aus der +Ebene gezogen werden. + +Um in diesem Koordinatensystem integrieren und differenzieren zu +können braucht es die Skalierungsfaktoren $h_{\tau}$, $h_{\sigma}$ und $h_{z}$. + +\dots + +Wird eine infinitessimal kleine Distanz $ds$ zwischen zwei Punkten betrachtet +kann dies im kartesischen Koordinatensystem mit +\begin{equation} + \left(ds\right)^2 = \left(dx\right)^2 + \left(dy\right)^2 + + \left(dz\right)^2 + \label{parzyl:eq:ds} +\end{equation} +ausgedrückt werden. +Das Skalierungsfaktoren werden so bestimmt, dass +\begin{equation} + \left(ds\right)^2 = \left(h_{\sigma}d\sigma\right)^2 + + \left(h_{\tau}d\tau\right)^2 + \left(h_z dz\right)^2 +\label{parzyl:eq:dspara} +\end{equation} +gilt. +Dafür werden $dx$, $dy$, und $dz$ in \eqref{parzyl:eq:ds} mit den Beziehungen +von \eqref{parzyl:coordRelationsa} - \eqref{parzyl:coordRelationse} als +\begin{align} + dx &= \frac{\partial x }{\partial \sigma} d\sigma + + \frac{\partial x }{\partial \tau} d\tau + + \frac{\partial x }{\partial \tilde{z}} d \tilde{z} + = \tau d\sigma + \sigma d \tau \\ + dy &= \frac{\partial y }{\partial \sigma} d\sigma + + \frac{\partial y }{\partial \tau} d\tau + + \frac{\partial y }{\partial \tilde{z}} d \tilde{z} + = \tau d\tau - \sigma d \sigma \\ + dz &= \frac{\partial \tilde{z} }{\partial \sigma} d\sigma + + \frac{\partial \tilde{z} }{\partial \tau} d\tau + + \frac{\partial \tilde{z} }{\partial \tilde{z}} d \tilde{z} + = d \tilde{z} \\ +\end{align} +substituiert. +Wird diese Gleichung in der Form von \eqref{parzyl:eq:dspara} +geschrieben, resultiert +\begin{equation} + \left(d s\right)^2 = + \left(\sigma^2 + \tau^2\right)\left(d\sigma\right)^2 + + \left(\sigma^2 + \tau^2\right)\left(d\tau\right)^2 + + \left(d \tilde{z}\right)^2. +\end{equation} +Daraus ergeben sich die Skalierungsfaktoren +\begin{align} + h_{\sigma} &= \sqrt{\sigma^2 + \tau^2}\\ + h_{\sigma} &= \sqrt{\sigma^2 + \tau^2}\\ + h_{z} &= 1. +\end{align} +\subsection{Differentialgleichung} +Möchte man eine Differentialgleichung im parabolischen +Zylinderkoordinatensystem aufstellen müssen die Skalierungsfaktoren +mitgerechnet werden. +Der Laplace Operator ist dadurch gegeben als +\begin{equation} + \Delta f = \frac{1}{\sigma^2 + \tau^2} + \left( + \frac{\partial^2 f}{\partial \sigma ^2} + + \frac{\partial^2 f}{\partial \tau ^2} + \right) + + \frac{\partial^2 f}{\partial z}. + \label{parzyl:eq:laplaceInParZylCor} +\end{equation} +\subsubsection{Lösung der Helmholtz-Gleichung im parabolischen Zylinderfunktion} +Die Differentialgleichungen, welche zu den parabolischen Zylinderfunktionen führen, tauchen +%, wie bereits erwähnt, +dann auf, wenn die Helmholtz-Gleichung +\begin{equation} + \Delta f(x,y,z) = \lambda f(x,y,z) +\end{equation} +im parabolischen Zylinderkoordinatensystem +\begin{equation} + \Delta f(\sigma,\tau,z) = \lambda f(\sigma,\tau,z) +\end{equation} +gelöst wird. +%Wobei der Laplace Operator $\Delta$ im parabolischen Zylinderkoordinatensystem gegeben ist als +%\begin{equation} +% \Delta +% = +% \frac{1}{\sigma^2 + \tau^2} +% \left ( +% \frac{\partial^2}{\partial \sigma^2} +% + +% \frac{\partial^2}{\partial \tau^2} +% \right ) +% + +% \frac{\partial^2}{\partial z^2}. +%\end{equation} +Mit dem Laplace Operator aus \eqref{parzyl:eq:laplaceInParZylCor} lautet die Helmholtz Gleichung +\begin{equation} + \Delta f(\sigma, \tau, z) + = + \frac{1}{\sigma^2 + \tau^2} + \left ( + \frac{\partial^2 f(\sigma,\tau,z)}{\partial \sigma^2} + + + \frac{\partial^2 f(\sigma,\tau,z)}{\partial \tau^2} + \right ) + + + \frac{\partial^2 f(\sigma,\tau,z)}{\partial z^2} + = + \lambda f(\sigma,\tau,z). +\end{equation} +Diese partielle Differentialgleichung kann mit Hilfe von Separation gelöst werden, dazu wird +\begin{equation} + f(\sigma,\tau,z) = g(\sigma)h(\tau)i(z) +\end{equation} +gesetzt. +Was dann schlussendlich zu den Differentialgleichungen +\begin{equation}\label{parzyl:sep_dgl_1} + g''(\sigma) + - + \left ( + \lambda\sigma^2 + + + \mu + \right ) + g(\sigma) + = + 0, +\end{equation} +\begin{equation}\label{parzyl:sep_dgl_2} + h''(\tau) + - + \left ( + \lambda\tau^2 + - + \mu + \right ) + h(\tau) + = + 0 +\end{equation} +und +\begin{equation}\label{parzyl:sep_dgl_3} + i''(z) + + + \left ( + \lambda + + + \mu + \right ) + i(\tau) + = + 0 +\end{equation} +führt. +Wobei die Lösung von \eqref{parzyl:sep_dgl_3} +\begin{equation} + i(z) + = + A\cos{ + \left ( + \sqrt{\lambda + \mu}z + \right )} + + + B\sin{ + \left ( + \sqrt{\lambda + \mu}z + \right )} +\end{equation} +ist und \eqref{parzyl:sep_dgl_1} und \eqref{parzyl:sep_dgl_2} die sogenannten Weberschen Differentialgleichungen sind, welche die parabolischen Zylinder Funktionen als Lösung haben. + diff --git a/buch/papers/parzyl/teil1.tex b/buch/papers/parzyl/teil1.tex index 9ea60e2..f297189 100644 --- a/buch/papers/parzyl/teil1.tex +++ b/buch/papers/parzyl/teil1.tex @@ -3,53 +3,26 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Lösung \label{parzyl:section:teil1}} \rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{parzyl:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +Die Differentialgleichungen \eqref{parzyl:sep_dgl_1} und \eqref{parzyl:sep_dgl_2} können mit einer Substitution +in die Whittaker Gleichung gelöst werden. +\begin{definition} + Die Funktion + \begin{equation*} + W_{k,m}(z) = + e^{-z/2} z^{m+1/2} \, + {}_{1} F_{1}(\frac{1}{2} + m - k, 1 + 2m; z) + \end{equation*} + heisst Whittaker Funktion und ist eine Lösung + von + \begin{equation} + \frac{d^2W}{d z^2} + + \left(-\frac{1}{4} + \frac{k}{z} + \frac{\frac{1}{4} - m^2}{z^2} \right) W = 0. + \end{equation} +\end{definition} -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{parzyl:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{parzyl:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{parzyl:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +Lösung Folgt\dots diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex index 75ba259..3f890d0 100644 --- a/buch/papers/parzyl/teil2.tex +++ b/buch/papers/parzyl/teil2.tex @@ -3,38 +3,89 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 +\section{Anwendung in der Physik \label{parzyl:section:teil2}} \rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? -\subsection{De finibus bonorum et malorum + +\subsection{Elektrisches Feld einer semi-infiniten Platte \label{parzyl:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld einer semi-infiniten Platte finden will. +Das dies so ist kann im zwei Dimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Wobei die Platte dann nur eine Linie ist. +Jede komplexe Funktion $F(z)$ kann geschrieben werden als +\begin{equation} + F(z) = U(x,y) + iV(x,y) \qquad z \in \mathbb{C}; x,y \in \mathbb{R}. +\end{equation} +Dabei muss gelten, falls die Funktion differenzierbar ist, dass +\begin{equation} + \frac{\partial U(x,y)}{\partial x} + = + \frac{\partial V(x,y)}{\partial y} + \qquad + \frac{\partial V(x,y)}{\partial x} + = + -\frac{\partial U(x,y)}{\partial y}. +\end{equation} +Aus dieser Bedingung folgt +\begin{equation} + \label{parzyl_e_feld_zweite_ab} + \underbrace{ + \frac{\partial^2 U(x,y)}{\partial x^2} + + + \frac{\partial^2 U(x,y)}{\partial y^2} + = + 0 + }_{\nabla^2U(x,y)=0} + \qquad + \underbrace{ + \frac{\partial^2 V(x,y)}{\partial x^2} + + + \frac{\partial^2 V(x,y)}{\partial y^2} + = + 0 + }_{\nabla^2V(x,y) = 0}. +\end{equation} +Zusätzlich zeigen diese Bedingungen auch, dass die zwei Funktionen $U(x,y)$ und $V(x,y)$ orthogonal zueinander sind. +Der Zusammenhang zum elektrischen Feld ist jetzt, dass das Potential an einem quellenfreien Punkt gegeben ist als +\begin{equation} + \nabla^2\phi(x,y) = 0. +\end{equation} +Da dies bei komplexen differenzierbaren Funktionen gilt, wie Gleichung \ref{parzyl_e_feld_zweite_ab} zeigt, kann entweder $U(x,y)$ oder $V(x,y)$ von einer solchen Funktion als das Potential angesehen werden. Im weiteren wird für das Potential $U(x,y)$ verwendet. +Da die Funktion, welche nicht das Potential beschreibt, in weiteren angenommen als $V(x,y)$, orthogonal zum Potential ist, zeigt dies das Verhalten des elektrischen Feldes. +Um nun zu den parabolische Zylinderkoordinaten zu gelangen muss nur noch eine geeignete komplexe Funktion $F(z)$ gefunden werden, welche eine semi-infinite Platte beschreiben kann. Man könnte natürlich auch nach anderen Funktionen suchen, welche andere Bedingungen erfüllen und würde dann auf andere Koordinatensysteme stossen. Die gesuchte Funktion in diesem Fall ist +\begin{equation} + F(z) + = + \sqrt{z} + = + \sqrt{x + iy}. +\end{equation} +Dies kann umgeformt werden zu +\begin{equation} + F(z) + = + \underbrace{\sqrt{\frac{\sqrt{x^2+y^2} + x}{2}}}_{U(x,y)} + + + i\underbrace{\sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}}_{V(x,y)} + . +\end{equation} +Die Äquipotentialflächen können nun betrachtet werden, indem man die Funktion welche das Potential beschreibt gleich eine Konstante setzt, +\begin{equation} + \sigma = U(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} + x}{2}}, +\end{equation} +und die Flächen mit der gleichen elektrischen Feldstärke können als +\begin{equation} + \tau = V(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} - x}{2}} +\end{equation} +beschrieben werden. Diese zwei Gleichungen zeigen nun wie man vom kartesischen Koordinatensystem ins parabolische Zylinderkoordinatensystem kommt. Werden diese Formeln nun nach x und y aufgelöst so beschreibe sie, wie man aus dem parabolischen Zylinderkoordinatensystem zurück ins kartesische rechnen kann +\begin{equation} + x = \sigma \tau, +\end{equation} +\begin{equation} + y = \frac{1}{2}\left ( \tau^2 - \sigma^2 \right ) +\end{equation} + + + diff --git a/buch/papers/parzyl/teil3.tex b/buch/papers/parzyl/teil3.tex index 72c23ca..4e44bd6 100644 --- a/buch/papers/parzyl/teil3.tex +++ b/buch/papers/parzyl/teil3.tex @@ -6,35 +6,3 @@ \section{Teil 3 \label{parzyl:section:teil3}} \rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{parzyl:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/Makefile.inc b/buch/papers/zeta/Makefile.inc index 11c7697..14babe2 100644 --- a/buch/papers/zeta/Makefile.inc +++ b/buch/papers/zeta/Makefile.inc @@ -7,8 +7,7 @@ dependencies-zeta = \ papers/zeta/packages.tex \ papers/zeta/main.tex \ papers/zeta/references.bib \ - papers/zeta/teil0.tex \ - papers/zeta/teil1.tex \ - papers/zeta/teil2.tex \ - papers/zeta/teil3.tex + papers/zeta/einleitung.tex \ + papers/zeta/analytic_continuation.tex \ + papers/zeta/zeta_gamma.tex \ diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex new file mode 100644 index 0000000..0ccc116 --- /dev/null +++ b/buch/papers/zeta/analytic_continuation.tex @@ -0,0 +1,477 @@ +\section{Analytische Fortsetzung} \label{zeta:section:analytische_fortsetzung} +\rhead{Analytische Fortsetzung} + +Die analytische Fortsetzung der Riemannschen Zetafunktion ist äusserst interessant. +Sie ermöglicht die Berechnung von $\zeta(-1)$ und weiterer spannender Werte. +So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = 0.5$. +Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung. + +Es werden zwei verschiedene Fortsetzungen benötigt. +Die erste erweitert die Zetafunktion auf $\Re(s) > 0$. +Die zweite verwendet eine Spiegelung an der $\Re(s) = 0.5$ Linie und erschliesst damit die ganze komplexe Ebene. +Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuation_overview} zu sehen. +\begin{figure} + \centering + \input{papers/zeta/continuation_overview.tikz.tex} + \caption{ + Die verschiedenen Abschnitte der Riemannschen Zetafunktion. + Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig. + Für den blauen Bereich gilt \eqref{zeta:equation:fortsetzung1}. + Um den roten Bereich zu bekommen verwendet die Funktionalgleichung \eqref{zeta:equation:functional} eine Spiegelung an $\Re(s) = 0.5$. + } + \label{zeta:fig:continuation_overview} +\end{figure} + +\subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0} +Zuerst definieren die Dirichletsche Etafunktion als +\begin{equation}\label{zeta:equation:eta} + \eta(s) + = + \sum_{n=1}^{\infty} + \frac{(-1)^{n-1}}{n^s}, +\end{equation} +wobei die Reihe bis auf die alternierenden Vorzeichen die selbe wie in der Zetafunktion ist. +Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) > 0$, da dann die einzelnen Glieder monoton fallend sind. + +Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion durch die Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. +Zuerst wiederholen wir zweimal die Definition der Zetafunktion \eqref{zeta:equation1}, wobei wir sie einmal durch $2^{s-1}$ teilen +\begin{align} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{1}{n^s} \label{zeta:align1} + \\ + \frac{1}{2^{s-1}} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{2}{(2n)^s}. \label{zeta:align2} +\end{align} +Durch Subtraktion der beiden Gleichungen \eqref{zeta:align1} minus \eqref{zeta:align2}, ergibt sich +\begin{align} + \left(1 - \frac{1}{2^{s-1}} \right) + \zeta(s) + &= + \frac{1}{1^s} + \underbrace{-\frac{2}{2^s} + \frac{1}{2^s}}_{-\frac{1}{2^s}} + + \frac{1}{3^s} + \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}} + \ldots + \\ + &= \eta(s). +\end{align} +Dies ist die Fortsetzung auf den noch unbekannten Bereich $0 < \Re(s) < 1$ +\begin{equation} \label{zeta:equation:fortsetzung1} + \zeta(s) + := + \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s). +\end{equation} + +\subsection{Fortsetzung auf ganz $\mathbb{C}$} \label{zeta:subsection:auf_ganz} +Für die Fortsetzung auf den Rest von $\mathbb{C}$, verwenden wir den Zusammenhang von Gamma- und Zetafunktion aus \ref{zeta:section:zusammenhang_mit_gammafunktion}. +Wir beginnen damit, die Gammafunktion für den halben Funktionswert zu berechnen als +\begin{equation} + \Gamma \left( \frac{s}{2} \right) + = + \int_0^{\infty} t^{\frac{s}{2}-1} e^{-t} dt. +\end{equation} +Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten +\begin{equation} + \Gamma \left( \frac{s}{2} \right) + = + \int_0^{\infty} + (\pi n^2)^{\frac{s}{2}} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + \,dx. +\end{equation} +Analog zum Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion} teilen wir durch $(\pi n^2)^{\frac{s}{2}}$ +\begin{equation} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} + = + \int_0^{\infty} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + \,dx, +\end{equation} +und finden Zeta durch die Summenbildung $\sum_{n=1}^{\infty}$ +\begin{equation} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + = + \int_0^{\infty} + x^{\frac{s}{2}-1} + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + \,dx. \label{zeta:equation:integral1} +\end{equation} +Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. +Im Abschnitt \ref{zeta:subsec:poisson_summation} wird die poissonsche Summenformel $\sum f(n) = \sum F(n)$ bewiesen. +In unserem Problem ist $f(n) = e^{-\pi n^2 x}$ und die zugehörige Fouriertransformation $F(n)$ ist +\begin{equation} + F(n) + = + \mathcal{F} + ( + e^{-\pi n^2 x} + ) + = + \frac{1}{\sqrt{x}} + e^{\frac{-n^2 \pi}{x}}. +\end{equation} +Dadurch ergibt sich +\begin{equation}\label{zeta:equation:psi} + \sum_{n=-\infty}^{\infty} + e^{-\pi n^2 x} + = + \frac{1}{\sqrt{x}} + \sum_{n=-\infty}^{\infty} + e^{\frac{-n^2 \pi}{x}}, +\end{equation} +wobei wir die Summen so verändern müssen, dass sie bei $n=1$ beginnen und wir $\psi(x)$ erhalten als +\begin{align} + 2 + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + + + 1 + &= + \frac{1}{\sqrt{x}} + \left( + 2 + \sum_{n=1}^{\infty} + e^{\frac{-n^2 \pi}{x}} + + + 1 + \right) + \\ + 2 + \psi(x) + + + 1 + &= + \frac{1}{\sqrt{x}} + \left( + 2 + \psi\left(\frac{1}{x}\right) + + + 1 + \right) + \\ + \psi(x) + &= + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}.\label{zeta:equation:psi} +\end{align} +Diese Gleichung wird später wichtig werden. + +Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als +\begin{equation}\label{zeta:equation:integral2} + \int_0^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + = + \underbrace{ + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + }_{I_1} + + + \underbrace{ + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + }_{I_2} + = + I_1 + I_2, +\end{equation} +wobei wir uns nun auf den ersten Teil $I_1$ konzentrieren werden. +Dabei setzen wir die Definition von $\psi(x)$ aus \eqref{zeta:equation:psi} ein und erhalten +\begin{align} + I_1 + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \left( + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}} + \right) + \,dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + + \frac{1}{2} + \biggl( + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \biggl) + \,dx + \\ + &= + \underbrace{ + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + \,dx + }_{I_3} + + + \underbrace{ + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \,dx + }_{I_4}. \label{zeta:equation:integral3} +\end{align} +Dabei kann das zweite Integral $I_4$ gelöst werden als +\begin{equation} + I_4 + = + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \,dx + = + \frac{1}{s(s-1)}. +\end{equation} +Das erste Integral $I_3$ aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. +Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. +Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. +Dies ergibt +\begin{align} + I_3 + = + \int_{\infty}^{1} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{-du}{u^2} + &= + \int_{1}^{\infty} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{du}{u^2} + \\ + &= + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + \,dx, +\end{align} +wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. +Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind. +Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten +\begin{equation} + I_1 + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + = + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + \,dx + + + \frac{1}{s(s-1)}. +\end{equation} +Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um schlussendlich +\begin{align} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + \nonumber + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + \,dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + \left( + x^{-\frac{s}{2}-\frac{1}{2}} + + + x^{\frac{s}{2}-1} + \right) + \psi(x) + \,dx + \\ + &= + \frac{-1}{s(1-s)} + + + \int_{1}^{\infty} + \left( + x^{\frac{1-s}{2}} + + + x^{\frac{s}{2}} + \right) + \frac{\psi(x)}{x} + \,dx, +\end{align} +zu erhalten. +Wenn wir dieses Resultat genau anschauen, erkennen wir dass sich nichts verändert wenn $s$ mit $1-s$ ersetzt wird. +Somit haben wir die analytische Fortsetzung gefunden als +\begin{equation}\label{zeta:equation:functional} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + = + \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}} + \zeta(1-s). +\end{equation} +%TODO Definitionen und Gleichungen klarer unterscheiden + +\subsection{Poissonsche Summenformel} \label{zeta:subsec:poisson_summation} + +Der Beweis für Gleichung \ref{zeta:equation:psi} folgt direkt durch die poissonsche Summenformel. +Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion. + +\begin{lemma} + Die Fourierreihe der periodischen Dirac Delta Funktion $\sum \delta(x - 2\pi k)$ ist + \begin{equation} \label{zeta:equation:fourier_dirac} + \sum_{k=-\infty}^{\infty} + \delta(x - 2\pi k) + = + \frac{1}{2\pi} + \sum_{n=-\infty}^{\infty} + e^{i n x}. + \end{equation} +\end{lemma} + +\begin{proof}[Beweis] + Eine Fourierreihe einer beliebigen periodischen Funktion $f(x)$ berechnet sich als + \begin{align} + f(x) + &= + \sum_{n=-\infty}^{\infty} + c_n + e^{i n x} \\ + c_n + &= + \frac{1}{2\pi} + \int_{-\pi}^{\pi} + f(x) + e^{-i n x} + \, dx. + \end{align} + Wenn $f(x)=\delta(x)$ eingesetz wird ergeben sich konstante Koeffizienten + \begin{equation} + c_n + = + \frac{1}{2\pi} + \int_{-\pi}^{\pi} + \delta(x) + e^{-i n x} + \, dx + = + \frac{1}{2\pi}, + \end{equation} + womit die sehr einfache Fourierreihe der Dirac Delta Funktion berechnet wäre. +\end{proof} + +\begin{satz}[Poissonsche Summernformel] + Die Summe einer Funktion $f(n)$ über alle ganzen Zahlen $n$ ist äquivalent zur Summe ihrer Fouriertransformation $F(k)$ über alle ganzen Zahlen $k$ + \begin{equation} + \sum_{n=-\infty}^{\infty} + f(n) + = + \sum_{k=-\infty}^{\infty} + F(k). + \end{equation} +\end{satz} + +\begin{proof}[Beweis] + Wir schreiben die Summe über die Fouriertransformation aus + \begin{align} + \sum_{k=-\infty}^{\infty} + F(k) + &= + \sum_{k=-\infty}^{\infty} + \int_{-\infty}^{\infty} + f(x) + e^{-i 2\pi x k} + \, dx + \\ + &= + \int_{-\infty}^{\infty} + f(x) + \underbrace{ + \sum_{k=-\infty}^{\infty} + e^{-i 2\pi x k} + }_{\text{\eqref{zeta:equation:fourier_dirac}}} + \, dx, + \end{align} + und verwenden die Fouriertransformation der Dirac Funktion aus \eqref{zeta:equation:fourier_dirac} + \begin{align} + \sum_{k=-\infty}^{\infty} + e^{-i 2\pi x k} + &= + 2 \pi + \sum_{k=-\infty}^{\infty} + \delta(-2\pi x - 2\pi k) + \\ + &= + \frac{2 \pi}{2 \pi} + \sum_{k=-\infty}^{\infty} + \delta(x + k). + \end{align} + Wenn wir dies einsetzen und erhalten wir den gesuchten Beweis für die poissonsche Summenformel + \begin{equation} + \sum_{k=-\infty}^{\infty} + F(k) + = + \int_{-\infty}^{\infty} + f(x) + \sum_{k=-\infty}^{\infty} + \delta(x + k) + \, dx + = + \sum_{k=-\infty}^{\infty} + \int_{-\infty}^{\infty} + f(x) + \delta(x + k) + \, dx + = + \sum_{k=-\infty}^{\infty} + f(k). + \end{equation} +\end{proof} diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/continuation_overview.tikz.tex new file mode 100644 index 0000000..836ab1d --- /dev/null +++ b/buch/papers/zeta/continuation_overview.tikz.tex @@ -0,0 +1,18 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=0.9cm, scale=2, + dot/.style={fill, circle, inner sep=0, minimum size=0.1cm}] + + \draw[->] (-2,0) -- (-1,0) node[dot]{} node[anchor=north]{$-1$} -- (0,0) node[anchor=north west]{$0$} -- (0.5,0) node[anchor=north west]{$0.5$}-- (1,0) node[anchor=north west]{$1$} -- (2,0) node[anchor=west]{$\Re(s)$}; + + \draw[->] (0,-1.2) -- (0,1.2) node[anchor=south]{$\Im(s)$}; + \begin{scope}[yscale=0.1] + \draw[] (1,-1) -- (1,1); + \end{scope} + \draw[dotted] (0.5,-1) -- (0.5,1); + + \begin{scope}[] + \fill[opacity=0.2, red] (-1.8,1) rectangle (0, -1); + \fill[opacity=0.2, blue] (0,1) rectangle (1, -1); + \fill[opacity=0.2, green] (1,1) rectangle (1.8, -1); + \end{scope} + +\end{tikzpicture} diff --git a/buch/papers/zeta/einleitung.tex b/buch/papers/zeta/einleitung.tex new file mode 100644 index 0000000..3b70531 --- /dev/null +++ b/buch/papers/zeta/einleitung.tex @@ -0,0 +1,11 @@ +\section{Einleitung} \label{zeta:section:einleitung} +\rhead{Einleitung} + +Die Riemannsche Zetafunktion ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als +\begin{equation}\label{zeta:equation1} + \zeta(s) + = + \sum_{n=1}^{\infty} + \frac{1}{n^s}. +\end{equation} + diff --git a/buch/papers/zeta/euler_product.tex b/buch/papers/zeta/euler_product.tex new file mode 100644 index 0000000..a6ed512 --- /dev/null +++ b/buch/papers/zeta/euler_product.tex @@ -0,0 +1,85 @@ +\section{Eulerprodukt} \label{zeta:section:eulerprodukt} +\rhead{Eulerprodukt} + +Das Eulerprodukt stellt die Verbindung der Zetafunktion und der Primzahlen her. +Diese Verbindung ist sehr wichtig, da durch sie eine Aussage zur Primzahlverteilung gemacht werden kann. +Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche eines der grössten ungelösten Probleme der Mathematik ist. + +\begin{satz} + Für alle Zahlen $s$ mit $\Re(s) > 1$ ist die Zetafunktion identisch mit dem unendlichen Eulerprodukt + \begin{equation}\label{zeta:eq:eulerprodukt} + \zeta(s) + = + \sum_{n=1}^\infty + \frac{1}{n^s} + = + \prod_{p \in P} + \frac{1}{1-p^{-s}} + \end{equation} + wobei $P$ die Menge aller Primzahlen darstellt. +\end{satz} + +\begin{proof}[Beweis] + Der Beweis startet mit dem Eulerprodukt und stellt dieses so um, dass die Zetafunktion erscheint. + Als erstes ersetzen wir die Faktoren durch geometrische Reihen + \begin{equation} + \prod_{i=1}^{\infty} + \frac{1}{1-p^{-s}} + = + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \left( + \frac{1}{p_i^s} + \right)^{k_i} + = + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \frac{1}{p_i^{s k_i}}, + \end{equation} + dabei iteriert der Index $i$ über alle Primzahlen $p_i$. + Durch Ausschreiben der Multiplikation und Ausklammern der Summen erhalten wir + \begin{align} + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \frac{1}{p_i^{s k_i}} + &= + \sum_{k_1=0}^{\infty} + \frac{1}{p_1^{s k_1}} + \sum_{k_2=0}^{\infty} + \frac{1}{p_2^{s k_2}} + \ldots + \nonumber \\ + &= + \sum_{k_1=0}^{\infty} + \sum_{k_2=0}^{\infty} + \ldots + \left( + \frac{1}{p_1^{k_1}} + \frac{1}{p_2^{k_2}} + \ldots + \right)^s. + \label{zeta:equation:eulerprodukt2} + \end{align} + Der Fundamentalsatz der Arithmetik (Primfaktorzerlegung) besagt, dass jede beliebige Zahl $n \in \mathbb{N}$ durch eine eindeutige Primfaktorzerlegung beschrieben werden kann + \begin{equation} + n = \prod_i p_i^{k_i} \quad \forall \quad n \in \mathbb{N}. + \end{equation} + Jeder Summand der Summen in \eqref{zeta:equation:eulerprodukt2} ist somit eine Zahl $n$. + Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir + \begin{equation} + \sum_{k_1=0}^{\infty} + \sum_{k_2=0}^{\infty} + \ldots + \left( + \frac{1}{p_1^{k_1}} + \frac{1}{p_2^{k_2}} + \ldots + \right)^s + = + \sum_{n=1}^\infty + \frac{1}{n^s} + = + \zeta(s) + \end{equation} +\end{proof} + diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex index 1d9e059..caddace 100644 --- a/buch/papers/zeta/main.tex +++ b/buch/papers/zeta/main.tex @@ -3,34 +3,17 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:zeta}} -\lhead{Thema} +\chapter{Riemannsche Zetafunktion\label{chapter:zeta}} +\lhead{Riemannsche Zetafunktion} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Raphael Unterer} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +%TODO Einleitung -\input{papers/zeta/teil0.tex} -\input{papers/zeta/teil1.tex} -\input{papers/zeta/teil2.tex} -\input{papers/zeta/teil3.tex} +\input{papers/zeta/einleitung.tex} +\input{papers/zeta/euler_product.tex} +\input{papers/zeta/zeta_gamma.tex} +\input{papers/zeta/analytic_continuation.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/zeta/teil0.tex b/buch/papers/zeta/teil0.tex deleted file mode 100644 index 56c0b1b..0000000 --- a/buch/papers/zeta/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{zeta:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{zeta:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/zeta/teil1.tex b/buch/papers/zeta/teil1.tex deleted file mode 100644 index 4017ee8..0000000 --- a/buch/papers/zeta/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{zeta:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{zeta:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{zeta:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{zeta:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/zeta/teil2.tex b/buch/papers/zeta/teil2.tex deleted file mode 100644 index 9e8a96e..0000000 --- a/buch/papers/zeta/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{zeta:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/teil3.tex b/buch/papers/zeta/teil3.tex deleted file mode 100644 index 6610cc3..0000000 --- a/buch/papers/zeta/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{zeta:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex new file mode 100644 index 0000000..db41676 --- /dev/null +++ b/buch/papers/zeta/zeta_gamma.tex @@ -0,0 +1,61 @@ +\section{Zusammenhang mit der Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion} +\rhead{Zusammenhang mit der Gammafunktion} + +In diesem Abschnitt wird gezeigt, wie sich die Zetafunktion durch die Gammafunktion $\Gamma(s)$ ausdrücken lässt. +Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ ist nicht nur interessant, er wird später auch für die Herleitung der analytischen Fortsetzung gebraucht. + +Wir erinnern uns an die Definition der Gammafunktion in \eqref{buch:rekursion:gamma:integralbeweis} +\begin{equation*} + \Gamma(s) + = + \int_0^{\infty} t^{s-1} e^{-t} \,dt, +\end{equation*} +wobei die Notation an die Zetafunktion angepasst ist. +Durch die Substitution von $t$ mit $t = nu$ und $dt = n\,du$ wird daraus +\begin{align*} + \Gamma(s) + &= + \int_0^{\infty} n^{s-1}u^{s-1} e^{-nu} n \,du \\ + &= + \int_0^{\infty} n^s u^{s-1} e^{-nu} \,du. +\end{align*} +Durch Division mit durch $n^s$ ergibt sich die Quotienten +\begin{equation*} + \frac{\Gamma(s)}{n^s} + = + \int_0^{\infty} u^{s-1} e^{-nu} \,du, +\end{equation*} +welche sich zur Zetafunktion summieren +\begin{equation} + \sum_{n=1}^{\infty} \frac{\Gamma(s)}{n^s} + = + \Gamma(s) \zeta(s) + = + \int_0^{\infty} u^{s-1} + \sum_{n=1}^{\infty}e^{-nu} + \,du. + \label{zeta:equation:zeta_gamma1} +\end{equation} +Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhalten +\begin{align} + \sum_{n=1}^{\infty}\left(e^{-u}\right)^n + &= + \sum_{n=0}^{\infty}\left(e^{-u}\right)^n + - + 1 + \\ + &= + \frac{1}{1 - e^{-u}} - 1 + \\ + &= + \frac{1}{e^u - 1}. +\end{align} +Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir den gewünschten Zusammenhang +\begin{equation}\label{zeta:equation:zeta_gamma_final} + \zeta(s) + = + \frac{1}{\Gamma(s)} + \int_0^{\infty} + \frac{u^{s-1}}{e^u -1} + du \qed +\end{equation} |
