diff options
Diffstat (limited to 'vorlesungen')
39 files changed, 5312 insertions, 39 deletions
diff --git a/vorlesungen/04_fresnel/common.tex b/vorlesungen/04_fresnel/common.tex index 418b7a5..f4d919b 100644 --- a/vorlesungen/04_fresnel/common.tex +++ b/vorlesungen/04_fresnel/common.tex @@ -9,8 +9,8 @@ \usetheme[hideothersubsections,hidetitle]{Hannover} } \beamertemplatenavigationsymbolsempty -\title[Klothoide]{Klothoide} -\author[N.~Eswararajah]{Nilakshan Eswararajah} +\title[Klothoide]{Fresnel-Integrale und Klothoide} +\author[A.~Müller]{Prof.~Dr.~Andreas Müller} \date[]{9.~Mai 2022} \newboolean{presentation} diff --git a/vorlesungen/04_fresnel/slides.tex b/vorlesungen/04_fresnel/slides.tex index 5a7cce2..a46fe9e 100644 --- a/vorlesungen/04_fresnel/slides.tex +++ b/vorlesungen/04_fresnel/slides.tex @@ -3,4 +3,8 @@ % % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{fresnel/test.tex} +\folie{fresnel/integrale.tex} +\folie{fresnel/numerik.tex} +\folie{fresnel/kruemmung.tex} +\folie{fresnel/klothoide.tex} +\folie{fresnel/apfel.tex} diff --git a/vorlesungen/12_dreieck/common.tex b/vorlesungen/12_dreieck/common.tex index 9414e42..1be1b4f 100644 --- a/vorlesungen/12_dreieck/common.tex +++ b/vorlesungen/12_dreieck/common.tex @@ -9,7 +9,7 @@ \usetheme[hideothersubsections,hidetitle]{Hannover} } \beamertemplatenavigationsymbolsempty -\title[Dreieckstest]{Dreieckstest} +\title[Ordnungsstatistik]{Ordnungsstatistik und Beta-Funktion} \author[A.~Müller]{Prof. Dr. Andreas Müller} \date[]{30.~Mai 2022} \newboolean{presentation} diff --git a/vorlesungen/12_dreieck/slides.tex b/vorlesungen/12_dreieck/slides.tex index 211a105..19b7417 100644 --- a/vorlesungen/12_dreieck/slides.tex +++ b/vorlesungen/12_dreieck/slides.tex @@ -6,3 +6,7 @@ \folie{dreieck/stichprobe.tex} \folie{dreieck/minmax.tex} \folie{dreieck/ordnungsstatistik.tex} +\folie{dreieck/dichte.tex} +\folie{dreieck/orderplot.tex} +\folie{dreieck/beta.tex} +\folie{dreieck/betaplot.tex} diff --git a/vorlesungen/18_hermiteintegrierbar/Makefile b/vorlesungen/18_hermiteintegrierbar/Makefile new file mode 100644 index 0000000..a2dfb87 --- /dev/null +++ b/vorlesungen/18_hermiteintegrierbar/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- hermiteintegrierbar +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: hermiteintegrierbar-handout.pdf MathSem-18-hermiteintegrierbar.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-18-hermiteintegrierbar.pdf: MathSem-18-hermiteintegrierbar.tex $(SOURCES) + pdflatex MathSem-18-hermiteintegrierbar.tex + +hermiteintegrierbar-handout.pdf: hermiteintegrierbar-handout.tex $(SOURCES) + pdflatex hermiteintegrierbar-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSem-18-hermiteintegrierbar.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-18-hermiteintegrierbar.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-18-hermiteintegrierbar.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-18-hermiteintegrierbar.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex b/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex new file mode 100644 index 0000000..7a3a647 --- /dev/null +++ b/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex @@ -0,0 +1,14 @@ +% +% MathSem-18-hermiteintegrierbar.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/18_hermiteintegrierbar/common.tex b/vorlesungen/18_hermiteintegrierbar/common.tex new file mode 100644 index 0000000..8b1c71f --- /dev/null +++ b/vorlesungen/18_hermiteintegrierbar/common.tex @@ -0,0 +1,17 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[$\int P(t)e^{-t^2}\,dt$]{Elementare Stammfunktion für +$\displaystyle\int P(t)e^{-t^2}\,dt$?} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex b/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex new file mode 100644 index 0000000..a466024 --- /dev/null +++ b/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex @@ -0,0 +1,11 @@ +% +% hermiteintegrierbar-handout.tex -- Handout XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/18_hermiteintegrierbar/slides.tex b/vorlesungen/18_hermiteintegrierbar/slides.tex new file mode 100644 index 0000000..cb3bbea --- /dev/null +++ b/vorlesungen/18_hermiteintegrierbar/slides.tex @@ -0,0 +1,11 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\folie{hermite/normalintegrale.tex} +\folie{hermite/normalhermite.tex} +\folie{hermite/hermiteentwicklung.tex} +\folie{hermite/loesung.tex} +\folie{hermite/skalarprodukt.tex} + diff --git a/vorlesungen/slides/dreieck/Makefile.inc b/vorlesungen/slides/dreieck/Makefile.inc index 0575397..bbc19b6 100644 --- a/vorlesungen/slides/dreieck/Makefile.inc +++ b/vorlesungen/slides/dreieck/Makefile.inc @@ -4,6 +4,11 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapterdreieck = \ + ../slides/dreieck/stichprobe.tex \ ../slides/dreieck/minmax.tex \ ../slides/dreieck/ordnungsstatistik.tex \ + ../slides/dreieck/orderplot.tex \ + ../slides/dreieck/dichte.tex \ + ../slides/dreieck/beta.tex \ + ../slides/dreieck/betaplot.tex \ ../slides/dreieck/test.tex diff --git a/vorlesungen/slides/dreieck/beta.tex b/vorlesungen/slides/dreieck/beta.tex new file mode 100644 index 0000000..fc3606a --- /dev/null +++ b/vorlesungen/slides/dreieck/beta.tex @@ -0,0 +1,70 @@ +% +% beta.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beta-Verteilung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Ordnungsstatistik} +\begin{align*} +\varphi(x) +&= +{\color{blue}N} x^{k-1} (1-x)^{n-k} +\\ +&\uncover<8->{ += +\beta_{k,n-k+1}(x) +} +\end{align*} +\end{block} +\uncover<8->{% +\begin{block}{Risch-Algorithmus} +Die Beta-Verteilungen haben ausser in Spezialfällen +keine Stammfunktion in geschlossener Form. +\end{block}} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<2->{% +\begin{definition} +Beta-Verteilung +\[ +\beta_{a,b}(x) += +\begin{cases} +\displaystyle +\uncover<7->{ +{\color{blue} +\frac{1}{B(a,b)} +} +} +x^{a-1}(1-x)^{b-1} +&0\le x\le 1 +\\ +0&\text{sonst} +\end{cases} +\] +\end{definition}} +\uncover<3->{% +\begin{block}{Normierung} +\begin{align*} +{\color{blue}\frac{1}{{N}}} +&\uncover<4->{= +\int_{-\infty}^\infty \beta_{a,b}(x)\,dx} +\\ +&\uncover<5->{= +\int_{0}^1 x^{a-1}(1-x)^{b-1}\,dx} +\\ +&\uncover<6->{= +B(a,b)} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/dreieck/betaplot.tex b/vorlesungen/slides/dreieck/betaplot.tex new file mode 100644 index 0000000..ee932e8 --- /dev/null +++ b/vorlesungen/slides/dreieck/betaplot.tex @@ -0,0 +1,38 @@ +% +% betaplot.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beta-Verteilungen} +\begin{center} +\begin{tikzpicture}[>=latex] + +\only<1>{ +\begin{scope} + \clip (-7,-3.2) rectangle (7,3.2); + \node at (0,-6.5) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}}; +\end{scope} +} + +\only<2>{ +\begin{scope} + \clip (-7,-3.2) rectangle (7,3.2); + \node at (0,-0) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}}; +\end{scope} +} + +\only<3>{ +\begin{scope} + \clip (-7,-3.2) rectangle (7,3.2); + \node at (0,6.5) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}}; +\end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/dreieck/chapter.tex b/vorlesungen/slides/dreieck/chapter.tex index 2c91eb5..0f58c4c 100644 --- a/vorlesungen/slides/dreieck/chapter.tex +++ b/vorlesungen/slides/dreieck/chapter.tex @@ -6,3 +6,6 @@ \folie{dreieck/test.tex} \folie{dreieck/minmax.tex} \folie{dreieck/ordnungsstatistik.tex} +\folie{dreieck/dichte.tex} +\folie{dreieck/beta.tex} +\folie{dreieck/betaplot.tex} diff --git a/vorlesungen/slides/dreieck/dichte.tex b/vorlesungen/slides/dreieck/dichte.tex new file mode 100644 index 0000000..168523a --- /dev/null +++ b/vorlesungen/slides/dreieck/dichte.tex @@ -0,0 +1,67 @@ +% +% dichte.tex -- Wahrscheinlichkeitsdichte +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wahrscheinlichkeitsdichte} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Definition} +\[ +\varphi_{X_{k:n}}(x) += +\frac{d}{dx} F_{X_{k:n}}(x) +\] +\end{block} +\end{column} +\begin{column}{0.60\textwidth} +\uncover<4->{% +\begin{block}{Gleichverteilung} +\[ +{\color{darkgreen}F(x)}=\begin{cases} +0&x \le 0\\ +x&0\le x \le 1,\\ +1&x\ge 1 +\end{cases} +\quad +\uncover<5->{ +{\color{red}\varphi(x)} += +\begin{cases} +1&0\le x \le 1\\ +0&\text{sonst} +\end{cases}} +\] +\end{block}} +\end{column} +\end{columns} +\uncover<2->{% +\begin{block}{Ordnungsstatistik} +nach einiger Rechnung: +\begin{align*} +\varphi_{X_{k:n}}(x) +&= +{\color<3->{red}\varphi_X(x)}\,k\binom{n}{k}{\color<3->{darkgreen}F_X(x)}^{k-1} +(1-{\color<3->{darkgreen}F_X(x)})^{n-k} +\intertext{\uncover<4->{für Gleichverteilung}} +\uncover<6->{ +\varphi_{X_{k:n}}(x) +&= +\begin{cases} +\displaystyle +{\color<7->{blue}k\binom{n}{k}}{\color{darkgreen}x}^{k-1}(1-{\color{darkgreen}x})^{n-k} +&0\le x \le 1\\ +0&\text{sonst} +\end{cases} +\qquad\uncover<7->{\text{({\color{blue}Normierung})}} +} +\end{align*} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/dreieck/minmax.tex b/vorlesungen/slides/dreieck/minmax.tex index 9ef8d1a..ff3a231 100644 --- a/vorlesungen/slides/dreieck/minmax.tex +++ b/vorlesungen/slides/dreieck/minmax.tex @@ -17,48 +17,66 @@ Verteilungsfunktion von Z=\operatorname{max}(X_1,\dots,X_n) \] \begin{align*} +\uncover<3->{ F_Z(x) &= -P(Z\le x) +P(Z\le x)} \\ +\uncover<4->{ &= P(X_1\le x\wedge\dots\wedge X_n\le x) +} \\ +\uncover<5->{ &= P(X_1\le x)\cdot \ldots\cdot P(X_n\le x) +} \\ +\uncover<6->{ &= F_X(x)^n +} \end{align*} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Minimum} Verteilungsfunktion von \[ Z=\operatorname{min}(X_1,\dots,X_n) \] \begin{align*} +\uncover<7->{ F_Z(x) &= P(Z\le x) +} \\ +\uncover<8->{ &=P(\overline{ X_1\le x\wedge\dots\wedge X_n \le x }) +} \\ +\uncover<9->{ &= 1-P( X_1> x\wedge\dots\wedge X_n > x ) +} \\ +\uncover<10->{ &= 1-(P(X_1>x)\cdot\ldots\cdot P(X_n>x)) +} \\ +\uncover<11->{ &= 1-(1-F_X(x))^n +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/dreieck/orderplot.tex b/vorlesungen/slides/dreieck/orderplot.tex new file mode 100644 index 0000000..7cf10c6 --- /dev/null +++ b/vorlesungen/slides/dreieck/orderplot.tex @@ -0,0 +1,16 @@ +% +% orderplot.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ordnungstatistik} +\vspace*{-18pt} +\begin{center} +\includegraphics[width=10cm]{../../buch/chapters/040-rekursion/images/order.pdf} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/dreieck/ordnungsstatistik.tex b/vorlesungen/slides/dreieck/ordnungsstatistik.tex index 6346953..c968e79 100644 --- a/vorlesungen/slides/dreieck/ordnungsstatistik.tex +++ b/vorlesungen/slides/dreieck/ordnungsstatistik.tex @@ -8,11 +8,76 @@ \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \frametitle{Ordnungstatistik} +\vspace{-10pt} +\begin{block}{Angeordnete Stichprobe} +\[ +X_{1:n} +\le +X_{2:n} +\le +\dots +\le +X_{(n-1):n} +\le +X_{n:n} +\] +$X_{k:n} = \mathstrut$der $k$-te von $n$ Werten +\end{block} \vspace{-20pt} \begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} +\begin{column}{0.44\textwidth} +\uncover<2->{% +\begin{block}{Verteilungsfunktion} +\begin{align*} +F_{X_{k:n}}(x) +&= +P(X_{k:n} \le x) +\\ +&\uncover<3->{= +P\bigl( +|\{i\;|\; {\color<4>{red}X_i\le x}\}| \ge k +\bigr)} +\\ +&\uncover<5->{= +P(\text{Anzahl $A_i$}\ge k)} +\\ +&\uncover<9->{= +P(K\ge k)} +\\ +\uncover<6->{ +F_{X_i}(x)&= P(X_i\le x)}\uncover<7->{ = P(A_i)}\uncover<10->{ = p} +} +\end{align*} +\uncover<4->{$A_i=\{X_i\le x\}$}\uncover<7->{ ist ein Beroulli- Experiment +\uncover<10->{mit Eintretens- wahrscheinlichkeit $p$} +\end{block}} \end{column} -\begin{column}{0.48\textwidth} +\begin{column}{0.52\textwidth} +\uncover<8->{% +\begin{block}{Wiederholtes Bernoulli-Experiment} +$K=\mathstrut$Anzahl $k$, für die $A$ eingetreten +ist\only<11->{, ist binomialverteilt:} +\begin{align*} +\uncover<12->{P(K=k) +&= +\phantom{\sum_{i=k}^n\mathstrut} +\binom{n}{k} p^k (1-p)^{n-k} +} +\\ +\uncover<13->{ +P(K\ge k) +&= +\sum_{i=k}^n +\binom{n}{i} p^i (1-p)^{n-i} +} +\\ +\uncover<14->{ +&= +\sum_{i=k}^n +\binom{n}{i} F_X(x)^i (1-F_X(x))^{n-i} +} +\end{align*} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/dreieck/stichprobe.tex b/vorlesungen/slides/dreieck/stichprobe.tex index da3a20e..4b2eff0 100644 --- a/vorlesungen/slides/dreieck/stichprobe.tex +++ b/vorlesungen/slides/dreieck/stichprobe.tex @@ -12,21 +12,22 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Zufallsvariable} -Gegeben eine Zufallsvariable $X$ mit +Gegeben eine Zufallsvariable $X$ \uncover<5->{mit Verteilungsfunktion \[ F_X(x) = P(X\le x) -\] -und +\]} +\uncover<6->{und Wahrscheinlichkeitsdichte \[ \varphi_X(x) = \frac{d}{dx} F_X(x) -\] +\]} \end{block} +\uncover<7->{% \begin{block}{Gleichverteilung} \[ F(x) = \begin{cases} @@ -34,6 +35,7 @@ F(x) = \begin{cases} x&\qquad 0\le x \le 1\\ 1&\qquad 1<x \end{cases} +\uncover<8->{ \qquad\Rightarrow\qquad \varphi(x) = @@ -41,19 +43,21 @@ x&\qquad 0\le x \le 1\\ 1&\qquad 0\le x \le 1\\ 0&\qquad\text{sonst}. \end{cases} +} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Stichprobe} $n$ Zufallsvariablen $X_1,\dots,X_n$ \begin{itemize} -\item +\item<3-> alle $X_i$ haben die gleiche Verteilung wie $X$ -\item +\item<4-> die $X_i$ sind unabhängig \end{itemize} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/fresnel/Makefile b/vorlesungen/slides/fresnel/Makefile new file mode 100644 index 0000000..77ad9a2 --- /dev/null +++ b/vorlesungen/slides/fresnel/Makefile @@ -0,0 +1,9 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: eulerpath.tex + +eulerpath.tex: eulerspirale.m + octave eulerspirale.m diff --git a/vorlesungen/slides/fresnel/Makefile.inc b/vorlesungen/slides/fresnel/Makefile.inc index c17b654..b6d11f0 100644 --- a/vorlesungen/slides/fresnel/Makefile.inc +++ b/vorlesungen/slides/fresnel/Makefile.inc @@ -4,4 +4,8 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapterfresnel = \ - ../slides/fresnel/test.tex + ../slides/fresnel/integrale.tex \ + ../slides/fresnel/kruemmung.tex \ + ../slides/fresnel/klothoide.tex \ + ../slides/fresnel/numerik.tex \ + ../slides/fresnel/apfel.tex diff --git a/vorlesungen/slides/fresnel/apfel.jpg b/vorlesungen/slides/fresnel/apfel.jpg Binary files differnew file mode 100644 index 0000000..96b975d --- /dev/null +++ b/vorlesungen/slides/fresnel/apfel.jpg diff --git a/vorlesungen/slides/fresnel/apfel.png b/vorlesungen/slides/fresnel/apfel.png Binary files differnew file mode 100644 index 0000000..f413852 --- /dev/null +++ b/vorlesungen/slides/fresnel/apfel.png diff --git a/vorlesungen/slides/fresnel/apfel.tex b/vorlesungen/slides/fresnel/apfel.tex new file mode 100644 index 0000000..090c3d5 --- /dev/null +++ b/vorlesungen/slides/fresnel/apfel.tex @@ -0,0 +1,32 @@ +% +% apfel.tex -- Apfelschale als Klothoide +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\input{../slides/fresnel/eulerpath.tex} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Apfelschale} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope} +\clip(-1,-1) rectangle (7,6); +\uncover<2->{ +\node at (3.1,2.2) [rotate=-3] + {\includegraphics[width=9.4cm]{../slides/fresnel/apfel.png}}; +} +\end{scope} +\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:$S(t)$}]; +\uncover<3->{ +\begin{scope}[scale=8] +\draw[color=red,opacity=0.5,line width=1.4pt] \fresnela; +\end{scope} +} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/fresnel/chapter.tex b/vorlesungen/slides/fresnel/chapter.tex index dc5d031..916a3a9 100644 --- a/vorlesungen/slides/fresnel/chapter.tex +++ b/vorlesungen/slides/fresnel/chapter.tex @@ -3,4 +3,8 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\folie{fresnel/test.tex} +\folie{fresnel/integrale.tex} +\folie{fresnel/kruemmung.tex} +\folie{fresnel/klothoide.tex} +\folie{fresnel/numerik.tex} +\folie{fresnel/apfel.tex} diff --git a/vorlesungen/slides/fresnel/eulerpath.tex b/vorlesungen/slides/fresnel/eulerpath.tex new file mode 100644 index 0000000..ecd0b2b --- /dev/null +++ b/vorlesungen/slides/fresnel/eulerpath.tex @@ -0,0 +1,4012 @@ +\def\fresnela{ (0,0) + -- (0.0100,0.0000) + -- (0.0200,0.0000) + -- (0.0300,0.0000) + -- (0.0400,0.0000) + -- (0.0500,0.0001) + -- (0.0600,0.0001) + -- (0.0700,0.0002) + -- (0.0800,0.0003) + -- (0.0900,0.0004) + -- (0.1000,0.0005) + -- (0.1100,0.0007) + -- (0.1200,0.0009) + -- (0.1300,0.0012) + -- (0.1400,0.0014) + -- (0.1500,0.0018) + -- (0.1600,0.0021) + -- (0.1700,0.0026) + -- (0.1800,0.0031) + -- (0.1899,0.0036) + -- (0.1999,0.0042) + -- (0.2099,0.0048) + -- (0.2199,0.0056) + -- (0.2298,0.0064) + -- (0.2398,0.0072) + -- (0.2498,0.0082) + -- (0.2597,0.0092) + -- (0.2696,0.0103) + -- (0.2796,0.0115) + -- (0.2895,0.0128) + -- (0.2994,0.0141) + -- (0.3093,0.0156) + -- (0.3192,0.0171) + -- (0.3290,0.0188) + -- (0.3389,0.0205) + -- (0.3487,0.0224) + -- (0.3585,0.0244) + -- (0.3683,0.0264) + -- (0.3780,0.0286) + -- (0.3878,0.0309) + -- (0.3975,0.0334) + -- (0.4072,0.0359) + -- (0.4168,0.0386) + -- (0.4264,0.0414) + -- (0.4359,0.0443) + -- (0.4455,0.0474) + -- (0.4549,0.0506) + -- (0.4644,0.0539) + -- (0.4738,0.0574) + -- (0.4831,0.0610) + -- (0.4923,0.0647) + -- (0.5016,0.0686) + -- (0.5107,0.0727) + -- (0.5198,0.0769) + -- (0.5288,0.0812) + -- (0.5377,0.0857) + -- (0.5466,0.0904) + -- (0.5553,0.0952) + -- (0.5640,0.1001) + -- (0.5726,0.1053) + -- (0.5811,0.1105) + -- (0.5895,0.1160) + -- (0.5978,0.1216) + -- (0.6059,0.1273) + -- (0.6140,0.1333) + -- (0.6219,0.1393) + -- (0.6298,0.1456) + -- (0.6374,0.1520) + -- (0.6450,0.1585) + -- (0.6524,0.1653) + -- (0.6597,0.1721) + -- (0.6668,0.1792) + -- (0.6737,0.1864) + -- (0.6805,0.1937) + -- (0.6871,0.2012) + -- (0.6935,0.2089) + -- (0.6998,0.2167) + -- (0.7058,0.2246) + -- (0.7117,0.2327) + -- (0.7174,0.2410) + -- (0.7228,0.2493) + -- (0.7281,0.2579) + -- (0.7331,0.2665) + -- (0.7379,0.2753) + -- (0.7425,0.2841) + -- (0.7469,0.2932) + -- (0.7510,0.3023) + -- (0.7548,0.3115) + -- (0.7584,0.3208) + -- (0.7617,0.3303) + -- (0.7648,0.3398) + -- (0.7676,0.3494) + -- (0.7702,0.3590) + -- (0.7724,0.3688) + -- (0.7744,0.3786) + -- (0.7760,0.3885) + -- (0.7774,0.3984) + -- (0.7785,0.4083) + -- (0.7793,0.4183) + -- (0.7797,0.4283) + -- (0.7799,0.4383) + -- (0.7797,0.4483) + -- (0.7793,0.4582) + -- (0.7785,0.4682) + -- (0.7774,0.4782) + -- (0.7759,0.4880) + -- (0.7741,0.4979) + -- (0.7721,0.5077) + -- (0.7696,0.5174) + -- (0.7669,0.5270) + -- (0.7638,0.5365) + -- (0.7604,0.5459) + -- (0.7567,0.5552) + -- (0.7526,0.5643) + -- (0.7482,0.5733) + -- (0.7436,0.5821) + -- (0.7385,0.5908) + -- (0.7332,0.5993) + -- (0.7276,0.6075) + -- (0.7217,0.6156) + -- (0.7154,0.6234) + -- (0.7089,0.6310) + -- (0.7021,0.6383) + -- (0.6950,0.6454) + -- (0.6877,0.6522) + -- (0.6801,0.6587) + -- (0.6722,0.6648) + -- (0.6641,0.6707) + -- (0.6558,0.6763) + -- (0.6473,0.6815) + -- (0.6386,0.6863) + -- (0.6296,0.6908) + -- (0.6205,0.6950) + -- (0.6112,0.6987) + -- (0.6018,0.7021) + -- (0.5923,0.7050) + -- (0.5826,0.7076) + -- (0.5728,0.7097) + -- (0.5630,0.7114) + -- (0.5531,0.7127) + -- (0.5431,0.7135) + -- (0.5331,0.7139) + -- (0.5231,0.7139) + -- (0.5131,0.7134) + -- (0.5032,0.7125) + -- (0.4933,0.7111) + -- (0.4834,0.7093) + -- (0.4737,0.7070) + -- (0.4641,0.7043) + -- (0.4546,0.7011) + -- (0.4453,0.6975) + -- (0.4361,0.6935) + -- (0.4272,0.6890) + -- (0.4185,0.6841) + -- (0.4100,0.6788) + -- (0.4018,0.6731) + -- (0.3939,0.6670) + -- (0.3862,0.6605) + -- (0.3790,0.6536) + -- (0.3720,0.6464) + -- (0.3655,0.6389) + -- (0.3593,0.6310) + -- (0.3535,0.6229) + -- (0.3482,0.6144) + -- (0.3433,0.6057) + -- (0.3388,0.5968) + -- (0.3348,0.5876) + -- (0.3313,0.5782) + -- (0.3283,0.5687) + -- (0.3258,0.5590) + -- (0.3238,0.5492) + -- (0.3224,0.5393) + -- (0.3214,0.5293) + -- (0.3211,0.5194) + -- (0.3212,0.5094) + -- (0.3219,0.4994) + -- (0.3232,0.4895) + -- (0.3250,0.4796) + -- (0.3273,0.4699) + -- (0.3302,0.4603) + -- (0.3336,0.4509) + -- (0.3376,0.4418) + -- (0.3420,0.4328) + -- (0.3470,0.4241) + -- (0.3524,0.4157) + -- (0.3584,0.4077) + -- (0.3648,0.4000) + -- (0.3716,0.3927) + -- (0.3788,0.3858) + -- (0.3865,0.3793) + -- (0.3945,0.3733) + -- (0.4028,0.3678) + -- (0.4115,0.3629) + -- (0.4204,0.3584) + -- (0.4296,0.3545) + -- (0.4391,0.3511) + -- (0.4487,0.3484) + -- (0.4584,0.3462) + -- (0.4683,0.3447) + -- (0.4783,0.3437) + -- (0.4883,0.3434) + -- (0.4982,0.3437) + -- (0.5082,0.3447) + -- (0.5181,0.3462) + -- (0.5278,0.3484) + -- (0.5374,0.3513) + -- (0.5468,0.3547) + -- (0.5560,0.3587) + -- (0.5648,0.3633) + -- (0.5734,0.3685) + -- (0.5816,0.3743) + -- (0.5894,0.3805) + -- (0.5967,0.3873) + -- (0.6036,0.3945) + -- (0.6100,0.4022) + -- (0.6159,0.4103) + -- (0.6212,0.4188) + -- (0.6259,0.4276) + -- (0.6300,0.4367) + -- (0.6335,0.4461) + -- (0.6363,0.4557) + -- (0.6384,0.4655) + -- (0.6399,0.4754) + -- (0.6407,0.4853) + -- (0.6408,0.4953) + -- (0.6401,0.5053) + -- (0.6388,0.5152) + -- (0.6368,0.5250) + -- (0.6340,0.5346) + -- (0.6306,0.5440) + -- (0.6266,0.5532) + -- (0.6218,0.5620) + -- (0.6165,0.5704) + -- (0.6105,0.5784) + -- (0.6040,0.5860) + -- (0.5970,0.5931) + -- (0.5894,0.5996) + -- (0.5814,0.6056) + -- (0.5729,0.6110) + -- (0.5641,0.6157) + -- (0.5550,0.6197) + -- (0.5455,0.6230) + -- (0.5359,0.6256) + -- (0.5261,0.6275) + -- (0.5161,0.6286) + -- (0.5061,0.6289) + -- (0.4961,0.6285) + -- (0.4862,0.6273) + -- (0.4764,0.6254) + -- (0.4668,0.6226) + -- (0.4574,0.6192) + -- (0.4483,0.6150) + -- (0.4396,0.6101) + -- (0.4313,0.6045) + -- (0.4235,0.5983) + -- (0.4161,0.5915) + -- (0.4094,0.5842) + -- (0.4033,0.5763) + -- (0.3978,0.5679) + -- (0.3930,0.5591) + -- (0.3889,0.5500) + -- (0.3856,0.5406) + -- (0.3831,0.5309) + -- (0.3814,0.5210) + -- (0.3805,0.5111) + -- (0.3805,0.5011) + -- (0.3812,0.4911) + -- (0.3828,0.4812) + -- (0.3853,0.4715) + -- (0.3885,0.4621) + -- (0.3925,0.4529) + -- (0.3973,0.4441) + -- (0.4028,0.4358) + -- (0.4090,0.4279) + -- (0.4158,0.4207) + -- (0.4233,0.4140) + -- (0.4313,0.4080) + -- (0.4397,0.4027) + -- (0.4487,0.3982) + -- (0.4579,0.3944) + -- (0.4675,0.3915) + -- (0.4773,0.3895) + -- (0.4872,0.3883) + -- (0.4972,0.3880) + -- (0.5072,0.3886) + -- (0.5171,0.3900) + -- (0.5268,0.3924) + -- (0.5362,0.3956) + -- (0.5454,0.3996) + -- (0.5541,0.4045) + -- (0.5624,0.4101) + -- (0.5701,0.4165) + -- (0.5772,0.4235) + -- (0.5836,0.4312) + -- (0.5893,0.4394) + -- (0.5942,0.4481) + -- (0.5983,0.4572) + -- (0.6015,0.4667) + -- (0.6038,0.4764) + -- (0.6053,0.4863) + -- (0.6057,0.4963) + -- (0.6052,0.5063) + -- (0.6038,0.5162) + -- (0.6015,0.5259) + -- (0.5982,0.5354) + -- (0.5941,0.5445) + -- (0.5891,0.5531) + -- (0.5833,0.5613) + -- (0.5767,0.5688) + -- (0.5695,0.5757) + -- (0.5616,0.5818) + -- (0.5531,0.5872) + -- (0.5442,0.5917) + -- (0.5349,0.5952) + -- (0.5253,0.5979) + -- (0.5154,0.5996) + -- (0.5054,0.6003) + -- (0.4954,0.6001) + -- (0.4855,0.5988) + -- (0.4758,0.5966) + -- (0.4663,0.5933) + -- (0.4572,0.5892) + -- (0.4486,0.5842) + -- (0.4405,0.5783) + -- (0.4331,0.5716) + -- (0.4263,0.5642) + -- (0.4204,0.5562) + -- (0.4153,0.5476) + -- (0.4111,0.5385) + -- (0.4079,0.5290) + -- (0.4057,0.5193) + -- (0.4045,0.5094) + -- (0.4043,0.4994) + -- (0.4052,0.4894) + -- (0.4071,0.4796) + -- (0.4100,0.4700) + -- (0.4139,0.4608) + -- (0.4188,0.4521) + -- (0.4246,0.4439) + -- (0.4311,0.4364) + -- (0.4385,0.4296) + -- (0.4465,0.4237) + -- (0.4551,0.4186) + -- (0.4643,0.4145) + -- (0.4738,0.4114) + -- (0.4835,0.4094) + -- (0.4935,0.4084) + -- (0.5035,0.4085) + -- (0.5134,0.4097) + -- (0.5231,0.4119) + -- (0.5326,0.4152) + -- (0.5416,0.4196) + -- (0.5501,0.4249) + -- (0.5579,0.4311) + -- (0.5650,0.4381) + -- (0.5713,0.4459) + -- (0.5767,0.4543) + -- (0.5811,0.4633) + -- (0.5845,0.4727) + -- (0.5868,0.4824) + -- (0.5880,0.4923) + -- (0.5880,0.5023) + -- (0.5869,0.5122) + -- (0.5848,0.5220) + -- (0.5815,0.5314) + -- (0.5771,0.5404) + -- (0.5718,0.5489) + -- (0.5655,0.5567) + -- (0.5584,0.5637) + -- (0.5505,0.5698) + -- (0.5419,0.5750) + -- (0.5329,0.5791) + -- (0.5233,0.5822) + -- (0.5135,0.5841) + -- (0.5036,0.5849) + -- (0.4936,0.5845) + -- (0.4837,0.5830) + -- (0.4741,0.5803) + -- (0.4649,0.5764) + -- (0.4562,0.5715) + -- (0.4481,0.5656) + -- (0.4408,0.5588) + -- (0.4343,0.5512) + -- (0.4289,0.5428) + -- (0.4244,0.5338) + -- (0.4211,0.5244) + -- (0.4189,0.5147) + -- (0.4180,0.5047) + -- (0.4182,0.4947) + -- (0.4197,0.4848) + -- (0.4223,0.4752) + -- (0.4261,0.4660) + -- (0.4311,0.4573) + -- (0.4370,0.4492) + -- (0.4439,0.4420) + -- (0.4516,0.4357) + -- (0.4601,0.4303) + -- (0.4691,0.4261) + -- (0.4786,0.4230) + -- (0.4885,0.4211) + -- (0.4984,0.4205) + -- (0.5084,0.4211) + -- (0.5182,0.4230) + -- (0.5277,0.4261) + -- (0.5368,0.4304) + -- (0.5452,0.4358) + -- (0.5528,0.4422) + -- (0.5596,0.4495) + -- (0.5654,0.4576) + -- (0.5701,0.4665) + -- (0.5737,0.4758) + -- (0.5760,0.4855) + -- (0.5771,0.4955) + -- (0.5768,0.5054) + -- (0.5753,0.5153) + -- (0.5725,0.5249) + -- (0.5684,0.5341) + -- (0.5633,0.5426) + -- (0.5570,0.5504) + -- (0.5498,0.5573) + -- (0.5417,0.5632) + -- (0.5329,0.5680) + -- (0.5236,0.5716) + -- (0.5139,0.5739) + -- (0.5040,0.5749) + -- (0.4940,0.5746) + -- (0.4841,0.5730) + -- (0.4746,0.5700) + -- (0.4655,0.5658) + -- (0.4571,0.5604) + -- (0.4494,0.5540) + -- (0.4428,0.5466) + -- (0.4371,0.5383) + -- (0.4327,0.5294) + -- 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(-0.4817,-0.5650) + -- (-0.4723,-0.5615) + -- (-0.4636,-0.5566) + -- (-0.4557,-0.5504) + -- (-0.4488,-0.5432) + -- (-0.4431,-0.5350) + -- (-0.4386,-0.5261) + -- (-0.4355,-0.5166) + -- (-0.4339,-0.5067) + -- (-0.4338,-0.4968) + -- (-0.4352,-0.4869) + -- (-0.4380,-0.4773) + -- (-0.4423,-0.4682) + -- (-0.4479,-0.4600) + -- (-0.4546,-0.4526) + -- (-0.4624,-0.4464) + -- (-0.4711,-0.4414) + -- (-0.4804,-0.4378) + -- (-0.4902,-0.4357) + -- (-0.5002,-0.4351) + -- (-0.5101,-0.4360) + -- (-0.5198,-0.4384) + -- (-0.5290,-0.4423) + -- (-0.5375,-0.4476) + -- (-0.5450,-0.4541) + -- (-0.5515,-0.4618) + -- (-0.5567,-0.4703) + -- (-0.5605,-0.4795) + -- (-0.5628,-0.4892) + -- (-0.5636,-0.4992) + -- (-0.5628,-0.5092) + -- (-0.5605,-0.5189) + -- (-0.5567,-0.5281) + -- (-0.5514,-0.5366) + -- (-0.5449,-0.5442) + -- (-0.5373,-0.5506) + -- (-0.5288,-0.5558) + -- (-0.5195,-0.5595) + -- (-0.5098,-0.5617) + -- (-0.4998,-0.5624) + -- (-0.4898,-0.5614) + -- (-0.4802,-0.5589) + -- (-0.4710,-0.5549) + -- (-0.4627,-0.5494) + -- (-0.4553,-0.5427) + -- (-0.4491,-0.5348) + -- (-0.4443,-0.5261) + -- (-0.4409,-0.5167) + -- (-0.4391,-0.5069) + -- (-0.4389,-0.4969) + -- (-0.4403,-0.4870) + -- (-0.4434,-0.4775) + -- (-0.4479,-0.4686) + -- (-0.4538,-0.4605) + -- (-0.4610,-0.4536) + -- (-0.4692,-0.4479) + -- (-0.4783,-0.4437) + -- (-0.4879,-0.4410) + -- (-0.4978,-0.4399) + -- (-0.5078,-0.4405) + -- (-0.5175,-0.4427) + -- (-0.5268,-0.4465) + -- (-0.5352,-0.4518) + -- (-0.5427,-0.4584) + -- (-0.5490,-0.4662) + -- (-0.5538,-0.4749) + -- (-0.5572,-0.4844) + -- (-0.5589,-0.4942) + -- (-0.5589,-0.5042) + -- (-0.5572,-0.5140) + -- (-0.5539,-0.5235) + -- (-0.5491,-0.5322) + -- (-0.5428,-0.5400) + -- (-0.5354,-0.5466) + -- (-0.5269,-0.5518) + -- (-0.5176,-0.5556) + -- (-0.5078,-0.5576) + -- (-0.4978,-0.5580) + -- (-0.4879,-0.5567) + -- (-0.4784,-0.5537) + -- (-0.4696,-0.5491) + -- (-0.4616,-0.5430) + -- (-0.4548,-0.5357) + -- (-0.4494,-0.5273) + -- (-0.4456,-0.5181) + -- (-0.4434,-0.5083) + -- (-0.4429,-0.4984) + -- (-0.4442,-0.4884) + -- (-0.4471,-0.4789) + -- (-0.4517,-0.4700) + -- (-0.4578,-0.4621) + -- (-0.4652,-0.4554) + -- (-0.4736,-0.4500) + -- (-0.4828,-0.4462) + -- (-0.4926,-0.4442) + -- (-0.5026,-0.4438) + -- (-0.5125,-0.4453) + -- (-0.5219,-0.4484) + -- (-0.5307,-0.4532) + -- (-0.5385,-0.4595) + -- (-0.5450,-0.4671) + -- (-0.5500,-0.4757) + -- (-0.5535,-0.4851) + -- (-0.5552,-0.4949) + -- (-0.5551,-0.5049) + -- (-0.5533,-0.5147) + -- (-0.5496,-0.5240) + -- (-0.5444,-0.5325) + -- (-0.5377,-0.5400) + -- (-0.5298,-0.5460) + -- (-0.5210,-0.5506) + -- (-0.5114,-0.5535) + -- (-0.5015,-0.5546) + -- (-0.4915,-0.5538) + -- (-0.4818,-0.5513) + -- (-0.4728,-0.5470) + -- (-0.4647,-0.5412) + -- (-0.4578,-0.5339) + -- (-0.4524,-0.5256) + -- (-0.4486,-0.5163) + -- (-0.4466,-0.5066) + -- (-0.4464,-0.4966) + -- (-0.4480,-0.4867) + -- (-0.4514,-0.4774) + -- (-0.4566,-0.4688) + -- (-0.4632,-0.4613) + -- (-0.4711,-0.4552) + -- (-0.4800,-0.4507) + -- (-0.4896,-0.4479) + -- (-0.4995,-0.4470) + -- (-0.5095,-0.4479) + -- (-0.5191,-0.4507) + -- (-0.5280,-0.4552) + -- (-0.5358,-0.4614) + -- (-0.5424,-0.4689) + -- (-0.5475,-0.4775) + -- (-0.5508,-0.4869) + -- (-0.5522,-0.4968) + -- (-0.5518,-0.5068) + -- (-0.5495,-0.5165) + -- (-0.5454,-0.5256) + -- (-0.5396,-0.5337) + -- (-0.5324,-0.5406) + -- (-0.5239,-0.5460) + -- (-0.5147,-0.5496) + -- (-0.5048,-0.5514) + -- (-0.4949,-0.5513) + -- (-0.4851,-0.5493) + -- (-0.4759,-0.5454) + -- (-0.4676,-0.5398) + -- (-0.4606,-0.5327) + -- (-0.4550,-0.5244) + -- (-0.4512,-0.5152) + -- (-0.4493,-0.5054) + -- (-0.4493,-0.4954) + -- (-0.4512,-0.4856) + -- (-0.4551,-0.4764) + -- (-0.4606,-0.4681) + -- (-0.4677,-0.4611) + -- (-0.4760,-0.4555) + -- (-0.4852,-0.4518) + -- (-0.4951,-0.4499) + -- (-0.5050,-0.4500) + -- (-0.5148,-0.4520) + -- (-0.5240,-0.4560) + -- (-0.5322,-0.4617) + -- (-0.5391,-0.4689) + -- (-0.5444,-0.4773) + -- (-0.5480,-0.4866) + -- (-0.5496,-0.4965) + -- (-0.5492,-0.5065) + -- (-0.5469,-0.5162) + -- (-0.5426,-0.5252) + -- (-0.5366,-0.5332) + -- (-0.5292,-0.5398) + -- (-0.5205,-0.5448) + -- (-0.5110,-0.5479) + -- (-0.5011,-0.5491) + -- (-0.4912,-0.5482) + -- (-0.4816,-0.5454) + -- (-0.4728,-0.5406) + -- (-0.4652,-0.5342) + -- (-0.4590,-0.5264) + -- (-0.4546,-0.5174) + -- (-0.4520,-0.5078) + -- (-0.4515,-0.4978) + -- (-0.4531,-0.4879) + -- (-0.4566,-0.4786) + -- (-0.4620,-0.4702) + -- (-0.4690,-0.4631) + -- (-0.4773,-0.4575) + -- (-0.4866,-0.4538) + -- (-0.4964,-0.4521) + -- (-0.5064,-0.4525) + -- (-0.5161,-0.4549) + -- (-0.5250,-0.4593) + -- (-0.5329,-0.4654) + -- (-0.5393,-0.4731) + -- (-0.5440,-0.4819) + -- (-0.5467,-0.4915) + -- (-0.5474,-0.5015) + -- (-0.5460,-0.5113) + -- (-0.5425,-0.5207) + -- (-0.5372,-0.5291) + -- (-0.5302,-0.5362) + -- (-0.5218,-0.5417) + -- (-0.5125,-0.5453) + -- (-0.5027,-0.5469) + -- (-0.4927,-0.5463) + -- (-0.4831,-0.5436) + -- (-0.4742,-0.5390) + -- (-0.4666,-0.5326) + -- (-0.4605,-0.5247) + -- (-0.4562,-0.5157) + -- (-0.4539,-0.5060) + -- (-0.4538,-0.4960) + -- (-0.4558,-0.4862) + -- (-0.4598,-0.4771) + -- (-0.4657,-0.4690) + -- (-0.4732,-0.4624) + -- (-0.4820,-0.4576) + -- (-0.4915,-0.4548) + -- (-0.5015,-0.4541) + -- (-0.5114,-0.4556) + -- (-0.5207,-0.4591) + -- (-0.5290,-0.4646) + -- (-0.5359,-0.4718) + -- (-0.5411,-0.4803) + -- (-0.5444,-0.4898) + -- (-0.5455,-0.4997) + -- (-0.5444,-0.5096) + -- (-0.5411,-0.5191) + -- (-0.5359,-0.5276) + -- (-0.5290,-0.5347) + -- (-0.5206,-0.5402) + -- (-0.5112,-0.5437) + -- (-0.5014,-0.5450) + -- (-0.4914,-0.5441) + -- (-0.4819,-0.5411) + -- (-0.4733,-0.5360) + -- (-0.4660,-0.5292) + -- (-0.4605,-0.5209) + -- (-0.4569,-0.5116) + -- (-0.4555,-0.5017) + -- (-0.4563,-0.4918) + -- (-0.4592,-0.4822) + -- (-0.4643,-0.4736) + -- (-0.4711,-0.4664) + -- (-0.4794,-0.4608) + -- (-0.4887,-0.4573) + -- (-0.4986,-0.4559) + -- (-0.5086,-0.4568) + -- (-0.5181,-0.4599) + -- (-0.5266,-0.4650) + -- (-0.5338,-0.4719) + -- (-0.5392,-0.4803) + -- (-0.5426,-0.4897) + -- (-0.5437,-0.4996) + -- (-0.5426,-0.5095) + -- (-0.5393,-0.5189) + -- (-0.5339,-0.5273) + -- (-0.5267,-0.5343) + -- (-0.5182,-0.5394) + -- (-0.5087,-0.5425) + -- (-0.4987,-0.5433) + -- (-0.4889,-0.5418) + -- (-0.4796,-0.5381) + -- (-0.4714,-0.5323) + -- (-0.4648,-0.5249) + -- (-0.4601,-0.5161) + -- (-0.4575,-0.5064) + -- (-0.4573,-0.4965) + -- (-0.4593,-0.4867) + -- (-0.4635,-0.4777) + -- (-0.4697,-0.4698) + -- (-0.4776,-0.4637) + -- (-0.4867,-0.4595) + -- (-0.4964,-0.4576) + -- (-0.5064,-0.4580) + -- (-0.5160,-0.4607) + -- (-0.5247,-0.4656) + -- (-0.5320,-0.4724) + -- (-0.5376,-0.4807) + -- (-0.5410,-0.4900) + -- (-0.5422,-0.4999) + -- (-0.5409,-0.5098) + -- (-0.5374,-0.5192) + -- (-0.5318,-0.5274) + -- (-0.5244,-0.5341) + -- (-0.5156,-0.5389) + -- (-0.5060,-0.5414) + -- (-0.4960,-0.5416) + -- (-0.4863,-0.5394) + -- (-0.4773,-0.5350) + -- (-0.4697,-0.5285) + -- (-0.4638,-0.5205) + -- (-0.4601,-0.5112) + -- (-0.4586,-0.5014) + -- (-0.4595,-0.4914) + -- (-0.4628,-0.4820) + -- (-0.4682,-0.4737) + -- (-0.4755,-0.4668) + -- (-0.4842,-0.4620) + -- (-0.4938,-0.4593) + -- (-0.5038,-0.4591) + -- (-0.5135,-0.4613) + -- (-0.5225,-0.4657) + -- (-0.5300,-0.4722) + -- (-0.5358,-0.4804) + -- (-0.5395,-0.4896) + -- (-0.5408,-0.4995) + -- (-0.5396,-0.5094) + -- (-0.5361,-0.5188) + -- (-0.5304,-0.5270) + -- (-0.5228,-0.5335) + -- (-0.5139,-0.5380) + -- (-0.5042,-0.5402) + -- (-0.4942,-0.5400) + -- (-0.4846,-0.5373) + -- (-0.4760,-0.5323) + -- (-0.4688,-0.5254) + -- (-0.4636,-0.5169) + -- (-0.4606,-0.5074) + -- (-0.4600,-0.4975) + -- (-0.4619,-0.4877) + -- (-0.4662,-0.4787) + -- (-0.4726,-0.4710) + -- (-0.4806,-0.4651) + -- (-0.4899,-0.4615) + -- (-0.4998,-0.4602) + -- (-0.5097,-0.4615) + -- (-0.5190,-0.4651) + -- (-0.5270,-0.4710) + -- (-0.5334,-0.4787) + -- (-0.5376,-0.4877) + -- (-0.5394,-0.4976) + -- (-0.5387,-0.5075) + -- (-0.5356,-0.5170) + -- (-0.5301,-0.5253) + -- (-0.5227,-0.5320) + -- (-0.5139,-0.5367) + -- (-0.5042,-0.5390) + -- (-0.4942,-0.5387) + -- (-0.4847,-0.5360) + -- (-0.4761,-0.5309) + -- (-0.4691,-0.5238) + -- (-0.4641,-0.5151) + -- (-0.4615,-0.5055) + -- (-0.4614,-0.4955) + -- (-0.4638,-0.4859) + -- (-0.4687,-0.4772) + -- (-0.4756,-0.4700) + -- (-0.4841,-0.4648) + -- (-0.4936,-0.4619) + -- (-0.5036,-0.4616) + -- (-0.5133,-0.4638) + -- (-0.5221,-0.4685) + -- (-0.5294,-0.4753) + -- (-0.5348,-0.4837) + -- (-0.5377,-0.4932) + -- (-0.5382,-0.5032) + -- (-0.5360,-0.5129) + -- (-0.5314,-0.5218) + -- (-0.5247,-0.5291) + -- (-0.5163,-0.5345) + -- (-0.5067,-0.5375) + -- (-0.4968,-0.5379) + -- (-0.4870,-0.5357) + -- (-0.4782,-0.5311) + -- (-0.4709,-0.5243) + -- (-0.4656,-0.5158) + -- (-0.4627,-0.5063) + -- (-0.4624,-0.4963) + -- (-0.4647,-0.4866) + -- (-0.4695,-0.4779) + -- (-0.4764,-0.4707) + -- (-0.4850,-0.4656) + -- (-0.4946,-0.4629) + -- (-0.5045,-0.4628) + -- (-0.5142,-0.4653) + -- (-0.5228,-0.4703) + -- (-0.5298,-0.4775) + -- (-0.5347,-0.4862) + -- (-0.5370,-0.4958) + -- (-0.5368,-0.5058) + -- (-0.5339,-0.5153) + -- (-0.5285,-0.5238) + -- (-0.5212,-0.5305) + -- (-0.5123,-0.5350) + -- (-0.5025,-0.5369) + -- (-0.4925,-0.5362) + -- (-0.4831,-0.5329) + -- (-0.4750,-0.5271) + -- (-0.4687,-0.5194) + -- (-0.4647,-0.5103) + -- (-0.4632,-0.5004) + -- (-0.4645,-0.4905) + -- (-0.4684,-0.4813) + -- (-0.4747,-0.4736) + -- (-0.4827,-0.4677) + -- (-0.4921,-0.4643) + -- (-0.5020,-0.4636) + -- (-0.5118,-0.4655) + -- (-0.5207,-0.4700) + -- (-0.5280,-0.4768) + -- (-0.5332,-0.4853) + -- (-0.5359,-0.4949) + -- (-0.5359,-0.5049) + -- (-0.5332,-0.5145) + -- (-0.5280,-0.5229) + -- (-0.5206,-0.5297) + -- (-0.5117,-0.5341) + -- (-0.5019,-0.5360) + -- (-0.4920,-0.5351) + -- (-0.4827,-0.5315) + -- (-0.4747,-0.5255) + -- (-0.4687,-0.5176) + -- (-0.4651,-0.5083) + -- (-0.4642,-0.4983) + -- (-0.4661,-0.4886) + -- (-0.4706,-0.4797) + -- (-0.4774,-0.4724) + -- (-0.4860,-0.4672) + -- (-0.4956,-0.4647) + -- (-0.5056,-0.4649) + -- (-0.5151,-0.4678) + -- (-0.5234,-0.4733) + -- (-0.5299,-0.4809) + -- (-0.5340,-0.4900) + -- (-0.5354,-0.4999) + -- (-0.5340,-0.5097) + -- (-0.5299,-0.5188) + -- (-0.5234,-0.5264) + -- (-0.5150,-0.5318) + -- (-0.5055,-0.5347) + -- (-0.4955,-0.5348) + -- (-0.4859,-0.5322) + -- (-0.4775,-0.5269) + -- (-0.4709,-0.5194) + -- (-0.4666,-0.5104) + -- (-0.4651,-0.5006) + -- (-0.4664,-0.4907) + -- (-0.4704,-0.4816) + -- (-0.4769,-0.4740) + -- (-0.4852,-0.4685) + -- (-0.4947,-0.4657) + -- (-0.5047,-0.4656) + -- (-0.5143,-0.4684) + -- (-0.5226,-0.4738) + -- (-0.5291,-0.4814) + -- (-0.5332,-0.4904) + -- (-0.5345,-0.5003) + -- (-0.5330,-0.5102) + -- (-0.5286,-0.5191) + -- (-0.5219,-0.5265) + -- (-0.5134,-0.5317) + -- (-0.5037,-0.5341) + -- (-0.4938,-0.5337) + -- (-0.4844,-0.5305) + -- (-0.4763,-0.5247) + -- (-0.4702,-0.5168) + -- (-0.4667,-0.5074) + -- (-0.4660,-0.4975) + -- (-0.4682,-0.4878) + -- (-0.4731,-0.4791) + -- (-0.4803,-0.4723) + -- (-0.4892,-0.4678) + -- (-0.4990,-0.4661) + -- (-0.5089,-0.4673) + -- (-0.5180,-0.4713) + -- (-0.5256,-0.4779) + -- (-0.5309,-0.4863) + -- (-0.5335,-0.4959) + -- (-0.5332,-0.5059) + -- (-0.5300,-0.5153) + -- (-0.5242,-0.5234) + -- (-0.5163,-0.5294) + -- (-0.5069,-0.5329) + -- (-0.4970,-0.5334) + -- (-0.4873,-0.5310) + -- (-0.4788,-0.5259) + -- (-0.4721,-0.5184) + -- (-0.4679,-0.5094) + -- (-0.4666,-0.4995) + -- (-0.4683,-0.4897) + -- (-0.4728,-0.4808) + -- (-0.4797,-0.4736) + -- (-0.4885,-0.4688) + -- (-0.4982,-0.4669) + -- (-0.5081,-0.4679) + -- (-0.5173,-0.4718) + -- (-0.5249,-0.4782) + -- (-0.5302,-0.4866) + -- (-0.5328,-0.4962) + -- (-0.5324,-0.5062) + -- (-0.5290,-0.5156) + -- (-0.5230,-0.5235) + -- (-0.5149,-0.5293) + -- (-0.5054,-0.5324) + -- (-0.4955,-0.5325) + -- (-0.4860,-0.5296) + -- (-0.4777,-0.5240) + -- (-0.4716,-0.5162) + -- (-0.4680,-0.5069) + -- (-0.4675,-0.4969) + -- (-0.4700,-0.4873) + -- (-0.4753,-0.4788) + -- (-0.4828,-0.4723) + -- (-0.4920,-0.4685) + -- (-0.5019,-0.4676) + -- (-0.5117,-0.4697) + -- (-0.5203,-0.4747) + -- (-0.5270,-0.4821) + -- (-0.5311,-0.4911) + -- (-0.5323,-0.5010) + -- (-0.5304,-0.5108) + -- (-0.5256,-0.5196) + -- (-0.5184,-0.5264) + -- (-0.5095,-0.5308) + -- (-0.4996,-0.5321) + -- (-0.4898,-0.5305) + -- (-0.4810,-0.5258) + -- (-0.4740,-0.5187) + -- (-0.4695,-0.5098) + -- (-0.4680,-0.5000) + -- (-0.4696,-0.4901) + -- (-0.4741,-0.4813) + -- (-0.4812,-0.4742) + -- (-0.4901,-0.4697) +} + +\def\Cplotright{ (0,0) + -- ({0.0100*\dx},{0.0100*\dy}) + -- ({0.0200*\dx},{0.0200*\dy}) + -- ({0.0300*\dx},{0.0300*\dy}) + -- ({0.0400*\dx},{0.0400*\dy}) + -- ({0.0500*\dx},{0.0500*\dy}) + -- ({0.0600*\dx},{0.0600*\dy}) + -- ({0.0700*\dx},{0.0700*\dy}) + -- ({0.0800*\dx},{0.0800*\dy}) + -- ({0.0900*\dx},{0.0900*\dy}) + -- ({0.1000*\dx},{0.1000*\dy}) + -- ({0.1100*\dx},{0.1100*\dy}) + -- ({0.1200*\dx},{0.1200*\dy}) + -- ({0.1300*\dx},{0.1300*\dy}) + -- ({0.1400*\dx},{0.1400*\dy}) + -- ({0.1500*\dx},{0.1500*\dy}) + -- ({0.1600*\dx},{0.1600*\dy}) + -- ({0.1700*\dx},{0.1700*\dy}) + -- ({0.1800*\dx},{0.1800*\dy}) + -- ({0.1900*\dx},{0.1899*\dy}) + -- ({0.2000*\dx},{0.1999*\dy}) + -- ({0.2100*\dx},{0.2099*\dy}) + -- ({0.2200*\dx},{0.2199*\dy}) + -- ({0.2300*\dx},{0.2298*\dy}) + -- ({0.2400*\dx},{0.2398*\dy}) + -- ({0.2500*\dx},{0.2498*\dy}) + -- ({0.2600*\dx},{0.2597*\dy}) + -- ({0.2700*\dx},{0.2696*\dy}) + -- ({0.2800*\dx},{0.2796*\dy}) + -- ({0.2900*\dx},{0.2895*\dy}) + -- ({0.3000*\dx},{0.2994*\dy}) + -- ({0.3100*\dx},{0.3093*\dy}) + -- ({0.3200*\dx},{0.3192*\dy}) + -- ({0.3300*\dx},{0.3290*\dy}) + -- ({0.3400*\dx},{0.3389*\dy}) + -- ({0.3500*\dx},{0.3487*\dy}) + -- ({0.3600*\dx},{0.3585*\dy}) + -- ({0.3700*\dx},{0.3683*\dy}) + -- ({0.3800*\dx},{0.3780*\dy}) + -- ({0.3900*\dx},{0.3878*\dy}) + -- ({0.4000*\dx},{0.3975*\dy}) + -- ({0.4100*\dx},{0.4072*\dy}) + -- ({0.4200*\dx},{0.4168*\dy}) + -- ({0.4300*\dx},{0.4264*\dy}) + -- ({0.4400*\dx},{0.4359*\dy}) + -- ({0.4500*\dx},{0.4455*\dy}) + -- ({0.4600*\dx},{0.4549*\dy}) + -- ({0.4700*\dx},{0.4644*\dy}) + -- ({0.4800*\dx},{0.4738*\dy}) + -- ({0.4900*\dx},{0.4831*\dy}) + -- ({0.5000*\dx},{0.4923*\dy}) + -- ({0.5100*\dx},{0.5016*\dy}) + -- ({0.5200*\dx},{0.5107*\dy}) + -- ({0.5300*\dx},{0.5198*\dy}) + -- ({0.5400*\dx},{0.5288*\dy}) + -- ({0.5500*\dx},{0.5377*\dy}) + -- ({0.5600*\dx},{0.5466*\dy}) + -- ({0.5700*\dx},{0.5553*\dy}) + -- ({0.5800*\dx},{0.5640*\dy}) + -- ({0.5900*\dx},{0.5726*\dy}) + -- ({0.6000*\dx},{0.5811*\dy}) + -- ({0.6100*\dx},{0.5895*\dy}) + -- ({0.6200*\dx},{0.5978*\dy}) + -- ({0.6300*\dx},{0.6059*\dy}) + -- ({0.6400*\dx},{0.6140*\dy}) + -- ({0.6500*\dx},{0.6219*\dy}) + -- ({0.6600*\dx},{0.6298*\dy}) + -- ({0.6700*\dx},{0.6374*\dy}) + -- ({0.6800*\dx},{0.6450*\dy}) + -- ({0.6900*\dx},{0.6524*\dy}) + -- ({0.7000*\dx},{0.6597*\dy}) + -- ({0.7100*\dx},{0.6668*\dy}) + -- ({0.7200*\dx},{0.6737*\dy}) + -- ({0.7300*\dx},{0.6805*\dy}) + -- ({0.7400*\dx},{0.6871*\dy}) + -- ({0.7500*\dx},{0.6935*\dy}) + -- ({0.7600*\dx},{0.6998*\dy}) + -- ({0.7700*\dx},{0.7058*\dy}) + -- ({0.7800*\dx},{0.7117*\dy}) + -- ({0.7900*\dx},{0.7174*\dy}) + -- ({0.8000*\dx},{0.7228*\dy}) + -- ({0.8100*\dx},{0.7281*\dy}) + -- ({0.8200*\dx},{0.7331*\dy}) + -- ({0.8300*\dx},{0.7379*\dy}) + -- ({0.8400*\dx},{0.7425*\dy}) + -- ({0.8500*\dx},{0.7469*\dy}) + -- ({0.8600*\dx},{0.7510*\dy}) + -- ({0.8700*\dx},{0.7548*\dy}) + -- ({0.8800*\dx},{0.7584*\dy}) + -- ({0.8900*\dx},{0.7617*\dy}) + -- ({0.9000*\dx},{0.7648*\dy}) + -- ({0.9100*\dx},{0.7676*\dy}) + -- ({0.9200*\dx},{0.7702*\dy}) + -- ({0.9300*\dx},{0.7724*\dy}) + -- ({0.9400*\dx},{0.7744*\dy}) + -- ({0.9500*\dx},{0.7760*\dy}) + -- ({0.9600*\dx},{0.7774*\dy}) + -- ({0.9700*\dx},{0.7785*\dy}) + -- ({0.9800*\dx},{0.7793*\dy}) + -- ({0.9900*\dx},{0.7797*\dy}) + -- ({1.0000*\dx},{0.7799*\dy}) + -- ({1.0100*\dx},{0.7797*\dy}) + -- ({1.0200*\dx},{0.7793*\dy}) + -- ({1.0300*\dx},{0.7785*\dy}) + -- ({1.0400*\dx},{0.7774*\dy}) + -- ({1.0500*\dx},{0.7759*\dy}) + -- ({1.0600*\dx},{0.7741*\dy}) + -- ({1.0700*\dx},{0.7721*\dy}) + -- ({1.0800*\dx},{0.7696*\dy}) + -- ({1.0900*\dx},{0.7669*\dy}) + -- ({1.1000*\dx},{0.7638*\dy}) + -- ({1.1100*\dx},{0.7604*\dy}) + -- ({1.1200*\dx},{0.7567*\dy}) + -- ({1.1300*\dx},{0.7526*\dy}) + -- ({1.1400*\dx},{0.7482*\dy}) + -- ({1.1500*\dx},{0.7436*\dy}) + -- ({1.1600*\dx},{0.7385*\dy}) + -- ({1.1700*\dx},{0.7332*\dy}) + -- ({1.1800*\dx},{0.7276*\dy}) + -- ({1.1900*\dx},{0.7217*\dy}) + -- ({1.2000*\dx},{0.7154*\dy}) + -- ({1.2100*\dx},{0.7089*\dy}) + -- ({1.2200*\dx},{0.7021*\dy}) + -- ({1.2300*\dx},{0.6950*\dy}) + -- ({1.2400*\dx},{0.6877*\dy}) + -- ({1.2500*\dx},{0.6801*\dy}) + -- ({1.2600*\dx},{0.6722*\dy}) + -- ({1.2700*\dx},{0.6641*\dy}) + -- ({1.2800*\dx},{0.6558*\dy}) + -- ({1.2900*\dx},{0.6473*\dy}) + -- ({1.3000*\dx},{0.6386*\dy}) + -- ({1.3100*\dx},{0.6296*\dy}) + -- ({1.3200*\dx},{0.6205*\dy}) + -- ({1.3300*\dx},{0.6112*\dy}) + -- ({1.3400*\dx},{0.6018*\dy}) + -- ({1.3500*\dx},{0.5923*\dy}) + -- ({1.3600*\dx},{0.5826*\dy}) + -- ({1.3700*\dx},{0.5728*\dy}) + -- ({1.3800*\dx},{0.5630*\dy}) + -- ({1.3900*\dx},{0.5531*\dy}) + -- ({1.4000*\dx},{0.5431*\dy}) + -- ({1.4100*\dx},{0.5331*\dy}) + -- ({1.4200*\dx},{0.5231*\dy}) + -- ({1.4300*\dx},{0.5131*\dy}) + -- ({1.4400*\dx},{0.5032*\dy}) + -- ({1.4500*\dx},{0.4933*\dy}) + -- ({1.4600*\dx},{0.4834*\dy}) + -- ({1.4700*\dx},{0.4737*\dy}) + -- ({1.4800*\dx},{0.4641*\dy}) + -- ({1.4900*\dx},{0.4546*\dy}) + -- ({1.5000*\dx},{0.4453*\dy}) + -- ({1.5100*\dx},{0.4361*\dy}) + -- ({1.5200*\dx},{0.4272*\dy}) + -- ({1.5300*\dx},{0.4185*\dy}) + -- ({1.5400*\dx},{0.4100*\dy}) + -- ({1.5500*\dx},{0.4018*\dy}) + -- ({1.5600*\dx},{0.3939*\dy}) + -- ({1.5700*\dx},{0.3862*\dy}) + -- ({1.5800*\dx},{0.3790*\dy}) + -- ({1.5900*\dx},{0.3720*\dy}) + -- ({1.6000*\dx},{0.3655*\dy}) + -- ({1.6100*\dx},{0.3593*\dy}) + -- ({1.6200*\dx},{0.3535*\dy}) + -- ({1.6300*\dx},{0.3482*\dy}) + -- ({1.6400*\dx},{0.3433*\dy}) + -- ({1.6500*\dx},{0.3388*\dy}) + -- ({1.6600*\dx},{0.3348*\dy}) + -- ({1.6700*\dx},{0.3313*\dy}) + -- ({1.6800*\dx},{0.3283*\dy}) + -- ({1.6900*\dx},{0.3258*\dy}) + -- ({1.7000*\dx},{0.3238*\dy}) + -- ({1.7100*\dx},{0.3224*\dy}) + -- ({1.7200*\dx},{0.3214*\dy}) + -- ({1.7300*\dx},{0.3211*\dy}) + -- ({1.7400*\dx},{0.3212*\dy}) + -- ({1.7500*\dx},{0.3219*\dy}) + -- ({1.7600*\dx},{0.3232*\dy}) + -- ({1.7700*\dx},{0.3250*\dy}) + -- ({1.7800*\dx},{0.3273*\dy}) + -- ({1.7900*\dx},{0.3302*\dy}) + -- ({1.8000*\dx},{0.3336*\dy}) + -- ({1.8100*\dx},{0.3376*\dy}) + -- ({1.8200*\dx},{0.3420*\dy}) + -- ({1.8300*\dx},{0.3470*\dy}) + -- ({1.8400*\dx},{0.3524*\dy}) + -- ({1.8500*\dx},{0.3584*\dy}) + -- ({1.8600*\dx},{0.3648*\dy}) + -- ({1.8700*\dx},{0.3716*\dy}) + -- ({1.8800*\dx},{0.3788*\dy}) + -- ({1.8900*\dx},{0.3865*\dy}) + -- ({1.9000*\dx},{0.3945*\dy}) + -- ({1.9100*\dx},{0.4028*\dy}) + -- ({1.9200*\dx},{0.4115*\dy}) + -- ({1.9300*\dx},{0.4204*\dy}) + -- ({1.9400*\dx},{0.4296*\dy}) + -- ({1.9500*\dx},{0.4391*\dy}) + -- ({1.9600*\dx},{0.4487*\dy}) + -- ({1.9700*\dx},{0.4584*\dy}) + -- ({1.9800*\dx},{0.4683*\dy}) + -- ({1.9900*\dx},{0.4783*\dy}) + -- ({2.0000*\dx},{0.4883*\dy}) + -- ({2.0100*\dx},{0.4982*\dy}) + -- ({2.0200*\dx},{0.5082*\dy}) + -- ({2.0300*\dx},{0.5181*\dy}) + -- ({2.0400*\dx},{0.5278*\dy}) + -- ({2.0500*\dx},{0.5374*\dy}) + -- ({2.0600*\dx},{0.5468*\dy}) + -- ({2.0700*\dx},{0.5560*\dy}) + -- ({2.0800*\dx},{0.5648*\dy}) + -- ({2.0900*\dx},{0.5734*\dy}) + -- ({2.1000*\dx},{0.5816*\dy}) + -- ({2.1100*\dx},{0.5894*\dy}) + -- ({2.1200*\dx},{0.5967*\dy}) + -- ({2.1300*\dx},{0.6036*\dy}) + -- ({2.1400*\dx},{0.6100*\dy}) + -- ({2.1500*\dx},{0.6159*\dy}) + -- ({2.1600*\dx},{0.6212*\dy}) + -- ({2.1700*\dx},{0.6259*\dy}) + -- ({2.1800*\dx},{0.6300*\dy}) + -- ({2.1900*\dx},{0.6335*\dy}) + -- ({2.2000*\dx},{0.6363*\dy}) + -- ({2.2100*\dx},{0.6384*\dy}) + -- ({2.2200*\dx},{0.6399*\dy}) + -- ({2.2300*\dx},{0.6407*\dy}) + -- ({2.2400*\dx},{0.6408*\dy}) + -- ({2.2500*\dx},{0.6401*\dy}) + -- ({2.2600*\dx},{0.6388*\dy}) + -- ({2.2700*\dx},{0.6368*\dy}) + -- ({2.2800*\dx},{0.6340*\dy}) + -- ({2.2900*\dx},{0.6306*\dy}) + -- ({2.3000*\dx},{0.6266*\dy}) + -- ({2.3100*\dx},{0.6218*\dy}) + -- ({2.3200*\dx},{0.6165*\dy}) + -- ({2.3300*\dx},{0.6105*\dy}) + -- ({2.3400*\dx},{0.6040*\dy}) + -- ({2.3500*\dx},{0.5970*\dy}) + -- ({2.3600*\dx},{0.5894*\dy}) + -- ({2.3700*\dx},{0.5814*\dy}) + -- ({2.3800*\dx},{0.5729*\dy}) + -- ({2.3900*\dx},{0.5641*\dy}) + -- ({2.4000*\dx},{0.5550*\dy}) + -- ({2.4100*\dx},{0.5455*\dy}) + -- ({2.4200*\dx},{0.5359*\dy}) + -- ({2.4300*\dx},{0.5261*\dy}) + -- ({2.4400*\dx},{0.5161*\dy}) + -- ({2.4500*\dx},{0.5061*\dy}) + -- ({2.4600*\dx},{0.4961*\dy}) + -- ({2.4700*\dx},{0.4862*\dy}) + -- ({2.4800*\dx},{0.4764*\dy}) + -- ({2.4900*\dx},{0.4668*\dy}) + -- ({2.5000*\dx},{0.4574*\dy}) + -- ({2.5100*\dx},{0.4483*\dy}) + -- ({2.5200*\dx},{0.4396*\dy}) + -- ({2.5300*\dx},{0.4313*\dy}) + -- ({2.5400*\dx},{0.4235*\dy}) + -- ({2.5500*\dx},{0.4161*\dy}) + -- ({2.5600*\dx},{0.4094*\dy}) + -- ({2.5700*\dx},{0.4033*\dy}) + -- ({2.5800*\dx},{0.3978*\dy}) + -- ({2.5900*\dx},{0.3930*\dy}) + -- ({2.6000*\dx},{0.3889*\dy}) + -- ({2.6100*\dx},{0.3856*\dy}) + -- ({2.6200*\dx},{0.3831*\dy}) + -- ({2.6300*\dx},{0.3814*\dy}) + -- ({2.6400*\dx},{0.3805*\dy}) + -- ({2.6500*\dx},{0.3805*\dy}) + -- ({2.6600*\dx},{0.3812*\dy}) + -- ({2.6700*\dx},{0.3828*\dy}) + -- ({2.6800*\dx},{0.3853*\dy}) + -- ({2.6900*\dx},{0.3885*\dy}) + -- ({2.7000*\dx},{0.3925*\dy}) + -- ({2.7100*\dx},{0.3973*\dy}) + -- ({2.7200*\dx},{0.4028*\dy}) + -- ({2.7300*\dx},{0.4090*\dy}) + -- ({2.7400*\dx},{0.4158*\dy}) + -- ({2.7500*\dx},{0.4233*\dy}) + -- ({2.7600*\dx},{0.4313*\dy}) + -- ({2.7700*\dx},{0.4397*\dy}) + -- ({2.7800*\dx},{0.4487*\dy}) + -- ({2.7900*\dx},{0.4579*\dy}) + -- ({2.8000*\dx},{0.4675*\dy}) + -- ({2.8100*\dx},{0.4773*\dy}) + -- ({2.8200*\dx},{0.4872*\dy}) + -- ({2.8300*\dx},{0.4972*\dy}) + -- ({2.8400*\dx},{0.5072*\dy}) + -- ({2.8500*\dx},{0.5171*\dy}) + -- ({2.8600*\dx},{0.5268*\dy}) + -- ({2.8700*\dx},{0.5362*\dy}) + -- ({2.8800*\dx},{0.5454*\dy}) + -- ({2.8900*\dx},{0.5541*\dy}) + -- ({2.9000*\dx},{0.5624*\dy}) + -- ({2.9100*\dx},{0.5701*\dy}) + -- ({2.9200*\dx},{0.5772*\dy}) + -- ({2.9300*\dx},{0.5836*\dy}) + -- ({2.9400*\dx},{0.5893*\dy}) + -- ({2.9500*\dx},{0.5942*\dy}) + -- ({2.9600*\dx},{0.5983*\dy}) + -- ({2.9700*\dx},{0.6015*\dy}) + -- ({2.9800*\dx},{0.6038*\dy}) + -- ({2.9900*\dx},{0.6053*\dy}) + -- ({3.0000*\dx},{0.6057*\dy}) + -- ({3.0100*\dx},{0.6052*\dy}) + -- ({3.0200*\dx},{0.6038*\dy}) + -- ({3.0300*\dx},{0.6015*\dy}) + -- ({3.0400*\dx},{0.5982*\dy}) + -- ({3.0500*\dx},{0.5941*\dy}) + -- ({3.0600*\dx},{0.5891*\dy}) + -- ({3.0700*\dx},{0.5833*\dy}) + -- ({3.0800*\dx},{0.5767*\dy}) + -- ({3.0900*\dx},{0.5695*\dy}) + -- ({3.1000*\dx},{0.5616*\dy}) + -- ({3.1100*\dx},{0.5531*\dy}) + -- ({3.1200*\dx},{0.5442*\dy}) + -- ({3.1300*\dx},{0.5349*\dy}) + -- ({3.1400*\dx},{0.5253*\dy}) + -- ({3.1500*\dx},{0.5154*\dy}) + -- ({3.1600*\dx},{0.5054*\dy}) + -- ({3.1700*\dx},{0.4954*\dy}) + -- ({3.1800*\dx},{0.4855*\dy}) + -- ({3.1900*\dx},{0.4758*\dy}) + -- ({3.2000*\dx},{0.4663*\dy}) + -- ({3.2100*\dx},{0.4572*\dy}) + -- ({3.2200*\dx},{0.4486*\dy}) + -- ({3.2300*\dx},{0.4405*\dy}) + -- ({3.2400*\dx},{0.4331*\dy}) + -- ({3.2500*\dx},{0.4263*\dy}) + -- ({3.2600*\dx},{0.4204*\dy}) + -- ({3.2700*\dx},{0.4153*\dy}) + -- ({3.2800*\dx},{0.4111*\dy}) + -- ({3.2900*\dx},{0.4079*\dy}) + -- ({3.3000*\dx},{0.4057*\dy}) + -- ({3.3100*\dx},{0.4045*\dy}) + -- ({3.3200*\dx},{0.4043*\dy}) + -- ({3.3300*\dx},{0.4052*\dy}) + -- ({3.3400*\dx},{0.4071*\dy}) + -- ({3.3500*\dx},{0.4100*\dy}) + -- ({3.3600*\dx},{0.4139*\dy}) + -- ({3.3700*\dx},{0.4188*\dy}) + -- ({3.3800*\dx},{0.4246*\dy}) + -- ({3.3900*\dx},{0.4311*\dy}) + -- ({3.4000*\dx},{0.4385*\dy}) + -- ({3.4100*\dx},{0.4465*\dy}) + -- ({3.4200*\dx},{0.4551*\dy}) + -- ({3.4300*\dx},{0.4643*\dy}) + -- ({3.4400*\dx},{0.4738*\dy}) + -- ({3.4500*\dx},{0.4835*\dy}) + -- ({3.4600*\dx},{0.4935*\dy}) + -- ({3.4700*\dx},{0.5035*\dy}) + -- ({3.4800*\dx},{0.5134*\dy}) + -- ({3.4900*\dx},{0.5231*\dy}) + -- ({3.5000*\dx},{0.5326*\dy}) + -- ({3.5100*\dx},{0.5416*\dy}) + -- ({3.5200*\dx},{0.5501*\dy}) + -- ({3.5300*\dx},{0.5579*\dy}) + -- ({3.5400*\dx},{0.5650*\dy}) + -- ({3.5500*\dx},{0.5713*\dy}) + -- ({3.5600*\dx},{0.5767*\dy}) + -- ({3.5700*\dx},{0.5811*\dy}) + -- ({3.5800*\dx},{0.5845*\dy}) + -- ({3.5900*\dx},{0.5868*\dy}) + -- ({3.6000*\dx},{0.5880*\dy}) + -- ({3.6100*\dx},{0.5880*\dy}) + -- ({3.6200*\dx},{0.5869*\dy}) + -- ({3.6300*\dx},{0.5848*\dy}) + -- ({3.6400*\dx},{0.5815*\dy}) + -- ({3.6500*\dx},{0.5771*\dy}) + -- ({3.6600*\dx},{0.5718*\dy}) + -- ({3.6700*\dx},{0.5655*\dy}) + -- ({3.6800*\dx},{0.5584*\dy}) + -- ({3.6900*\dx},{0.5505*\dy}) + -- ({3.7000*\dx},{0.5419*\dy}) + -- ({3.7100*\dx},{0.5329*\dy}) + -- ({3.7200*\dx},{0.5233*\dy}) + -- ({3.7300*\dx},{0.5135*\dy}) + -- ({3.7400*\dx},{0.5036*\dy}) + -- ({3.7500*\dx},{0.4936*\dy}) + -- ({3.7600*\dx},{0.4837*\dy}) + -- ({3.7700*\dx},{0.4741*\dy}) + -- ({3.7800*\dx},{0.4649*\dy}) + -- ({3.7900*\dx},{0.4562*\dy}) + -- ({3.8000*\dx},{0.4481*\dy}) + -- ({3.8100*\dx},{0.4408*\dy}) + -- ({3.8200*\dx},{0.4343*\dy}) + -- ({3.8300*\dx},{0.4289*\dy}) + -- ({3.8400*\dx},{0.4244*\dy}) + -- ({3.8500*\dx},{0.4211*\dy}) + -- ({3.8600*\dx},{0.4189*\dy}) + -- ({3.8700*\dx},{0.4180*\dy}) + -- ({3.8800*\dx},{0.4182*\dy}) + -- ({3.8900*\dx},{0.4197*\dy}) + -- ({3.9000*\dx},{0.4223*\dy}) + -- ({3.9100*\dx},{0.4261*\dy}) + -- ({3.9200*\dx},{0.4311*\dy}) + -- ({3.9300*\dx},{0.4370*\dy}) + -- ({3.9400*\dx},{0.4439*\dy}) + -- ({3.9500*\dx},{0.4516*\dy}) + -- ({3.9600*\dx},{0.4601*\dy}) + -- ({3.9700*\dx},{0.4691*\dy}) + -- ({3.9800*\dx},{0.4786*\dy}) + -- ({3.9900*\dx},{0.4885*\dy}) + -- ({4.0000*\dx},{0.4984*\dy}) + -- ({4.0100*\dx},{0.5084*\dy}) + -- ({4.0200*\dx},{0.5182*\dy}) + -- ({4.0300*\dx},{0.5277*\dy}) + -- ({4.0400*\dx},{0.5368*\dy}) + -- ({4.0500*\dx},{0.5452*\dy}) + -- ({4.0600*\dx},{0.5528*\dy}) + -- ({4.0700*\dx},{0.5596*\dy}) + -- ({4.0800*\dx},{0.5654*\dy}) + -- ({4.0900*\dx},{0.5701*\dy}) + -- ({4.1000*\dx},{0.5737*\dy}) + -- ({4.1100*\dx},{0.5760*\dy}) + -- ({4.1200*\dx},{0.5771*\dy}) + -- ({4.1300*\dx},{0.5768*\dy}) + -- ({4.1400*\dx},{0.5753*\dy}) + -- ({4.1500*\dx},{0.5725*\dy}) + -- ({4.1600*\dx},{0.5684*\dy}) + -- ({4.1700*\dx},{0.5633*\dy}) + -- ({4.1800*\dx},{0.5570*\dy}) + -- ({4.1900*\dx},{0.5498*\dy}) + -- ({4.2000*\dx},{0.5417*\dy}) + -- ({4.2100*\dx},{0.5329*\dy}) + -- ({4.2200*\dx},{0.5236*\dy}) + -- ({4.2300*\dx},{0.5139*\dy}) + -- ({4.2400*\dx},{0.5040*\dy}) + -- ({4.2500*\dx},{0.4940*\dy}) + -- ({4.2600*\dx},{0.4841*\dy}) + -- ({4.2700*\dx},{0.4746*\dy}) + -- ({4.2800*\dx},{0.4655*\dy}) + -- ({4.2900*\dx},{0.4571*\dy}) + -- ({4.3000*\dx},{0.4494*\dy}) + -- ({4.3100*\dx},{0.4428*\dy}) + -- ({4.3200*\dx},{0.4371*\dy}) + -- ({4.3300*\dx},{0.4327*\dy}) + -- ({4.3400*\dx},{0.4295*\dy}) + -- ({4.3500*\dx},{0.4276*\dy}) + -- ({4.3600*\dx},{0.4270*\dy}) + -- ({4.3700*\dx},{0.4279*\dy}) + -- ({4.3800*\dx},{0.4301*\dy}) + -- ({4.3900*\dx},{0.4336*\dy}) + -- ({4.4000*\dx},{0.4383*\dy}) + -- ({4.4100*\dx},{0.4443*\dy}) + -- ({4.4200*\dx},{0.4512*\dy}) + -- ({4.4300*\dx},{0.4591*\dy}) + -- ({4.4400*\dx},{0.4678*\dy}) + -- ({4.4500*\dx},{0.4771*\dy}) + -- ({4.4600*\dx},{0.4868*\dy}) + -- ({4.4700*\dx},{0.4967*\dy}) + -- ({4.4800*\dx},{0.5067*\dy}) + -- ({4.4900*\dx},{0.5165*\dy}) + -- ({4.5000*\dx},{0.5260*\dy}) + -- ({4.5100*\dx},{0.5350*\dy}) + -- ({4.5200*\dx},{0.5432*\dy}) + -- ({4.5300*\dx},{0.5505*\dy}) + -- ({4.5400*\dx},{0.5568*\dy}) + -- ({4.5500*\dx},{0.5619*\dy}) + -- ({4.5600*\dx},{0.5658*\dy}) + -- ({4.5700*\dx},{0.5683*\dy}) + -- ({4.5800*\dx},{0.5694*\dy}) + -- ({4.5900*\dx},{0.5690*\dy}) + -- ({4.6000*\dx},{0.5672*\dy}) + -- ({4.6100*\dx},{0.5641*\dy}) + -- ({4.6200*\dx},{0.5595*\dy}) + -- ({4.6300*\dx},{0.5538*\dy}) + -- ({4.6400*\dx},{0.5469*\dy}) + -- ({4.6500*\dx},{0.5391*\dy}) + -- ({4.6600*\dx},{0.5304*\dy}) + -- ({4.6700*\dx},{0.5211*\dy}) + -- ({4.6800*\dx},{0.5114*\dy}) + -- ({4.6900*\dx},{0.5014*\dy}) + -- ({4.7000*\dx},{0.4914*\dy}) + -- ({4.7100*\dx},{0.4817*\dy}) + -- ({4.7200*\dx},{0.4723*\dy}) + -- ({4.7300*\dx},{0.4636*\dy}) + -- ({4.7400*\dx},{0.4557*\dy}) + -- ({4.7500*\dx},{0.4488*\dy}) + -- ({4.7600*\dx},{0.4431*\dy}) + -- ({4.7700*\dx},{0.4386*\dy}) + -- ({4.7800*\dx},{0.4355*\dy}) + -- ({4.7900*\dx},{0.4339*\dy}) + -- ({4.8000*\dx},{0.4338*\dy}) + -- ({4.8100*\dx},{0.4352*\dy}) + -- ({4.8200*\dx},{0.4380*\dy}) + -- ({4.8300*\dx},{0.4423*\dy}) + -- ({4.8400*\dx},{0.4479*\dy}) + -- ({4.8500*\dx},{0.4546*\dy}) + -- ({4.8600*\dx},{0.4624*\dy}) + -- ({4.8700*\dx},{0.4711*\dy}) + -- ({4.8800*\dx},{0.4804*\dy}) + -- ({4.8900*\dx},{0.4902*\dy}) + -- ({4.9000*\dx},{0.5002*\dy}) + -- ({4.9100*\dx},{0.5101*\dy}) + -- ({4.9200*\dx},{0.5198*\dy}) + -- ({4.9300*\dx},{0.5290*\dy}) + -- ({4.9400*\dx},{0.5375*\dy}) + -- ({4.9500*\dx},{0.5450*\dy}) + -- ({4.9600*\dx},{0.5515*\dy}) + -- ({4.9700*\dx},{0.5567*\dy}) + -- ({4.9800*\dx},{0.5605*\dy}) + -- ({4.9900*\dx},{0.5628*\dy}) +} + +\def\Cplotleft{ (0,0) + -- ({-0.0100*\dx},{-0.0100*\dy}) + -- ({-0.0200*\dx},{-0.0200*\dy}) + -- ({-0.0300*\dx},{-0.0300*\dy}) + -- ({-0.0400*\dx},{-0.0400*\dy}) + -- ({-0.0500*\dx},{-0.0500*\dy}) + -- ({-0.0600*\dx},{-0.0600*\dy}) + -- ({-0.0700*\dx},{-0.0700*\dy}) + -- ({-0.0800*\dx},{-0.0800*\dy}) + -- ({-0.0900*\dx},{-0.0900*\dy}) + -- ({-0.1000*\dx},{-0.1000*\dy}) + -- ({-0.1100*\dx},{-0.1100*\dy}) + -- ({-0.1200*\dx},{-0.1200*\dy}) + -- ({-0.1300*\dx},{-0.1300*\dy}) + -- ({-0.1400*\dx},{-0.1400*\dy}) + -- ({-0.1500*\dx},{-0.1500*\dy}) + -- ({-0.1600*\dx},{-0.1600*\dy}) + -- ({-0.1700*\dx},{-0.1700*\dy}) + -- ({-0.1800*\dx},{-0.1800*\dy}) + -- ({-0.1900*\dx},{-0.1899*\dy}) + -- ({-0.2000*\dx},{-0.1999*\dy}) + -- ({-0.2100*\dx},{-0.2099*\dy}) + -- ({-0.2200*\dx},{-0.2199*\dy}) + -- ({-0.2300*\dx},{-0.2298*\dy}) + -- ({-0.2400*\dx},{-0.2398*\dy}) + -- ({-0.2500*\dx},{-0.2498*\dy}) + -- ({-0.2600*\dx},{-0.2597*\dy}) + -- ({-0.2700*\dx},{-0.2696*\dy}) + -- ({-0.2800*\dx},{-0.2796*\dy}) + -- ({-0.2900*\dx},{-0.2895*\dy}) + -- ({-0.3000*\dx},{-0.2994*\dy}) + -- ({-0.3100*\dx},{-0.3093*\dy}) + -- ({-0.3200*\dx},{-0.3192*\dy}) + -- ({-0.3300*\dx},{-0.3290*\dy}) + -- ({-0.3400*\dx},{-0.3389*\dy}) + -- ({-0.3500*\dx},{-0.3487*\dy}) + -- ({-0.3600*\dx},{-0.3585*\dy}) + -- ({-0.3700*\dx},{-0.3683*\dy}) + -- ({-0.3800*\dx},{-0.3780*\dy}) + -- ({-0.3900*\dx},{-0.3878*\dy}) + -- ({-0.4000*\dx},{-0.3975*\dy}) + -- ({-0.4100*\dx},{-0.4072*\dy}) + -- ({-0.4200*\dx},{-0.4168*\dy}) + -- ({-0.4300*\dx},{-0.4264*\dy}) + -- ({-0.4400*\dx},{-0.4359*\dy}) + -- ({-0.4500*\dx},{-0.4455*\dy}) + -- ({-0.4600*\dx},{-0.4549*\dy}) + -- ({-0.4700*\dx},{-0.4644*\dy}) + -- ({-0.4800*\dx},{-0.4738*\dy}) + -- ({-0.4900*\dx},{-0.4831*\dy}) + -- ({-0.5000*\dx},{-0.4923*\dy}) + -- ({-0.5100*\dx},{-0.5016*\dy}) + -- ({-0.5200*\dx},{-0.5107*\dy}) + -- ({-0.5300*\dx},{-0.5198*\dy}) + -- ({-0.5400*\dx},{-0.5288*\dy}) + -- ({-0.5500*\dx},{-0.5377*\dy}) + -- ({-0.5600*\dx},{-0.5466*\dy}) + -- ({-0.5700*\dx},{-0.5553*\dy}) + -- ({-0.5800*\dx},{-0.5640*\dy}) + -- ({-0.5900*\dx},{-0.5726*\dy}) + -- ({-0.6000*\dx},{-0.5811*\dy}) + -- ({-0.6100*\dx},{-0.5895*\dy}) + -- ({-0.6200*\dx},{-0.5978*\dy}) + -- ({-0.6300*\dx},{-0.6059*\dy}) + -- ({-0.6400*\dx},{-0.6140*\dy}) + -- ({-0.6500*\dx},{-0.6219*\dy}) + -- ({-0.6600*\dx},{-0.6298*\dy}) + -- ({-0.6700*\dx},{-0.6374*\dy}) + -- ({-0.6800*\dx},{-0.6450*\dy}) + -- ({-0.6900*\dx},{-0.6524*\dy}) + -- ({-0.7000*\dx},{-0.6597*\dy}) + -- ({-0.7100*\dx},{-0.6668*\dy}) + -- ({-0.7200*\dx},{-0.6737*\dy}) + -- ({-0.7300*\dx},{-0.6805*\dy}) + -- ({-0.7400*\dx},{-0.6871*\dy}) + -- ({-0.7500*\dx},{-0.6935*\dy}) + -- ({-0.7600*\dx},{-0.6998*\dy}) + -- ({-0.7700*\dx},{-0.7058*\dy}) + -- ({-0.7800*\dx},{-0.7117*\dy}) + -- ({-0.7900*\dx},{-0.7174*\dy}) + -- ({-0.8000*\dx},{-0.7228*\dy}) + -- ({-0.8100*\dx},{-0.7281*\dy}) + -- ({-0.8200*\dx},{-0.7331*\dy}) + -- ({-0.8300*\dx},{-0.7379*\dy}) + -- ({-0.8400*\dx},{-0.7425*\dy}) + -- ({-0.8500*\dx},{-0.7469*\dy}) + -- ({-0.8600*\dx},{-0.7510*\dy}) + -- ({-0.8700*\dx},{-0.7548*\dy}) + -- ({-0.8800*\dx},{-0.7584*\dy}) + -- ({-0.8900*\dx},{-0.7617*\dy}) + -- ({-0.9000*\dx},{-0.7648*\dy}) + -- ({-0.9100*\dx},{-0.7676*\dy}) + -- ({-0.9200*\dx},{-0.7702*\dy}) + -- ({-0.9300*\dx},{-0.7724*\dy}) + -- ({-0.9400*\dx},{-0.7744*\dy}) + -- ({-0.9500*\dx},{-0.7760*\dy}) + -- ({-0.9600*\dx},{-0.7774*\dy}) + -- ({-0.9700*\dx},{-0.7785*\dy}) + -- ({-0.9800*\dx},{-0.7793*\dy}) + -- ({-0.9900*\dx},{-0.7797*\dy}) + -- ({-1.0000*\dx},{-0.7799*\dy}) + -- ({-1.0100*\dx},{-0.7797*\dy}) + -- ({-1.0200*\dx},{-0.7793*\dy}) + -- ({-1.0300*\dx},{-0.7785*\dy}) + -- ({-1.0400*\dx},{-0.7774*\dy}) + -- ({-1.0500*\dx},{-0.7759*\dy}) + -- ({-1.0600*\dx},{-0.7741*\dy}) + -- ({-1.0700*\dx},{-0.7721*\dy}) + -- ({-1.0800*\dx},{-0.7696*\dy}) + -- ({-1.0900*\dx},{-0.7669*\dy}) + -- ({-1.1000*\dx},{-0.7638*\dy}) + -- ({-1.1100*\dx},{-0.7604*\dy}) + -- ({-1.1200*\dx},{-0.7567*\dy}) + -- ({-1.1300*\dx},{-0.7526*\dy}) + -- ({-1.1400*\dx},{-0.7482*\dy}) + -- ({-1.1500*\dx},{-0.7436*\dy}) + -- ({-1.1600*\dx},{-0.7385*\dy}) + -- ({-1.1700*\dx},{-0.7332*\dy}) + -- ({-1.1800*\dx},{-0.7276*\dy}) + -- ({-1.1900*\dx},{-0.7217*\dy}) + -- ({-1.2000*\dx},{-0.7154*\dy}) + -- ({-1.2100*\dx},{-0.7089*\dy}) + -- ({-1.2200*\dx},{-0.7021*\dy}) + -- ({-1.2300*\dx},{-0.6950*\dy}) + -- ({-1.2400*\dx},{-0.6877*\dy}) + -- ({-1.2500*\dx},{-0.6801*\dy}) + -- ({-1.2600*\dx},{-0.6722*\dy}) + -- ({-1.2700*\dx},{-0.6641*\dy}) + -- ({-1.2800*\dx},{-0.6558*\dy}) + -- ({-1.2900*\dx},{-0.6473*\dy}) + -- ({-1.3000*\dx},{-0.6386*\dy}) + -- ({-1.3100*\dx},{-0.6296*\dy}) + -- ({-1.3200*\dx},{-0.6205*\dy}) + -- ({-1.3300*\dx},{-0.6112*\dy}) + -- ({-1.3400*\dx},{-0.6018*\dy}) + -- ({-1.3500*\dx},{-0.5923*\dy}) + -- ({-1.3600*\dx},{-0.5826*\dy}) + -- ({-1.3700*\dx},{-0.5728*\dy}) + -- ({-1.3800*\dx},{-0.5630*\dy}) + -- ({-1.3900*\dx},{-0.5531*\dy}) + -- ({-1.4000*\dx},{-0.5431*\dy}) + -- ({-1.4100*\dx},{-0.5331*\dy}) + -- ({-1.4200*\dx},{-0.5231*\dy}) + -- ({-1.4300*\dx},{-0.5131*\dy}) + -- ({-1.4400*\dx},{-0.5032*\dy}) + -- ({-1.4500*\dx},{-0.4933*\dy}) + -- ({-1.4600*\dx},{-0.4834*\dy}) + -- ({-1.4700*\dx},{-0.4737*\dy}) + -- ({-1.4800*\dx},{-0.4641*\dy}) + -- ({-1.4900*\dx},{-0.4546*\dy}) + -- ({-1.5000*\dx},{-0.4453*\dy}) + -- ({-1.5100*\dx},{-0.4361*\dy}) + -- ({-1.5200*\dx},{-0.4272*\dy}) + -- ({-1.5300*\dx},{-0.4185*\dy}) + -- ({-1.5400*\dx},{-0.4100*\dy}) + -- ({-1.5500*\dx},{-0.4018*\dy}) + -- ({-1.5600*\dx},{-0.3939*\dy}) + -- ({-1.5700*\dx},{-0.3862*\dy}) + -- ({-1.5800*\dx},{-0.3790*\dy}) + -- ({-1.5900*\dx},{-0.3720*\dy}) + -- ({-1.6000*\dx},{-0.3655*\dy}) + -- ({-1.6100*\dx},{-0.3593*\dy}) + -- ({-1.6200*\dx},{-0.3535*\dy}) + -- ({-1.6300*\dx},{-0.3482*\dy}) + -- ({-1.6400*\dx},{-0.3433*\dy}) + -- ({-1.6500*\dx},{-0.3388*\dy}) + -- ({-1.6600*\dx},{-0.3348*\dy}) + -- ({-1.6700*\dx},{-0.3313*\dy}) + -- ({-1.6800*\dx},{-0.3283*\dy}) + -- ({-1.6900*\dx},{-0.3258*\dy}) + -- ({-1.7000*\dx},{-0.3238*\dy}) + -- ({-1.7100*\dx},{-0.3224*\dy}) + -- ({-1.7200*\dx},{-0.3214*\dy}) + -- ({-1.7300*\dx},{-0.3211*\dy}) + -- ({-1.7400*\dx},{-0.3212*\dy}) + -- ({-1.7500*\dx},{-0.3219*\dy}) + -- ({-1.7600*\dx},{-0.3232*\dy}) + -- ({-1.7700*\dx},{-0.3250*\dy}) + -- ({-1.7800*\dx},{-0.3273*\dy}) + -- ({-1.7900*\dx},{-0.3302*\dy}) + -- ({-1.8000*\dx},{-0.3336*\dy}) + -- ({-1.8100*\dx},{-0.3376*\dy}) + -- ({-1.8200*\dx},{-0.3420*\dy}) + -- ({-1.8300*\dx},{-0.3470*\dy}) + -- ({-1.8400*\dx},{-0.3524*\dy}) + -- ({-1.8500*\dx},{-0.3584*\dy}) + -- ({-1.8600*\dx},{-0.3648*\dy}) + -- ({-1.8700*\dx},{-0.3716*\dy}) + -- ({-1.8800*\dx},{-0.3788*\dy}) + -- ({-1.8900*\dx},{-0.3865*\dy}) + -- ({-1.9000*\dx},{-0.3945*\dy}) + -- ({-1.9100*\dx},{-0.4028*\dy}) + -- ({-1.9200*\dx},{-0.4115*\dy}) + -- ({-1.9300*\dx},{-0.4204*\dy}) + -- ({-1.9400*\dx},{-0.4296*\dy}) + -- ({-1.9500*\dx},{-0.4391*\dy}) + -- ({-1.9600*\dx},{-0.4487*\dy}) + -- ({-1.9700*\dx},{-0.4584*\dy}) + -- ({-1.9800*\dx},{-0.4683*\dy}) + -- ({-1.9900*\dx},{-0.4783*\dy}) + -- ({-2.0000*\dx},{-0.4883*\dy}) + -- ({-2.0100*\dx},{-0.4982*\dy}) + -- ({-2.0200*\dx},{-0.5082*\dy}) + -- ({-2.0300*\dx},{-0.5181*\dy}) + -- ({-2.0400*\dx},{-0.5278*\dy}) + -- ({-2.0500*\dx},{-0.5374*\dy}) + -- ({-2.0600*\dx},{-0.5468*\dy}) + -- ({-2.0700*\dx},{-0.5560*\dy}) + -- ({-2.0800*\dx},{-0.5648*\dy}) + -- ({-2.0900*\dx},{-0.5734*\dy}) + -- ({-2.1000*\dx},{-0.5816*\dy}) + -- ({-2.1100*\dx},{-0.5894*\dy}) + -- ({-2.1200*\dx},{-0.5967*\dy}) + -- ({-2.1300*\dx},{-0.6036*\dy}) + -- ({-2.1400*\dx},{-0.6100*\dy}) + -- ({-2.1500*\dx},{-0.6159*\dy}) + -- ({-2.1600*\dx},{-0.6212*\dy}) + -- ({-2.1700*\dx},{-0.6259*\dy}) + -- ({-2.1800*\dx},{-0.6300*\dy}) + -- ({-2.1900*\dx},{-0.6335*\dy}) + -- ({-2.2000*\dx},{-0.6363*\dy}) + -- ({-2.2100*\dx},{-0.6384*\dy}) + -- ({-2.2200*\dx},{-0.6399*\dy}) + -- ({-2.2300*\dx},{-0.6407*\dy}) + -- ({-2.2400*\dx},{-0.6408*\dy}) + -- ({-2.2500*\dx},{-0.6401*\dy}) + -- ({-2.2600*\dx},{-0.6388*\dy}) + -- ({-2.2700*\dx},{-0.6368*\dy}) + -- ({-2.2800*\dx},{-0.6340*\dy}) + -- ({-2.2900*\dx},{-0.6306*\dy}) + -- ({-2.3000*\dx},{-0.6266*\dy}) + -- ({-2.3100*\dx},{-0.6218*\dy}) + -- ({-2.3200*\dx},{-0.6165*\dy}) + -- ({-2.3300*\dx},{-0.6105*\dy}) + -- ({-2.3400*\dx},{-0.6040*\dy}) + -- ({-2.3500*\dx},{-0.5970*\dy}) + -- ({-2.3600*\dx},{-0.5894*\dy}) + -- ({-2.3700*\dx},{-0.5814*\dy}) + -- ({-2.3800*\dx},{-0.5729*\dy}) + -- ({-2.3900*\dx},{-0.5641*\dy}) + -- ({-2.4000*\dx},{-0.5550*\dy}) + -- ({-2.4100*\dx},{-0.5455*\dy}) + -- ({-2.4200*\dx},{-0.5359*\dy}) + -- ({-2.4300*\dx},{-0.5261*\dy}) + -- ({-2.4400*\dx},{-0.5161*\dy}) + -- ({-2.4500*\dx},{-0.5061*\dy}) + -- ({-2.4600*\dx},{-0.4961*\dy}) + -- ({-2.4700*\dx},{-0.4862*\dy}) + -- ({-2.4800*\dx},{-0.4764*\dy}) + -- ({-2.4900*\dx},{-0.4668*\dy}) + -- ({-2.5000*\dx},{-0.4574*\dy}) + -- ({-2.5100*\dx},{-0.4483*\dy}) + -- ({-2.5200*\dx},{-0.4396*\dy}) + -- ({-2.5300*\dx},{-0.4313*\dy}) + -- ({-2.5400*\dx},{-0.4235*\dy}) + -- ({-2.5500*\dx},{-0.4161*\dy}) + -- ({-2.5600*\dx},{-0.4094*\dy}) + -- ({-2.5700*\dx},{-0.4033*\dy}) + -- ({-2.5800*\dx},{-0.3978*\dy}) + -- ({-2.5900*\dx},{-0.3930*\dy}) + -- ({-2.6000*\dx},{-0.3889*\dy}) + -- ({-2.6100*\dx},{-0.3856*\dy}) + -- ({-2.6200*\dx},{-0.3831*\dy}) + -- ({-2.6300*\dx},{-0.3814*\dy}) + -- ({-2.6400*\dx},{-0.3805*\dy}) + -- ({-2.6500*\dx},{-0.3805*\dy}) + -- ({-2.6600*\dx},{-0.3812*\dy}) + -- ({-2.6700*\dx},{-0.3828*\dy}) + -- ({-2.6800*\dx},{-0.3853*\dy}) + -- ({-2.6900*\dx},{-0.3885*\dy}) + -- ({-2.7000*\dx},{-0.3925*\dy}) + -- ({-2.7100*\dx},{-0.3973*\dy}) + -- ({-2.7200*\dx},{-0.4028*\dy}) + -- ({-2.7300*\dx},{-0.4090*\dy}) + -- ({-2.7400*\dx},{-0.4158*\dy}) + -- ({-2.7500*\dx},{-0.4233*\dy}) + -- ({-2.7600*\dx},{-0.4313*\dy}) + -- ({-2.7700*\dx},{-0.4397*\dy}) + -- ({-2.7800*\dx},{-0.4487*\dy}) + -- ({-2.7900*\dx},{-0.4579*\dy}) + -- ({-2.8000*\dx},{-0.4675*\dy}) + -- ({-2.8100*\dx},{-0.4773*\dy}) + -- ({-2.8200*\dx},{-0.4872*\dy}) + -- ({-2.8300*\dx},{-0.4972*\dy}) + -- ({-2.8400*\dx},{-0.5072*\dy}) + -- ({-2.8500*\dx},{-0.5171*\dy}) + -- ({-2.8600*\dx},{-0.5268*\dy}) + -- ({-2.8700*\dx},{-0.5362*\dy}) + -- ({-2.8800*\dx},{-0.5454*\dy}) + -- ({-2.8900*\dx},{-0.5541*\dy}) + -- ({-2.9000*\dx},{-0.5624*\dy}) + -- ({-2.9100*\dx},{-0.5701*\dy}) + -- ({-2.9200*\dx},{-0.5772*\dy}) + -- ({-2.9300*\dx},{-0.5836*\dy}) + -- ({-2.9400*\dx},{-0.5893*\dy}) + -- ({-2.9500*\dx},{-0.5942*\dy}) + -- ({-2.9600*\dx},{-0.5983*\dy}) + -- ({-2.9700*\dx},{-0.6015*\dy}) + -- ({-2.9800*\dx},{-0.6038*\dy}) + -- ({-2.9900*\dx},{-0.6053*\dy}) + -- ({-3.0000*\dx},{-0.6057*\dy}) + -- ({-3.0100*\dx},{-0.6052*\dy}) + -- ({-3.0200*\dx},{-0.6038*\dy}) + -- ({-3.0300*\dx},{-0.6015*\dy}) + -- ({-3.0400*\dx},{-0.5982*\dy}) + -- ({-3.0500*\dx},{-0.5941*\dy}) + -- ({-3.0600*\dx},{-0.5891*\dy}) + -- ({-3.0700*\dx},{-0.5833*\dy}) + -- ({-3.0800*\dx},{-0.5767*\dy}) + -- ({-3.0900*\dx},{-0.5695*\dy}) + -- ({-3.1000*\dx},{-0.5616*\dy}) + -- ({-3.1100*\dx},{-0.5531*\dy}) + -- ({-3.1200*\dx},{-0.5442*\dy}) + -- ({-3.1300*\dx},{-0.5349*\dy}) + -- ({-3.1400*\dx},{-0.5253*\dy}) + -- ({-3.1500*\dx},{-0.5154*\dy}) + -- ({-3.1600*\dx},{-0.5054*\dy}) + -- ({-3.1700*\dx},{-0.4954*\dy}) + -- ({-3.1800*\dx},{-0.4855*\dy}) + -- ({-3.1900*\dx},{-0.4758*\dy}) + -- ({-3.2000*\dx},{-0.4663*\dy}) + -- ({-3.2100*\dx},{-0.4572*\dy}) + -- ({-3.2200*\dx},{-0.4486*\dy}) + -- ({-3.2300*\dx},{-0.4405*\dy}) + -- ({-3.2400*\dx},{-0.4331*\dy}) + -- ({-3.2500*\dx},{-0.4263*\dy}) + -- ({-3.2600*\dx},{-0.4204*\dy}) + -- ({-3.2700*\dx},{-0.4153*\dy}) + -- ({-3.2800*\dx},{-0.4111*\dy}) + -- ({-3.2900*\dx},{-0.4079*\dy}) + -- ({-3.3000*\dx},{-0.4057*\dy}) + -- ({-3.3100*\dx},{-0.4045*\dy}) + -- ({-3.3200*\dx},{-0.4043*\dy}) + -- ({-3.3300*\dx},{-0.4052*\dy}) + -- ({-3.3400*\dx},{-0.4071*\dy}) + -- ({-3.3500*\dx},{-0.4100*\dy}) + -- ({-3.3600*\dx},{-0.4139*\dy}) + -- ({-3.3700*\dx},{-0.4188*\dy}) + -- ({-3.3800*\dx},{-0.4246*\dy}) + -- ({-3.3900*\dx},{-0.4311*\dy}) + -- ({-3.4000*\dx},{-0.4385*\dy}) + -- ({-3.4100*\dx},{-0.4465*\dy}) + -- ({-3.4200*\dx},{-0.4551*\dy}) + -- ({-3.4300*\dx},{-0.4643*\dy}) + -- ({-3.4400*\dx},{-0.4738*\dy}) + -- ({-3.4500*\dx},{-0.4835*\dy}) + -- ({-3.4600*\dx},{-0.4935*\dy}) + -- ({-3.4700*\dx},{-0.5035*\dy}) + -- ({-3.4800*\dx},{-0.5134*\dy}) + -- ({-3.4900*\dx},{-0.5231*\dy}) + -- ({-3.5000*\dx},{-0.5326*\dy}) + -- ({-3.5100*\dx},{-0.5416*\dy}) + -- ({-3.5200*\dx},{-0.5501*\dy}) + -- ({-3.5300*\dx},{-0.5579*\dy}) + -- ({-3.5400*\dx},{-0.5650*\dy}) + -- ({-3.5500*\dx},{-0.5713*\dy}) + -- ({-3.5600*\dx},{-0.5767*\dy}) + -- ({-3.5700*\dx},{-0.5811*\dy}) + -- ({-3.5800*\dx},{-0.5845*\dy}) + -- ({-3.5900*\dx},{-0.5868*\dy}) + -- ({-3.6000*\dx},{-0.5880*\dy}) + -- ({-3.6100*\dx},{-0.5880*\dy}) + -- ({-3.6200*\dx},{-0.5869*\dy}) + -- ({-3.6300*\dx},{-0.5848*\dy}) + -- ({-3.6400*\dx},{-0.5815*\dy}) + -- ({-3.6500*\dx},{-0.5771*\dy}) + -- ({-3.6600*\dx},{-0.5718*\dy}) + -- ({-3.6700*\dx},{-0.5655*\dy}) + -- ({-3.6800*\dx},{-0.5584*\dy}) + -- ({-3.6900*\dx},{-0.5505*\dy}) + -- ({-3.7000*\dx},{-0.5419*\dy}) + -- ({-3.7100*\dx},{-0.5329*\dy}) + -- ({-3.7200*\dx},{-0.5233*\dy}) + -- ({-3.7300*\dx},{-0.5135*\dy}) + -- ({-3.7400*\dx},{-0.5036*\dy}) + -- ({-3.7500*\dx},{-0.4936*\dy}) + -- ({-3.7600*\dx},{-0.4837*\dy}) + -- ({-3.7700*\dx},{-0.4741*\dy}) + -- ({-3.7800*\dx},{-0.4649*\dy}) + -- ({-3.7900*\dx},{-0.4562*\dy}) + -- ({-3.8000*\dx},{-0.4481*\dy}) + -- ({-3.8100*\dx},{-0.4408*\dy}) + -- ({-3.8200*\dx},{-0.4343*\dy}) + -- ({-3.8300*\dx},{-0.4289*\dy}) + -- ({-3.8400*\dx},{-0.4244*\dy}) + -- ({-3.8500*\dx},{-0.4211*\dy}) + -- ({-3.8600*\dx},{-0.4189*\dy}) + -- ({-3.8700*\dx},{-0.4180*\dy}) + -- ({-3.8800*\dx},{-0.4182*\dy}) + -- ({-3.8900*\dx},{-0.4197*\dy}) + -- ({-3.9000*\dx},{-0.4223*\dy}) + -- ({-3.9100*\dx},{-0.4261*\dy}) + -- ({-3.9200*\dx},{-0.4311*\dy}) + -- ({-3.9300*\dx},{-0.4370*\dy}) + -- ({-3.9400*\dx},{-0.4439*\dy}) + -- ({-3.9500*\dx},{-0.4516*\dy}) + -- ({-3.9600*\dx},{-0.4601*\dy}) + -- ({-3.9700*\dx},{-0.4691*\dy}) + -- ({-3.9800*\dx},{-0.4786*\dy}) + -- ({-3.9900*\dx},{-0.4885*\dy}) + -- ({-4.0000*\dx},{-0.4984*\dy}) + -- ({-4.0100*\dx},{-0.5084*\dy}) + -- ({-4.0200*\dx},{-0.5182*\dy}) + -- ({-4.0300*\dx},{-0.5277*\dy}) + -- ({-4.0400*\dx},{-0.5368*\dy}) + -- ({-4.0500*\dx},{-0.5452*\dy}) + -- ({-4.0600*\dx},{-0.5528*\dy}) + -- ({-4.0700*\dx},{-0.5596*\dy}) + -- ({-4.0800*\dx},{-0.5654*\dy}) + -- ({-4.0900*\dx},{-0.5701*\dy}) + -- ({-4.1000*\dx},{-0.5737*\dy}) + -- ({-4.1100*\dx},{-0.5760*\dy}) + -- ({-4.1200*\dx},{-0.5771*\dy}) + -- ({-4.1300*\dx},{-0.5768*\dy}) + -- ({-4.1400*\dx},{-0.5753*\dy}) + -- ({-4.1500*\dx},{-0.5725*\dy}) + -- ({-4.1600*\dx},{-0.5684*\dy}) + -- ({-4.1700*\dx},{-0.5633*\dy}) + -- ({-4.1800*\dx},{-0.5570*\dy}) + -- ({-4.1900*\dx},{-0.5498*\dy}) + -- ({-4.2000*\dx},{-0.5417*\dy}) + -- ({-4.2100*\dx},{-0.5329*\dy}) + -- ({-4.2200*\dx},{-0.5236*\dy}) + -- ({-4.2300*\dx},{-0.5139*\dy}) + -- ({-4.2400*\dx},{-0.5040*\dy}) + -- ({-4.2500*\dx},{-0.4940*\dy}) + -- ({-4.2600*\dx},{-0.4841*\dy}) + -- ({-4.2700*\dx},{-0.4746*\dy}) + -- ({-4.2800*\dx},{-0.4655*\dy}) + -- ({-4.2900*\dx},{-0.4571*\dy}) + -- ({-4.3000*\dx},{-0.4494*\dy}) + -- ({-4.3100*\dx},{-0.4428*\dy}) + -- ({-4.3200*\dx},{-0.4371*\dy}) + -- ({-4.3300*\dx},{-0.4327*\dy}) + -- ({-4.3400*\dx},{-0.4295*\dy}) + -- ({-4.3500*\dx},{-0.4276*\dy}) + -- ({-4.3600*\dx},{-0.4270*\dy}) + -- ({-4.3700*\dx},{-0.4279*\dy}) + -- ({-4.3800*\dx},{-0.4301*\dy}) + -- ({-4.3900*\dx},{-0.4336*\dy}) + -- ({-4.4000*\dx},{-0.4383*\dy}) + -- ({-4.4100*\dx},{-0.4443*\dy}) + -- ({-4.4200*\dx},{-0.4512*\dy}) + -- ({-4.4300*\dx},{-0.4591*\dy}) + -- ({-4.4400*\dx},{-0.4678*\dy}) + -- ({-4.4500*\dx},{-0.4771*\dy}) + -- ({-4.4600*\dx},{-0.4868*\dy}) + -- ({-4.4700*\dx},{-0.4967*\dy}) + -- ({-4.4800*\dx},{-0.5067*\dy}) + -- ({-4.4900*\dx},{-0.5165*\dy}) + -- ({-4.5000*\dx},{-0.5260*\dy}) + -- ({-4.5100*\dx},{-0.5350*\dy}) + -- ({-4.5200*\dx},{-0.5432*\dy}) + -- ({-4.5300*\dx},{-0.5505*\dy}) + -- ({-4.5400*\dx},{-0.5568*\dy}) + -- ({-4.5500*\dx},{-0.5619*\dy}) + -- ({-4.5600*\dx},{-0.5658*\dy}) + -- ({-4.5700*\dx},{-0.5683*\dy}) + -- ({-4.5800*\dx},{-0.5694*\dy}) + -- ({-4.5900*\dx},{-0.5690*\dy}) + -- ({-4.6000*\dx},{-0.5672*\dy}) + -- ({-4.6100*\dx},{-0.5641*\dy}) + -- ({-4.6200*\dx},{-0.5595*\dy}) + -- ({-4.6300*\dx},{-0.5538*\dy}) + -- ({-4.6400*\dx},{-0.5469*\dy}) + -- ({-4.6500*\dx},{-0.5391*\dy}) + -- ({-4.6600*\dx},{-0.5304*\dy}) + -- ({-4.6700*\dx},{-0.5211*\dy}) + -- ({-4.6800*\dx},{-0.5114*\dy}) + -- ({-4.6900*\dx},{-0.5014*\dy}) + -- ({-4.7000*\dx},{-0.4914*\dy}) + -- ({-4.7100*\dx},{-0.4817*\dy}) + -- ({-4.7200*\dx},{-0.4723*\dy}) + -- ({-4.7300*\dx},{-0.4636*\dy}) + -- ({-4.7400*\dx},{-0.4557*\dy}) + -- ({-4.7500*\dx},{-0.4488*\dy}) + -- ({-4.7600*\dx},{-0.4431*\dy}) + -- ({-4.7700*\dx},{-0.4386*\dy}) + -- ({-4.7800*\dx},{-0.4355*\dy}) + -- ({-4.7900*\dx},{-0.4339*\dy}) + -- ({-4.8000*\dx},{-0.4338*\dy}) + -- ({-4.8100*\dx},{-0.4352*\dy}) + -- ({-4.8200*\dx},{-0.4380*\dy}) + -- ({-4.8300*\dx},{-0.4423*\dy}) + -- ({-4.8400*\dx},{-0.4479*\dy}) + -- ({-4.8500*\dx},{-0.4546*\dy}) + -- ({-4.8600*\dx},{-0.4624*\dy}) + -- ({-4.8700*\dx},{-0.4711*\dy}) + -- ({-4.8800*\dx},{-0.4804*\dy}) + -- ({-4.8900*\dx},{-0.4902*\dy}) + -- ({-4.9000*\dx},{-0.5002*\dy}) + -- ({-4.9100*\dx},{-0.5101*\dy}) + -- ({-4.9200*\dx},{-0.5198*\dy}) + -- ({-4.9300*\dx},{-0.5290*\dy}) + -- ({-4.9400*\dx},{-0.5375*\dy}) + -- ({-4.9500*\dx},{-0.5450*\dy}) + -- ({-4.9600*\dx},{-0.5515*\dy}) + -- ({-4.9700*\dx},{-0.5567*\dy}) + -- ({-4.9800*\dx},{-0.5605*\dy}) + -- ({-4.9900*\dx},{-0.5628*\dy}) +} + +\def\Splotright{ (0,0) + -- ({0.0100*\dx},{0.0000*\dy}) + -- ({0.0200*\dx},{0.0000*\dy}) + -- ({0.0300*\dx},{0.0000*\dy}) + -- ({0.0400*\dx},{0.0000*\dy}) + -- ({0.0500*\dx},{0.0001*\dy}) + -- ({0.0600*\dx},{0.0001*\dy}) + -- ({0.0700*\dx},{0.0002*\dy}) + -- ({0.0800*\dx},{0.0003*\dy}) + -- ({0.0900*\dx},{0.0004*\dy}) + -- ({0.1000*\dx},{0.0005*\dy}) + -- ({0.1100*\dx},{0.0007*\dy}) + -- ({0.1200*\dx},{0.0009*\dy}) + -- ({0.1300*\dx},{0.0012*\dy}) + -- ({0.1400*\dx},{0.0014*\dy}) + -- ({0.1500*\dx},{0.0018*\dy}) + -- ({0.1600*\dx},{0.0021*\dy}) + -- ({0.1700*\dx},{0.0026*\dy}) + -- ({0.1800*\dx},{0.0031*\dy}) + -- ({0.1900*\dx},{0.0036*\dy}) + -- ({0.2000*\dx},{0.0042*\dy}) + -- ({0.2100*\dx},{0.0048*\dy}) + -- ({0.2200*\dx},{0.0056*\dy}) + -- ({0.2300*\dx},{0.0064*\dy}) + -- ({0.2400*\dx},{0.0072*\dy}) + -- ({0.2500*\dx},{0.0082*\dy}) + -- ({0.2600*\dx},{0.0092*\dy}) + -- ({0.2700*\dx},{0.0103*\dy}) + -- ({0.2800*\dx},{0.0115*\dy}) + -- ({0.2900*\dx},{0.0128*\dy}) + -- ({0.3000*\dx},{0.0141*\dy}) + -- ({0.3100*\dx},{0.0156*\dy}) + -- ({0.3200*\dx},{0.0171*\dy}) + -- ({0.3300*\dx},{0.0188*\dy}) + -- ({0.3400*\dx},{0.0205*\dy}) + -- ({0.3500*\dx},{0.0224*\dy}) + -- ({0.3600*\dx},{0.0244*\dy}) + -- ({0.3700*\dx},{0.0264*\dy}) + -- ({0.3800*\dx},{0.0286*\dy}) + -- ({0.3900*\dx},{0.0309*\dy}) + -- ({0.4000*\dx},{0.0334*\dy}) + -- ({0.4100*\dx},{0.0359*\dy}) + -- ({0.4200*\dx},{0.0386*\dy}) + -- ({0.4300*\dx},{0.0414*\dy}) + -- ({0.4400*\dx},{0.0443*\dy}) + -- ({0.4500*\dx},{0.0474*\dy}) + -- ({0.4600*\dx},{0.0506*\dy}) + -- ({0.4700*\dx},{0.0539*\dy}) + -- ({0.4800*\dx},{0.0574*\dy}) + -- ({0.4900*\dx},{0.0610*\dy}) + -- ({0.5000*\dx},{0.0647*\dy}) + -- ({0.5100*\dx},{0.0686*\dy}) + -- ({0.5200*\dx},{0.0727*\dy}) + -- ({0.5300*\dx},{0.0769*\dy}) + -- ({0.5400*\dx},{0.0812*\dy}) + -- ({0.5500*\dx},{0.0857*\dy}) + -- ({0.5600*\dx},{0.0904*\dy}) + -- ({0.5700*\dx},{0.0952*\dy}) + -- ({0.5800*\dx},{0.1001*\dy}) + -- ({0.5900*\dx},{0.1053*\dy}) + -- ({0.6000*\dx},{0.1105*\dy}) + -- ({0.6100*\dx},{0.1160*\dy}) + -- ({0.6200*\dx},{0.1216*\dy}) + -- ({0.6300*\dx},{0.1273*\dy}) + -- ({0.6400*\dx},{0.1333*\dy}) + -- ({0.6500*\dx},{0.1393*\dy}) + -- ({0.6600*\dx},{0.1456*\dy}) + -- ({0.6700*\dx},{0.1520*\dy}) + -- ({0.6800*\dx},{0.1585*\dy}) + -- ({0.6900*\dx},{0.1653*\dy}) + -- ({0.7000*\dx},{0.1721*\dy}) + -- ({0.7100*\dx},{0.1792*\dy}) + -- ({0.7200*\dx},{0.1864*\dy}) + -- ({0.7300*\dx},{0.1937*\dy}) + -- ({0.7400*\dx},{0.2012*\dy}) + -- ({0.7500*\dx},{0.2089*\dy}) + -- ({0.7600*\dx},{0.2167*\dy}) + -- ({0.7700*\dx},{0.2246*\dy}) + -- ({0.7800*\dx},{0.2327*\dy}) + -- ({0.7900*\dx},{0.2410*\dy}) + -- ({0.8000*\dx},{0.2493*\dy}) + -- ({0.8100*\dx},{0.2579*\dy}) + -- ({0.8200*\dx},{0.2665*\dy}) + -- ({0.8300*\dx},{0.2753*\dy}) + -- ({0.8400*\dx},{0.2841*\dy}) + -- ({0.8500*\dx},{0.2932*\dy}) + -- ({0.8600*\dx},{0.3023*\dy}) + -- ({0.8700*\dx},{0.3115*\dy}) + -- ({0.8800*\dx},{0.3208*\dy}) + -- ({0.8900*\dx},{0.3303*\dy}) + -- ({0.9000*\dx},{0.3398*\dy}) + -- ({0.9100*\dx},{0.3494*\dy}) + -- ({0.9200*\dx},{0.3590*\dy}) + -- ({0.9300*\dx},{0.3688*\dy}) + -- ({0.9400*\dx},{0.3786*\dy}) + -- ({0.9500*\dx},{0.3885*\dy}) + -- ({0.9600*\dx},{0.3984*\dy}) + -- ({0.9700*\dx},{0.4083*\dy}) + -- ({0.9800*\dx},{0.4183*\dy}) + -- ({0.9900*\dx},{0.4283*\dy}) + -- ({1.0000*\dx},{0.4383*\dy}) + -- ({1.0100*\dx},{0.4483*\dy}) + -- ({1.0200*\dx},{0.4582*\dy}) + -- ({1.0300*\dx},{0.4682*\dy}) + -- ({1.0400*\dx},{0.4782*\dy}) + -- ({1.0500*\dx},{0.4880*\dy}) + -- ({1.0600*\dx},{0.4979*\dy}) + -- ({1.0700*\dx},{0.5077*\dy}) + -- ({1.0800*\dx},{0.5174*\dy}) + -- ({1.0900*\dx},{0.5270*\dy}) + -- ({1.1000*\dx},{0.5365*\dy}) + -- ({1.1100*\dx},{0.5459*\dy}) + -- ({1.1200*\dx},{0.5552*\dy}) + -- ({1.1300*\dx},{0.5643*\dy}) + -- ({1.1400*\dx},{0.5733*\dy}) + -- ({1.1500*\dx},{0.5821*\dy}) + -- ({1.1600*\dx},{0.5908*\dy}) + -- ({1.1700*\dx},{0.5993*\dy}) + -- ({1.1800*\dx},{0.6075*\dy}) + -- ({1.1900*\dx},{0.6156*\dy}) + -- ({1.2000*\dx},{0.6234*\dy}) + -- ({1.2100*\dx},{0.6310*\dy}) + -- ({1.2200*\dx},{0.6383*\dy}) + -- ({1.2300*\dx},{0.6454*\dy}) + -- ({1.2400*\dx},{0.6522*\dy}) + -- ({1.2500*\dx},{0.6587*\dy}) + -- ({1.2600*\dx},{0.6648*\dy}) + -- ({1.2700*\dx},{0.6707*\dy}) + -- ({1.2800*\dx},{0.6763*\dy}) + -- ({1.2900*\dx},{0.6815*\dy}) + -- ({1.3000*\dx},{0.6863*\dy}) + -- ({1.3100*\dx},{0.6908*\dy}) + -- ({1.3200*\dx},{0.6950*\dy}) + -- ({1.3300*\dx},{0.6987*\dy}) + -- ({1.3400*\dx},{0.7021*\dy}) + -- ({1.3500*\dx},{0.7050*\dy}) + -- ({1.3600*\dx},{0.7076*\dy}) + -- ({1.3700*\dx},{0.7097*\dy}) + -- ({1.3800*\dx},{0.7114*\dy}) + -- ({1.3900*\dx},{0.7127*\dy}) + -- ({1.4000*\dx},{0.7135*\dy}) + -- ({1.4100*\dx},{0.7139*\dy}) + -- ({1.4200*\dx},{0.7139*\dy}) + -- ({1.4300*\dx},{0.7134*\dy}) + -- ({1.4400*\dx},{0.7125*\dy}) + -- ({1.4500*\dx},{0.7111*\dy}) + -- ({1.4600*\dx},{0.7093*\dy}) + -- ({1.4700*\dx},{0.7070*\dy}) + -- ({1.4800*\dx},{0.7043*\dy}) + -- ({1.4900*\dx},{0.7011*\dy}) + -- ({1.5000*\dx},{0.6975*\dy}) + -- ({1.5100*\dx},{0.6935*\dy}) + -- ({1.5200*\dx},{0.6890*\dy}) + -- ({1.5300*\dx},{0.6841*\dy}) + -- ({1.5400*\dx},{0.6788*\dy}) + -- ({1.5500*\dx},{0.6731*\dy}) + -- ({1.5600*\dx},{0.6670*\dy}) + -- ({1.5700*\dx},{0.6605*\dy}) + -- ({1.5800*\dx},{0.6536*\dy}) + -- ({1.5900*\dx},{0.6464*\dy}) + -- ({1.6000*\dx},{0.6389*\dy}) + -- ({1.6100*\dx},{0.6310*\dy}) + -- ({1.6200*\dx},{0.6229*\dy}) + -- ({1.6300*\dx},{0.6144*\dy}) + -- ({1.6400*\dx},{0.6057*\dy}) + -- ({1.6500*\dx},{0.5968*\dy}) + -- ({1.6600*\dx},{0.5876*\dy}) + -- ({1.6700*\dx},{0.5782*\dy}) + -- ({1.6800*\dx},{0.5687*\dy}) + -- ({1.6900*\dx},{0.5590*\dy}) + -- ({1.7000*\dx},{0.5492*\dy}) + -- ({1.7100*\dx},{0.5393*\dy}) + -- ({1.7200*\dx},{0.5293*\dy}) + -- ({1.7300*\dx},{0.5194*\dy}) + -- ({1.7400*\dx},{0.5094*\dy}) + -- ({1.7500*\dx},{0.4994*\dy}) + -- ({1.7600*\dx},{0.4895*\dy}) + -- ({1.7700*\dx},{0.4796*\dy}) + -- ({1.7800*\dx},{0.4699*\dy}) + -- ({1.7900*\dx},{0.4603*\dy}) + -- ({1.8000*\dx},{0.4509*\dy}) + -- ({1.8100*\dx},{0.4418*\dy}) + -- ({1.8200*\dx},{0.4328*\dy}) + -- ({1.8300*\dx},{0.4241*\dy}) + -- ({1.8400*\dx},{0.4157*\dy}) + -- ({1.8500*\dx},{0.4077*\dy}) + -- ({1.8600*\dx},{0.4000*\dy}) + -- ({1.8700*\dx},{0.3927*\dy}) + -- ({1.8800*\dx},{0.3858*\dy}) + -- ({1.8900*\dx},{0.3793*\dy}) + -- ({1.9000*\dx},{0.3733*\dy}) + -- ({1.9100*\dx},{0.3678*\dy}) + -- ({1.9200*\dx},{0.3629*\dy}) + -- ({1.9300*\dx},{0.3584*\dy}) + -- ({1.9400*\dx},{0.3545*\dy}) + -- ({1.9500*\dx},{0.3511*\dy}) + -- ({1.9600*\dx},{0.3484*\dy}) + -- ({1.9700*\dx},{0.3462*\dy}) + -- ({1.9800*\dx},{0.3447*\dy}) + -- ({1.9900*\dx},{0.3437*\dy}) + -- ({2.0000*\dx},{0.3434*\dy}) + -- ({2.0100*\dx},{0.3437*\dy}) + -- ({2.0200*\dx},{0.3447*\dy}) + -- ({2.0300*\dx},{0.3462*\dy}) + -- ({2.0400*\dx},{0.3484*\dy}) + -- ({2.0500*\dx},{0.3513*\dy}) + -- ({2.0600*\dx},{0.3547*\dy}) + -- ({2.0700*\dx},{0.3587*\dy}) + -- ({2.0800*\dx},{0.3633*\dy}) + -- ({2.0900*\dx},{0.3685*\dy}) + -- ({2.1000*\dx},{0.3743*\dy}) + -- ({2.1100*\dx},{0.3805*\dy}) + -- ({2.1200*\dx},{0.3873*\dy}) + -- ({2.1300*\dx},{0.3945*\dy}) + -- ({2.1400*\dx},{0.4022*\dy}) + -- ({2.1500*\dx},{0.4103*\dy}) + -- ({2.1600*\dx},{0.4188*\dy}) + -- ({2.1700*\dx},{0.4276*\dy}) + -- ({2.1800*\dx},{0.4367*\dy}) + -- ({2.1900*\dx},{0.4461*\dy}) + -- ({2.2000*\dx},{0.4557*\dy}) + -- ({2.2100*\dx},{0.4655*\dy}) + -- ({2.2200*\dx},{0.4754*\dy}) + -- ({2.2300*\dx},{0.4853*\dy}) + -- ({2.2400*\dx},{0.4953*\dy}) + -- ({2.2500*\dx},{0.5053*\dy}) + -- ({2.2600*\dx},{0.5152*\dy}) + -- ({2.2700*\dx},{0.5250*\dy}) + -- ({2.2800*\dx},{0.5346*\dy}) + -- ({2.2900*\dx},{0.5440*\dy}) + -- ({2.3000*\dx},{0.5532*\dy}) + -- ({2.3100*\dx},{0.5620*\dy}) + -- ({2.3200*\dx},{0.5704*\dy}) + -- ({2.3300*\dx},{0.5784*\dy}) + -- ({2.3400*\dx},{0.5860*\dy}) + -- ({2.3500*\dx},{0.5931*\dy}) + -- ({2.3600*\dx},{0.5996*\dy}) + -- ({2.3700*\dx},{0.6056*\dy}) + -- ({2.3800*\dx},{0.6110*\dy}) + -- ({2.3900*\dx},{0.6157*\dy}) + -- ({2.4000*\dx},{0.6197*\dy}) + -- ({2.4100*\dx},{0.6230*\dy}) + -- ({2.4200*\dx},{0.6256*\dy}) + -- ({2.4300*\dx},{0.6275*\dy}) + -- ({2.4400*\dx},{0.6286*\dy}) + -- ({2.4500*\dx},{0.6289*\dy}) + -- ({2.4600*\dx},{0.6285*\dy}) + -- ({2.4700*\dx},{0.6273*\dy}) + -- ({2.4800*\dx},{0.6254*\dy}) + -- ({2.4900*\dx},{0.6226*\dy}) + -- ({2.5000*\dx},{0.6192*\dy}) + -- ({2.5100*\dx},{0.6150*\dy}) + -- ({2.5200*\dx},{0.6101*\dy}) + -- ({2.5300*\dx},{0.6045*\dy}) + -- ({2.5400*\dx},{0.5983*\dy}) + -- ({2.5500*\dx},{0.5915*\dy}) + -- ({2.5600*\dx},{0.5842*\dy}) + -- ({2.5700*\dx},{0.5763*\dy}) + -- ({2.5800*\dx},{0.5679*\dy}) + -- ({2.5900*\dx},{0.5591*\dy}) + -- ({2.6000*\dx},{0.5500*\dy}) + -- ({2.6100*\dx},{0.5406*\dy}) + -- ({2.6200*\dx},{0.5309*\dy}) + -- ({2.6300*\dx},{0.5210*\dy}) + -- ({2.6400*\dx},{0.5111*\dy}) + -- ({2.6500*\dx},{0.5011*\dy}) + -- ({2.6600*\dx},{0.4911*\dy}) + -- ({2.6700*\dx},{0.4812*\dy}) + -- ({2.6800*\dx},{0.4715*\dy}) + -- ({2.6900*\dx},{0.4621*\dy}) + -- ({2.7000*\dx},{0.4529*\dy}) + -- ({2.7100*\dx},{0.4441*\dy}) + -- ({2.7200*\dx},{0.4358*\dy}) + -- ({2.7300*\dx},{0.4279*\dy}) + -- ({2.7400*\dx},{0.4207*\dy}) + -- ({2.7500*\dx},{0.4140*\dy}) + -- ({2.7600*\dx},{0.4080*\dy}) + -- ({2.7700*\dx},{0.4027*\dy}) + -- ({2.7800*\dx},{0.3982*\dy}) + -- ({2.7900*\dx},{0.3944*\dy}) + -- ({2.8000*\dx},{0.3915*\dy}) + -- ({2.8100*\dx},{0.3895*\dy}) + -- ({2.8200*\dx},{0.3883*\dy}) + -- ({2.8300*\dx},{0.3880*\dy}) + -- ({2.8400*\dx},{0.3886*\dy}) + -- ({2.8500*\dx},{0.3900*\dy}) + -- ({2.8600*\dx},{0.3924*\dy}) + -- ({2.8700*\dx},{0.3956*\dy}) + -- ({2.8800*\dx},{0.3996*\dy}) + -- ({2.8900*\dx},{0.4045*\dy}) + -- ({2.9000*\dx},{0.4101*\dy}) + -- ({2.9100*\dx},{0.4165*\dy}) + -- ({2.9200*\dx},{0.4235*\dy}) + -- ({2.9300*\dx},{0.4312*\dy}) + -- ({2.9400*\dx},{0.4394*\dy}) + -- ({2.9500*\dx},{0.4481*\dy}) + -- ({2.9600*\dx},{0.4572*\dy}) + -- ({2.9700*\dx},{0.4667*\dy}) + -- ({2.9800*\dx},{0.4764*\dy}) + -- ({2.9900*\dx},{0.4863*\dy}) + -- ({3.0000*\dx},{0.4963*\dy}) + -- ({3.0100*\dx},{0.5063*\dy}) + -- ({3.0200*\dx},{0.5162*\dy}) + -- ({3.0300*\dx},{0.5259*\dy}) + -- ({3.0400*\dx},{0.5354*\dy}) + -- ({3.0500*\dx},{0.5445*\dy}) + -- ({3.0600*\dx},{0.5531*\dy}) + -- ({3.0700*\dx},{0.5613*\dy}) + -- ({3.0800*\dx},{0.5688*\dy}) + -- ({3.0900*\dx},{0.5757*\dy}) + -- ({3.1000*\dx},{0.5818*\dy}) + -- ({3.1100*\dx},{0.5872*\dy}) + -- ({3.1200*\dx},{0.5917*\dy}) + -- ({3.1300*\dx},{0.5952*\dy}) + -- ({3.1400*\dx},{0.5979*\dy}) + -- ({3.1500*\dx},{0.5996*\dy}) + -- ({3.1600*\dx},{0.6003*\dy}) + -- ({3.1700*\dx},{0.6001*\dy}) + -- ({3.1800*\dx},{0.5988*\dy}) + -- ({3.1900*\dx},{0.5966*\dy}) + -- ({3.2000*\dx},{0.5933*\dy}) + -- ({3.2100*\dx},{0.5892*\dy}) + -- ({3.2200*\dx},{0.5842*\dy}) + -- ({3.2300*\dx},{0.5783*\dy}) + -- ({3.2400*\dx},{0.5716*\dy}) + -- ({3.2500*\dx},{0.5642*\dy}) + -- ({3.2600*\dx},{0.5562*\dy}) + -- ({3.2700*\dx},{0.5476*\dy}) + -- ({3.2800*\dx},{0.5385*\dy}) + -- ({3.2900*\dx},{0.5290*\dy}) + -- ({3.3000*\dx},{0.5193*\dy}) + -- ({3.3100*\dx},{0.5094*\dy}) + -- ({3.3200*\dx},{0.4994*\dy}) + -- ({3.3300*\dx},{0.4894*\dy}) + -- ({3.3400*\dx},{0.4796*\dy}) + -- ({3.3500*\dx},{0.4700*\dy}) + -- ({3.3600*\dx},{0.4608*\dy}) + -- ({3.3700*\dx},{0.4521*\dy}) + -- ({3.3800*\dx},{0.4439*\dy}) + -- ({3.3900*\dx},{0.4364*\dy}) + -- ({3.4000*\dx},{0.4296*\dy}) + -- ({3.4100*\dx},{0.4237*\dy}) + -- ({3.4200*\dx},{0.4186*\dy}) + -- ({3.4300*\dx},{0.4145*\dy}) + -- ({3.4400*\dx},{0.4114*\dy}) + -- ({3.4500*\dx},{0.4094*\dy}) + -- ({3.4600*\dx},{0.4084*\dy}) + -- ({3.4700*\dx},{0.4085*\dy}) + -- ({3.4800*\dx},{0.4097*\dy}) + -- ({3.4900*\dx},{0.4119*\dy}) + -- ({3.5000*\dx},{0.4152*\dy}) + -- ({3.5100*\dx},{0.4196*\dy}) + -- ({3.5200*\dx},{0.4249*\dy}) + -- ({3.5300*\dx},{0.4311*\dy}) + -- ({3.5400*\dx},{0.4381*\dy}) + -- ({3.5500*\dx},{0.4459*\dy}) + -- ({3.5600*\dx},{0.4543*\dy}) + -- ({3.5700*\dx},{0.4633*\dy}) + -- ({3.5800*\dx},{0.4727*\dy}) + -- ({3.5900*\dx},{0.4824*\dy}) + -- ({3.6000*\dx},{0.4923*\dy}) + -- ({3.6100*\dx},{0.5023*\dy}) + -- ({3.6200*\dx},{0.5122*\dy}) + -- ({3.6300*\dx},{0.5220*\dy}) + -- ({3.6400*\dx},{0.5314*\dy}) + -- ({3.6500*\dx},{0.5404*\dy}) + -- ({3.6600*\dx},{0.5489*\dy}) + -- ({3.6700*\dx},{0.5567*\dy}) + -- ({3.6800*\dx},{0.5637*\dy}) + -- ({3.6900*\dx},{0.5698*\dy}) + -- ({3.7000*\dx},{0.5750*\dy}) + -- ({3.7100*\dx},{0.5791*\dy}) + -- ({3.7200*\dx},{0.5822*\dy}) + -- ({3.7300*\dx},{0.5841*\dy}) + -- ({3.7400*\dx},{0.5849*\dy}) + -- ({3.7500*\dx},{0.5845*\dy}) + -- ({3.7600*\dx},{0.5830*\dy}) + -- ({3.7700*\dx},{0.5803*\dy}) + -- ({3.7800*\dx},{0.5764*\dy}) + -- ({3.7900*\dx},{0.5715*\dy}) + -- ({3.8000*\dx},{0.5656*\dy}) + -- ({3.8100*\dx},{0.5588*\dy}) + -- ({3.8200*\dx},{0.5512*\dy}) + -- ({3.8300*\dx},{0.5428*\dy}) + -- ({3.8400*\dx},{0.5338*\dy}) + -- ({3.8500*\dx},{0.5244*\dy}) + -- ({3.8600*\dx},{0.5147*\dy}) + -- ({3.8700*\dx},{0.5047*\dy}) + -- ({3.8800*\dx},{0.4947*\dy}) + -- ({3.8900*\dx},{0.4848*\dy}) + -- ({3.9000*\dx},{0.4752*\dy}) + -- ({3.9100*\dx},{0.4660*\dy}) + -- ({3.9200*\dx},{0.4573*\dy}) + -- ({3.9300*\dx},{0.4492*\dy}) + -- ({3.9400*\dx},{0.4420*\dy}) + -- ({3.9500*\dx},{0.4357*\dy}) + -- ({3.9600*\dx},{0.4303*\dy}) + -- ({3.9700*\dx},{0.4261*\dy}) + -- ({3.9800*\dx},{0.4230*\dy}) + -- ({3.9900*\dx},{0.4211*\dy}) + -- ({4.0000*\dx},{0.4205*\dy}) + -- ({4.0100*\dx},{0.4211*\dy}) + -- ({4.0200*\dx},{0.4230*\dy}) + -- ({4.0300*\dx},{0.4261*\dy}) + -- ({4.0400*\dx},{0.4304*\dy}) + -- ({4.0500*\dx},{0.4358*\dy}) + -- ({4.0600*\dx},{0.4422*\dy}) + -- ({4.0700*\dx},{0.4495*\dy}) + -- ({4.0800*\dx},{0.4576*\dy}) + -- ({4.0900*\dx},{0.4665*\dy}) + -- ({4.1000*\dx},{0.4758*\dy}) + -- ({4.1100*\dx},{0.4855*\dy}) + -- ({4.1200*\dx},{0.4955*\dy}) + -- ({4.1300*\dx},{0.5054*\dy}) + -- ({4.1400*\dx},{0.5153*\dy}) + -- ({4.1500*\dx},{0.5249*\dy}) + -- ({4.1600*\dx},{0.5341*\dy}) + -- ({4.1700*\dx},{0.5426*\dy}) + -- ({4.1800*\dx},{0.5504*\dy}) + -- ({4.1900*\dx},{0.5573*\dy}) + -- ({4.2000*\dx},{0.5632*\dy}) + -- ({4.2100*\dx},{0.5680*\dy}) + -- ({4.2200*\dx},{0.5716*\dy}) + -- ({4.2300*\dx},{0.5739*\dy}) + -- ({4.2400*\dx},{0.5749*\dy}) + -- ({4.2500*\dx},{0.5746*\dy}) + -- ({4.2600*\dx},{0.5730*\dy}) + -- ({4.2700*\dx},{0.5700*\dy}) + -- ({4.2800*\dx},{0.5658*\dy}) + -- ({4.2900*\dx},{0.5604*\dy}) + -- ({4.3000*\dx},{0.5540*\dy}) + -- ({4.3100*\dx},{0.5466*\dy}) + -- ({4.3200*\dx},{0.5383*\dy}) + -- ({4.3300*\dx},{0.5294*\dy}) + -- ({4.3400*\dx},{0.5199*\dy}) + -- ({4.3500*\dx},{0.5101*\dy}) + -- ({4.3600*\dx},{0.5001*\dy}) + -- ({4.3700*\dx},{0.4902*\dy}) + -- ({4.3800*\dx},{0.4804*\dy}) + -- ({4.3900*\dx},{0.4711*\dy}) + -- ({4.4000*\dx},{0.4623*\dy}) + -- ({4.4100*\dx},{0.4542*\dy}) + -- ({4.4200*\dx},{0.4471*\dy}) + -- ({4.4300*\dx},{0.4410*\dy}) + -- ({4.4400*\dx},{0.4360*\dy}) + -- ({4.4500*\dx},{0.4323*\dy}) + -- ({4.4600*\dx},{0.4299*\dy}) + -- ({4.4700*\dx},{0.4289*\dy}) + -- ({4.4800*\dx},{0.4293*\dy}) + -- ({4.4900*\dx},{0.4311*\dy}) + -- ({4.5000*\dx},{0.4343*\dy}) + -- ({4.5100*\dx},{0.4387*\dy}) + -- ({4.5200*\dx},{0.4444*\dy}) + -- ({4.5300*\dx},{0.4512*\dy}) + -- ({4.5400*\dx},{0.4590*\dy}) + -- ({4.5500*\dx},{0.4676*\dy}) + -- ({4.5600*\dx},{0.4768*\dy}) + -- ({4.5700*\dx},{0.4864*\dy}) + -- ({4.5800*\dx},{0.4964*\dy}) + -- ({4.5900*\dx},{0.5064*\dy}) + -- ({4.6000*\dx},{0.5162*\dy}) + -- ({4.6100*\dx},{0.5257*\dy}) + -- ({4.6200*\dx},{0.5346*\dy}) + -- ({4.6300*\dx},{0.5427*\dy}) + -- ({4.6400*\dx},{0.5500*\dy}) + -- ({4.6500*\dx},{0.5562*\dy}) + -- ({4.6600*\dx},{0.5611*\dy}) + -- ({4.6700*\dx},{0.5648*\dy}) + -- ({4.6800*\dx},{0.5670*\dy}) + -- ({4.6900*\dx},{0.5678*\dy}) + -- ({4.7000*\dx},{0.5671*\dy}) + -- ({4.7100*\dx},{0.5650*\dy}) + -- ({4.7200*\dx},{0.5615*\dy}) + -- ({4.7300*\dx},{0.5566*\dy}) + -- ({4.7400*\dx},{0.5504*\dy}) + -- ({4.7500*\dx},{0.5432*\dy}) + -- ({4.7600*\dx},{0.5350*\dy}) + -- ({4.7700*\dx},{0.5261*\dy}) + -- ({4.7800*\dx},{0.5166*\dy}) + -- ({4.7900*\dx},{0.5067*\dy}) + -- ({4.8000*\dx},{0.4968*\dy}) + -- ({4.8100*\dx},{0.4869*\dy}) + -- ({4.8200*\dx},{0.4773*\dy}) + -- ({4.8300*\dx},{0.4682*\dy}) + -- ({4.8400*\dx},{0.4600*\dy}) + -- ({4.8500*\dx},{0.4526*\dy}) + -- ({4.8600*\dx},{0.4464*\dy}) + -- ({4.8700*\dx},{0.4414*\dy}) + -- ({4.8800*\dx},{0.4378*\dy}) + -- ({4.8900*\dx},{0.4357*\dy}) + -- ({4.9000*\dx},{0.4351*\dy}) + -- ({4.9100*\dx},{0.4360*\dy}) + -- ({4.9200*\dx},{0.4384*\dy}) + -- ({4.9300*\dx},{0.4423*\dy}) + -- ({4.9400*\dx},{0.4476*\dy}) + -- ({4.9500*\dx},{0.4541*\dy}) + -- ({4.9600*\dx},{0.4618*\dy}) + -- ({4.9700*\dx},{0.4703*\dy}) + -- ({4.9800*\dx},{0.4795*\dy}) + -- ({4.9900*\dx},{0.4892*\dy}) +} + +\def\Splotleft{ (0,0) + -- ({-0.0100*\dx},{-0.0000*\dy}) + -- ({-0.0200*\dx},{-0.0000*\dy}) + -- ({-0.0300*\dx},{-0.0000*\dy}) + -- ({-0.0400*\dx},{-0.0000*\dy}) + -- ({-0.0500*\dx},{-0.0001*\dy}) + -- ({-0.0600*\dx},{-0.0001*\dy}) + -- ({-0.0700*\dx},{-0.0002*\dy}) + -- ({-0.0800*\dx},{-0.0003*\dy}) + -- ({-0.0900*\dx},{-0.0004*\dy}) + -- ({-0.1000*\dx},{-0.0005*\dy}) + -- ({-0.1100*\dx},{-0.0007*\dy}) + -- ({-0.1200*\dx},{-0.0009*\dy}) + -- ({-0.1300*\dx},{-0.0012*\dy}) + -- ({-0.1400*\dx},{-0.0014*\dy}) + -- ({-0.1500*\dx},{-0.0018*\dy}) + -- ({-0.1600*\dx},{-0.0021*\dy}) + -- ({-0.1700*\dx},{-0.0026*\dy}) + -- ({-0.1800*\dx},{-0.0031*\dy}) + -- ({-0.1900*\dx},{-0.0036*\dy}) + -- ({-0.2000*\dx},{-0.0042*\dy}) + -- ({-0.2100*\dx},{-0.0048*\dy}) + -- ({-0.2200*\dx},{-0.0056*\dy}) + -- ({-0.2300*\dx},{-0.0064*\dy}) + -- ({-0.2400*\dx},{-0.0072*\dy}) + -- ({-0.2500*\dx},{-0.0082*\dy}) + -- ({-0.2600*\dx},{-0.0092*\dy}) + -- ({-0.2700*\dx},{-0.0103*\dy}) + -- ({-0.2800*\dx},{-0.0115*\dy}) + -- ({-0.2900*\dx},{-0.0128*\dy}) + -- ({-0.3000*\dx},{-0.0141*\dy}) + -- ({-0.3100*\dx},{-0.0156*\dy}) + -- ({-0.3200*\dx},{-0.0171*\dy}) + -- ({-0.3300*\dx},{-0.0188*\dy}) + -- ({-0.3400*\dx},{-0.0205*\dy}) + -- ({-0.3500*\dx},{-0.0224*\dy}) + -- ({-0.3600*\dx},{-0.0244*\dy}) + -- ({-0.3700*\dx},{-0.0264*\dy}) + -- ({-0.3800*\dx},{-0.0286*\dy}) + -- ({-0.3900*\dx},{-0.0309*\dy}) + -- ({-0.4000*\dx},{-0.0334*\dy}) + -- ({-0.4100*\dx},{-0.0359*\dy}) + -- ({-0.4200*\dx},{-0.0386*\dy}) + -- ({-0.4300*\dx},{-0.0414*\dy}) + -- ({-0.4400*\dx},{-0.0443*\dy}) + -- ({-0.4500*\dx},{-0.0474*\dy}) + -- ({-0.4600*\dx},{-0.0506*\dy}) + -- ({-0.4700*\dx},{-0.0539*\dy}) + -- ({-0.4800*\dx},{-0.0574*\dy}) + -- ({-0.4900*\dx},{-0.0610*\dy}) + -- ({-0.5000*\dx},{-0.0647*\dy}) + -- ({-0.5100*\dx},{-0.0686*\dy}) + -- ({-0.5200*\dx},{-0.0727*\dy}) + -- ({-0.5300*\dx},{-0.0769*\dy}) + -- ({-0.5400*\dx},{-0.0812*\dy}) + -- ({-0.5500*\dx},{-0.0857*\dy}) + -- ({-0.5600*\dx},{-0.0904*\dy}) + -- ({-0.5700*\dx},{-0.0952*\dy}) + -- ({-0.5800*\dx},{-0.1001*\dy}) + -- ({-0.5900*\dx},{-0.1053*\dy}) + -- ({-0.6000*\dx},{-0.1105*\dy}) + -- ({-0.6100*\dx},{-0.1160*\dy}) + -- ({-0.6200*\dx},{-0.1216*\dy}) + -- ({-0.6300*\dx},{-0.1273*\dy}) + -- ({-0.6400*\dx},{-0.1333*\dy}) + -- ({-0.6500*\dx},{-0.1393*\dy}) + -- ({-0.6600*\dx},{-0.1456*\dy}) + -- ({-0.6700*\dx},{-0.1520*\dy}) + -- ({-0.6800*\dx},{-0.1585*\dy}) + -- ({-0.6900*\dx},{-0.1653*\dy}) + -- ({-0.7000*\dx},{-0.1721*\dy}) + -- ({-0.7100*\dx},{-0.1792*\dy}) + -- ({-0.7200*\dx},{-0.1864*\dy}) + -- ({-0.7300*\dx},{-0.1937*\dy}) + -- ({-0.7400*\dx},{-0.2012*\dy}) + -- ({-0.7500*\dx},{-0.2089*\dy}) + -- ({-0.7600*\dx},{-0.2167*\dy}) + -- ({-0.7700*\dx},{-0.2246*\dy}) + -- ({-0.7800*\dx},{-0.2327*\dy}) + -- ({-0.7900*\dx},{-0.2410*\dy}) + -- ({-0.8000*\dx},{-0.2493*\dy}) + -- ({-0.8100*\dx},{-0.2579*\dy}) + -- ({-0.8200*\dx},{-0.2665*\dy}) + -- ({-0.8300*\dx},{-0.2753*\dy}) + -- ({-0.8400*\dx},{-0.2841*\dy}) + -- ({-0.8500*\dx},{-0.2932*\dy}) + -- ({-0.8600*\dx},{-0.3023*\dy}) + -- ({-0.8700*\dx},{-0.3115*\dy}) + -- ({-0.8800*\dx},{-0.3208*\dy}) + -- ({-0.8900*\dx},{-0.3303*\dy}) + -- ({-0.9000*\dx},{-0.3398*\dy}) + -- ({-0.9100*\dx},{-0.3494*\dy}) + -- ({-0.9200*\dx},{-0.3590*\dy}) + -- ({-0.9300*\dx},{-0.3688*\dy}) + -- ({-0.9400*\dx},{-0.3786*\dy}) + -- ({-0.9500*\dx},{-0.3885*\dy}) + -- ({-0.9600*\dx},{-0.3984*\dy}) + -- ({-0.9700*\dx},{-0.4083*\dy}) + -- ({-0.9800*\dx},{-0.4183*\dy}) + -- ({-0.9900*\dx},{-0.4283*\dy}) + -- ({-1.0000*\dx},{-0.4383*\dy}) + -- ({-1.0100*\dx},{-0.4483*\dy}) + -- ({-1.0200*\dx},{-0.4582*\dy}) + -- ({-1.0300*\dx},{-0.4682*\dy}) + -- ({-1.0400*\dx},{-0.4782*\dy}) + -- ({-1.0500*\dx},{-0.4880*\dy}) + -- ({-1.0600*\dx},{-0.4979*\dy}) + -- ({-1.0700*\dx},{-0.5077*\dy}) + -- ({-1.0800*\dx},{-0.5174*\dy}) + -- ({-1.0900*\dx},{-0.5270*\dy}) + -- ({-1.1000*\dx},{-0.5365*\dy}) + -- ({-1.1100*\dx},{-0.5459*\dy}) + -- ({-1.1200*\dx},{-0.5552*\dy}) + -- ({-1.1300*\dx},{-0.5643*\dy}) + -- ({-1.1400*\dx},{-0.5733*\dy}) + -- ({-1.1500*\dx},{-0.5821*\dy}) + -- ({-1.1600*\dx},{-0.5908*\dy}) + -- ({-1.1700*\dx},{-0.5993*\dy}) + -- ({-1.1800*\dx},{-0.6075*\dy}) + -- ({-1.1900*\dx},{-0.6156*\dy}) + -- ({-1.2000*\dx},{-0.6234*\dy}) + -- ({-1.2100*\dx},{-0.6310*\dy}) + -- ({-1.2200*\dx},{-0.6383*\dy}) + -- ({-1.2300*\dx},{-0.6454*\dy}) + -- ({-1.2400*\dx},{-0.6522*\dy}) + -- ({-1.2500*\dx},{-0.6587*\dy}) + -- ({-1.2600*\dx},{-0.6648*\dy}) + -- ({-1.2700*\dx},{-0.6707*\dy}) + -- ({-1.2800*\dx},{-0.6763*\dy}) + -- ({-1.2900*\dx},{-0.6815*\dy}) + -- ({-1.3000*\dx},{-0.6863*\dy}) + -- ({-1.3100*\dx},{-0.6908*\dy}) + -- ({-1.3200*\dx},{-0.6950*\dy}) + -- ({-1.3300*\dx},{-0.6987*\dy}) + -- ({-1.3400*\dx},{-0.7021*\dy}) + -- ({-1.3500*\dx},{-0.7050*\dy}) + -- ({-1.3600*\dx},{-0.7076*\dy}) + -- ({-1.3700*\dx},{-0.7097*\dy}) + -- ({-1.3800*\dx},{-0.7114*\dy}) + -- ({-1.3900*\dx},{-0.7127*\dy}) + -- ({-1.4000*\dx},{-0.7135*\dy}) + -- ({-1.4100*\dx},{-0.7139*\dy}) + -- ({-1.4200*\dx},{-0.7139*\dy}) + -- ({-1.4300*\dx},{-0.7134*\dy}) + -- ({-1.4400*\dx},{-0.7125*\dy}) + -- ({-1.4500*\dx},{-0.7111*\dy}) + -- ({-1.4600*\dx},{-0.7093*\dy}) + -- ({-1.4700*\dx},{-0.7070*\dy}) + -- ({-1.4800*\dx},{-0.7043*\dy}) + -- ({-1.4900*\dx},{-0.7011*\dy}) + -- ({-1.5000*\dx},{-0.6975*\dy}) + -- ({-1.5100*\dx},{-0.6935*\dy}) + -- ({-1.5200*\dx},{-0.6890*\dy}) + -- ({-1.5300*\dx},{-0.6841*\dy}) + -- ({-1.5400*\dx},{-0.6788*\dy}) + -- ({-1.5500*\dx},{-0.6731*\dy}) + -- ({-1.5600*\dx},{-0.6670*\dy}) + -- ({-1.5700*\dx},{-0.6605*\dy}) + -- ({-1.5800*\dx},{-0.6536*\dy}) + -- ({-1.5900*\dx},{-0.6464*\dy}) + -- ({-1.6000*\dx},{-0.6389*\dy}) + -- ({-1.6100*\dx},{-0.6310*\dy}) + -- ({-1.6200*\dx},{-0.6229*\dy}) + -- ({-1.6300*\dx},{-0.6144*\dy}) + -- ({-1.6400*\dx},{-0.6057*\dy}) + -- ({-1.6500*\dx},{-0.5968*\dy}) + -- ({-1.6600*\dx},{-0.5876*\dy}) + -- ({-1.6700*\dx},{-0.5782*\dy}) + -- ({-1.6800*\dx},{-0.5687*\dy}) + -- ({-1.6900*\dx},{-0.5590*\dy}) + -- ({-1.7000*\dx},{-0.5492*\dy}) + -- ({-1.7100*\dx},{-0.5393*\dy}) + -- ({-1.7200*\dx},{-0.5293*\dy}) + -- ({-1.7300*\dx},{-0.5194*\dy}) + -- ({-1.7400*\dx},{-0.5094*\dy}) + -- ({-1.7500*\dx},{-0.4994*\dy}) + -- ({-1.7600*\dx},{-0.4895*\dy}) + -- ({-1.7700*\dx},{-0.4796*\dy}) + -- ({-1.7800*\dx},{-0.4699*\dy}) + -- ({-1.7900*\dx},{-0.4603*\dy}) + -- ({-1.8000*\dx},{-0.4509*\dy}) + -- ({-1.8100*\dx},{-0.4418*\dy}) + -- ({-1.8200*\dx},{-0.4328*\dy}) + -- ({-1.8300*\dx},{-0.4241*\dy}) + -- ({-1.8400*\dx},{-0.4157*\dy}) + -- ({-1.8500*\dx},{-0.4077*\dy}) + -- ({-1.8600*\dx},{-0.4000*\dy}) + -- ({-1.8700*\dx},{-0.3927*\dy}) + -- ({-1.8800*\dx},{-0.3858*\dy}) + -- ({-1.8900*\dx},{-0.3793*\dy}) + -- ({-1.9000*\dx},{-0.3733*\dy}) + -- ({-1.9100*\dx},{-0.3678*\dy}) + -- ({-1.9200*\dx},{-0.3629*\dy}) + -- ({-1.9300*\dx},{-0.3584*\dy}) + -- ({-1.9400*\dx},{-0.3545*\dy}) + -- ({-1.9500*\dx},{-0.3511*\dy}) + -- ({-1.9600*\dx},{-0.3484*\dy}) + -- ({-1.9700*\dx},{-0.3462*\dy}) + -- ({-1.9800*\dx},{-0.3447*\dy}) + -- ({-1.9900*\dx},{-0.3437*\dy}) + -- ({-2.0000*\dx},{-0.3434*\dy}) + -- ({-2.0100*\dx},{-0.3437*\dy}) + -- ({-2.0200*\dx},{-0.3447*\dy}) + -- ({-2.0300*\dx},{-0.3462*\dy}) + -- ({-2.0400*\dx},{-0.3484*\dy}) + -- ({-2.0500*\dx},{-0.3513*\dy}) + -- ({-2.0600*\dx},{-0.3547*\dy}) + -- ({-2.0700*\dx},{-0.3587*\dy}) + -- ({-2.0800*\dx},{-0.3633*\dy}) + -- ({-2.0900*\dx},{-0.3685*\dy}) + -- ({-2.1000*\dx},{-0.3743*\dy}) + -- ({-2.1100*\dx},{-0.3805*\dy}) + -- ({-2.1200*\dx},{-0.3873*\dy}) + -- ({-2.1300*\dx},{-0.3945*\dy}) + -- ({-2.1400*\dx},{-0.4022*\dy}) + -- ({-2.1500*\dx},{-0.4103*\dy}) + -- ({-2.1600*\dx},{-0.4188*\dy}) + -- ({-2.1700*\dx},{-0.4276*\dy}) + -- ({-2.1800*\dx},{-0.4367*\dy}) + -- ({-2.1900*\dx},{-0.4461*\dy}) + -- ({-2.2000*\dx},{-0.4557*\dy}) + -- ({-2.2100*\dx},{-0.4655*\dy}) + -- ({-2.2200*\dx},{-0.4754*\dy}) + -- ({-2.2300*\dx},{-0.4853*\dy}) + -- ({-2.2400*\dx},{-0.4953*\dy}) + -- ({-2.2500*\dx},{-0.5053*\dy}) + -- ({-2.2600*\dx},{-0.5152*\dy}) + -- ({-2.2700*\dx},{-0.5250*\dy}) + -- ({-2.2800*\dx},{-0.5346*\dy}) + -- ({-2.2900*\dx},{-0.5440*\dy}) + -- ({-2.3000*\dx},{-0.5532*\dy}) + -- ({-2.3100*\dx},{-0.5620*\dy}) + -- ({-2.3200*\dx},{-0.5704*\dy}) + -- ({-2.3300*\dx},{-0.5784*\dy}) + -- ({-2.3400*\dx},{-0.5860*\dy}) + -- ({-2.3500*\dx},{-0.5931*\dy}) + -- ({-2.3600*\dx},{-0.5996*\dy}) + -- ({-2.3700*\dx},{-0.6056*\dy}) + -- ({-2.3800*\dx},{-0.6110*\dy}) + -- ({-2.3900*\dx},{-0.6157*\dy}) + -- ({-2.4000*\dx},{-0.6197*\dy}) + -- ({-2.4100*\dx},{-0.6230*\dy}) + -- ({-2.4200*\dx},{-0.6256*\dy}) + -- ({-2.4300*\dx},{-0.6275*\dy}) + -- ({-2.4400*\dx},{-0.6286*\dy}) + -- ({-2.4500*\dx},{-0.6289*\dy}) + -- ({-2.4600*\dx},{-0.6285*\dy}) + -- ({-2.4700*\dx},{-0.6273*\dy}) + -- ({-2.4800*\dx},{-0.6254*\dy}) + -- ({-2.4900*\dx},{-0.6226*\dy}) + -- ({-2.5000*\dx},{-0.6192*\dy}) + -- ({-2.5100*\dx},{-0.6150*\dy}) + -- ({-2.5200*\dx},{-0.6101*\dy}) + -- ({-2.5300*\dx},{-0.6045*\dy}) + -- ({-2.5400*\dx},{-0.5983*\dy}) + -- ({-2.5500*\dx},{-0.5915*\dy}) + -- ({-2.5600*\dx},{-0.5842*\dy}) + -- ({-2.5700*\dx},{-0.5763*\dy}) + -- ({-2.5800*\dx},{-0.5679*\dy}) + -- ({-2.5900*\dx},{-0.5591*\dy}) + -- ({-2.6000*\dx},{-0.5500*\dy}) + -- ({-2.6100*\dx},{-0.5406*\dy}) + -- ({-2.6200*\dx},{-0.5309*\dy}) + -- ({-2.6300*\dx},{-0.5210*\dy}) + -- ({-2.6400*\dx},{-0.5111*\dy}) + -- ({-2.6500*\dx},{-0.5011*\dy}) + -- ({-2.6600*\dx},{-0.4911*\dy}) + -- ({-2.6700*\dx},{-0.4812*\dy}) + -- ({-2.6800*\dx},{-0.4715*\dy}) + -- ({-2.6900*\dx},{-0.4621*\dy}) + -- ({-2.7000*\dx},{-0.4529*\dy}) + -- ({-2.7100*\dx},{-0.4441*\dy}) + -- ({-2.7200*\dx},{-0.4358*\dy}) + -- ({-2.7300*\dx},{-0.4279*\dy}) + -- ({-2.7400*\dx},{-0.4207*\dy}) + -- ({-2.7500*\dx},{-0.4140*\dy}) + -- ({-2.7600*\dx},{-0.4080*\dy}) + -- ({-2.7700*\dx},{-0.4027*\dy}) + -- ({-2.7800*\dx},{-0.3982*\dy}) + -- ({-2.7900*\dx},{-0.3944*\dy}) + -- ({-2.8000*\dx},{-0.3915*\dy}) + -- ({-2.8100*\dx},{-0.3895*\dy}) + -- ({-2.8200*\dx},{-0.3883*\dy}) + -- ({-2.8300*\dx},{-0.3880*\dy}) + -- ({-2.8400*\dx},{-0.3886*\dy}) + -- ({-2.8500*\dx},{-0.3900*\dy}) + -- ({-2.8600*\dx},{-0.3924*\dy}) + -- ({-2.8700*\dx},{-0.3956*\dy}) + -- ({-2.8800*\dx},{-0.3996*\dy}) + -- ({-2.8900*\dx},{-0.4045*\dy}) + -- ({-2.9000*\dx},{-0.4101*\dy}) + -- ({-2.9100*\dx},{-0.4165*\dy}) + -- ({-2.9200*\dx},{-0.4235*\dy}) + -- ({-2.9300*\dx},{-0.4312*\dy}) + -- ({-2.9400*\dx},{-0.4394*\dy}) + -- ({-2.9500*\dx},{-0.4481*\dy}) + -- ({-2.9600*\dx},{-0.4572*\dy}) + -- ({-2.9700*\dx},{-0.4667*\dy}) + -- ({-2.9800*\dx},{-0.4764*\dy}) + -- ({-2.9900*\dx},{-0.4863*\dy}) + -- ({-3.0000*\dx},{-0.4963*\dy}) + -- ({-3.0100*\dx},{-0.5063*\dy}) + -- ({-3.0200*\dx},{-0.5162*\dy}) + -- ({-3.0300*\dx},{-0.5259*\dy}) + -- ({-3.0400*\dx},{-0.5354*\dy}) + -- ({-3.0500*\dx},{-0.5445*\dy}) + -- ({-3.0600*\dx},{-0.5531*\dy}) + -- ({-3.0700*\dx},{-0.5613*\dy}) + -- ({-3.0800*\dx},{-0.5688*\dy}) + -- ({-3.0900*\dx},{-0.5757*\dy}) + -- ({-3.1000*\dx},{-0.5818*\dy}) + -- ({-3.1100*\dx},{-0.5872*\dy}) + -- ({-3.1200*\dx},{-0.5917*\dy}) + -- ({-3.1300*\dx},{-0.5952*\dy}) + -- ({-3.1400*\dx},{-0.5979*\dy}) + -- ({-3.1500*\dx},{-0.5996*\dy}) + -- ({-3.1600*\dx},{-0.6003*\dy}) + -- ({-3.1700*\dx},{-0.6001*\dy}) + -- ({-3.1800*\dx},{-0.5988*\dy}) + -- ({-3.1900*\dx},{-0.5966*\dy}) + -- ({-3.2000*\dx},{-0.5933*\dy}) + -- ({-3.2100*\dx},{-0.5892*\dy}) + -- ({-3.2200*\dx},{-0.5842*\dy}) + -- ({-3.2300*\dx},{-0.5783*\dy}) + -- ({-3.2400*\dx},{-0.5716*\dy}) + -- ({-3.2500*\dx},{-0.5642*\dy}) + -- ({-3.2600*\dx},{-0.5562*\dy}) + -- ({-3.2700*\dx},{-0.5476*\dy}) + -- ({-3.2800*\dx},{-0.5385*\dy}) + -- ({-3.2900*\dx},{-0.5290*\dy}) + -- ({-3.3000*\dx},{-0.5193*\dy}) + -- ({-3.3100*\dx},{-0.5094*\dy}) + -- ({-3.3200*\dx},{-0.4994*\dy}) + -- ({-3.3300*\dx},{-0.4894*\dy}) + -- ({-3.3400*\dx},{-0.4796*\dy}) + -- ({-3.3500*\dx},{-0.4700*\dy}) + -- ({-3.3600*\dx},{-0.4608*\dy}) + -- ({-3.3700*\dx},{-0.4521*\dy}) + -- ({-3.3800*\dx},{-0.4439*\dy}) + -- ({-3.3900*\dx},{-0.4364*\dy}) + -- ({-3.4000*\dx},{-0.4296*\dy}) + -- ({-3.4100*\dx},{-0.4237*\dy}) + -- ({-3.4200*\dx},{-0.4186*\dy}) + -- ({-3.4300*\dx},{-0.4145*\dy}) + -- ({-3.4400*\dx},{-0.4114*\dy}) + -- ({-3.4500*\dx},{-0.4094*\dy}) + -- ({-3.4600*\dx},{-0.4084*\dy}) + -- ({-3.4700*\dx},{-0.4085*\dy}) + -- ({-3.4800*\dx},{-0.4097*\dy}) + -- ({-3.4900*\dx},{-0.4119*\dy}) + -- ({-3.5000*\dx},{-0.4152*\dy}) + -- ({-3.5100*\dx},{-0.4196*\dy}) + -- ({-3.5200*\dx},{-0.4249*\dy}) + -- ({-3.5300*\dx},{-0.4311*\dy}) + -- ({-3.5400*\dx},{-0.4381*\dy}) + -- ({-3.5500*\dx},{-0.4459*\dy}) + -- ({-3.5600*\dx},{-0.4543*\dy}) + -- ({-3.5700*\dx},{-0.4633*\dy}) + -- ({-3.5800*\dx},{-0.4727*\dy}) + -- ({-3.5900*\dx},{-0.4824*\dy}) + -- ({-3.6000*\dx},{-0.4923*\dy}) + -- ({-3.6100*\dx},{-0.5023*\dy}) + -- ({-3.6200*\dx},{-0.5122*\dy}) + -- ({-3.6300*\dx},{-0.5220*\dy}) + -- ({-3.6400*\dx},{-0.5314*\dy}) + -- ({-3.6500*\dx},{-0.5404*\dy}) + -- ({-3.6600*\dx},{-0.5489*\dy}) + -- ({-3.6700*\dx},{-0.5567*\dy}) + -- ({-3.6800*\dx},{-0.5637*\dy}) + -- ({-3.6900*\dx},{-0.5698*\dy}) + -- ({-3.7000*\dx},{-0.5750*\dy}) + -- ({-3.7100*\dx},{-0.5791*\dy}) + -- ({-3.7200*\dx},{-0.5822*\dy}) + -- ({-3.7300*\dx},{-0.5841*\dy}) + -- ({-3.7400*\dx},{-0.5849*\dy}) + -- ({-3.7500*\dx},{-0.5845*\dy}) + -- ({-3.7600*\dx},{-0.5830*\dy}) + -- ({-3.7700*\dx},{-0.5803*\dy}) + -- ({-3.7800*\dx},{-0.5764*\dy}) + -- ({-3.7900*\dx},{-0.5715*\dy}) + -- ({-3.8000*\dx},{-0.5656*\dy}) + -- ({-3.8100*\dx},{-0.5588*\dy}) + -- ({-3.8200*\dx},{-0.5512*\dy}) + -- ({-3.8300*\dx},{-0.5428*\dy}) + -- ({-3.8400*\dx},{-0.5338*\dy}) + -- ({-3.8500*\dx},{-0.5244*\dy}) + -- ({-3.8600*\dx},{-0.5147*\dy}) + -- ({-3.8700*\dx},{-0.5047*\dy}) + -- ({-3.8800*\dx},{-0.4947*\dy}) + -- ({-3.8900*\dx},{-0.4848*\dy}) + -- ({-3.9000*\dx},{-0.4752*\dy}) + -- ({-3.9100*\dx},{-0.4660*\dy}) + -- ({-3.9200*\dx},{-0.4573*\dy}) + -- ({-3.9300*\dx},{-0.4492*\dy}) + -- ({-3.9400*\dx},{-0.4420*\dy}) + -- ({-3.9500*\dx},{-0.4357*\dy}) + -- ({-3.9600*\dx},{-0.4303*\dy}) + -- ({-3.9700*\dx},{-0.4261*\dy}) + -- ({-3.9800*\dx},{-0.4230*\dy}) + -- ({-3.9900*\dx},{-0.4211*\dy}) + -- ({-4.0000*\dx},{-0.4205*\dy}) + -- ({-4.0100*\dx},{-0.4211*\dy}) + -- ({-4.0200*\dx},{-0.4230*\dy}) + -- ({-4.0300*\dx},{-0.4261*\dy}) + -- ({-4.0400*\dx},{-0.4304*\dy}) + -- ({-4.0500*\dx},{-0.4358*\dy}) + -- ({-4.0600*\dx},{-0.4422*\dy}) + -- ({-4.0700*\dx},{-0.4495*\dy}) + -- ({-4.0800*\dx},{-0.4576*\dy}) + -- ({-4.0900*\dx},{-0.4665*\dy}) + -- ({-4.1000*\dx},{-0.4758*\dy}) + -- ({-4.1100*\dx},{-0.4855*\dy}) + -- ({-4.1200*\dx},{-0.4955*\dy}) + -- ({-4.1300*\dx},{-0.5054*\dy}) + -- ({-4.1400*\dx},{-0.5153*\dy}) + -- ({-4.1500*\dx},{-0.5249*\dy}) + -- ({-4.1600*\dx},{-0.5341*\dy}) + -- ({-4.1700*\dx},{-0.5426*\dy}) + -- ({-4.1800*\dx},{-0.5504*\dy}) + -- ({-4.1900*\dx},{-0.5573*\dy}) + -- ({-4.2000*\dx},{-0.5632*\dy}) + -- ({-4.2100*\dx},{-0.5680*\dy}) + -- ({-4.2200*\dx},{-0.5716*\dy}) + -- ({-4.2300*\dx},{-0.5739*\dy}) + -- ({-4.2400*\dx},{-0.5749*\dy}) + -- ({-4.2500*\dx},{-0.5746*\dy}) + -- ({-4.2600*\dx},{-0.5730*\dy}) + -- ({-4.2700*\dx},{-0.5700*\dy}) + -- ({-4.2800*\dx},{-0.5658*\dy}) + -- ({-4.2900*\dx},{-0.5604*\dy}) + -- ({-4.3000*\dx},{-0.5540*\dy}) + -- ({-4.3100*\dx},{-0.5466*\dy}) + -- ({-4.3200*\dx},{-0.5383*\dy}) + -- ({-4.3300*\dx},{-0.5294*\dy}) + -- ({-4.3400*\dx},{-0.5199*\dy}) + -- ({-4.3500*\dx},{-0.5101*\dy}) + -- ({-4.3600*\dx},{-0.5001*\dy}) + -- ({-4.3700*\dx},{-0.4902*\dy}) + -- ({-4.3800*\dx},{-0.4804*\dy}) + -- ({-4.3900*\dx},{-0.4711*\dy}) + -- ({-4.4000*\dx},{-0.4623*\dy}) + -- ({-4.4100*\dx},{-0.4542*\dy}) + -- ({-4.4200*\dx},{-0.4471*\dy}) + -- ({-4.4300*\dx},{-0.4410*\dy}) + -- ({-4.4400*\dx},{-0.4360*\dy}) + -- ({-4.4500*\dx},{-0.4323*\dy}) + -- ({-4.4600*\dx},{-0.4299*\dy}) + -- ({-4.4700*\dx},{-0.4289*\dy}) + -- ({-4.4800*\dx},{-0.4293*\dy}) + -- ({-4.4900*\dx},{-0.4311*\dy}) + -- ({-4.5000*\dx},{-0.4343*\dy}) + -- ({-4.5100*\dx},{-0.4387*\dy}) + -- ({-4.5200*\dx},{-0.4444*\dy}) + -- ({-4.5300*\dx},{-0.4512*\dy}) + -- ({-4.5400*\dx},{-0.4590*\dy}) + -- ({-4.5500*\dx},{-0.4676*\dy}) + -- ({-4.5600*\dx},{-0.4768*\dy}) + -- ({-4.5700*\dx},{-0.4864*\dy}) + -- ({-4.5800*\dx},{-0.4964*\dy}) + -- ({-4.5900*\dx},{-0.5064*\dy}) + -- ({-4.6000*\dx},{-0.5162*\dy}) + -- ({-4.6100*\dx},{-0.5257*\dy}) + -- ({-4.6200*\dx},{-0.5346*\dy}) + -- ({-4.6300*\dx},{-0.5427*\dy}) + -- ({-4.6400*\dx},{-0.5500*\dy}) + -- ({-4.6500*\dx},{-0.5562*\dy}) + -- ({-4.6600*\dx},{-0.5611*\dy}) + -- ({-4.6700*\dx},{-0.5648*\dy}) + -- ({-4.6800*\dx},{-0.5670*\dy}) + -- ({-4.6900*\dx},{-0.5678*\dy}) + -- ({-4.7000*\dx},{-0.5671*\dy}) + -- ({-4.7100*\dx},{-0.5650*\dy}) + -- ({-4.7200*\dx},{-0.5615*\dy}) + -- ({-4.7300*\dx},{-0.5566*\dy}) + -- ({-4.7400*\dx},{-0.5504*\dy}) + -- ({-4.7500*\dx},{-0.5432*\dy}) + -- ({-4.7600*\dx},{-0.5350*\dy}) + -- ({-4.7700*\dx},{-0.5261*\dy}) + -- ({-4.7800*\dx},{-0.5166*\dy}) + -- ({-4.7900*\dx},{-0.5067*\dy}) + -- ({-4.8000*\dx},{-0.4968*\dy}) + -- ({-4.8100*\dx},{-0.4869*\dy}) + -- ({-4.8200*\dx},{-0.4773*\dy}) + -- ({-4.8300*\dx},{-0.4682*\dy}) + -- ({-4.8400*\dx},{-0.4600*\dy}) + -- ({-4.8500*\dx},{-0.4526*\dy}) + -- ({-4.8600*\dx},{-0.4464*\dy}) + -- ({-4.8700*\dx},{-0.4414*\dy}) + -- ({-4.8800*\dx},{-0.4378*\dy}) + -- ({-4.8900*\dx},{-0.4357*\dy}) + -- ({-4.9000*\dx},{-0.4351*\dy}) + -- ({-4.9100*\dx},{-0.4360*\dy}) + -- ({-4.9200*\dx},{-0.4384*\dy}) + -- ({-4.9300*\dx},{-0.4423*\dy}) + -- ({-4.9400*\dx},{-0.4476*\dy}) + -- ({-4.9500*\dx},{-0.4541*\dy}) + -- ({-4.9600*\dx},{-0.4618*\dy}) + -- ({-4.9700*\dx},{-0.4703*\dy}) + -- ({-4.9800*\dx},{-0.4795*\dy}) + -- ({-4.9900*\dx},{-0.4892*\dy}) +} + diff --git a/vorlesungen/slides/fresnel/eulerspirale.m b/vorlesungen/slides/fresnel/eulerspirale.m new file mode 100644 index 0000000..84e3696 --- /dev/null +++ b/vorlesungen/slides/fresnel/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/vorlesungen/slides/fresnel/integrale.tex b/vorlesungen/slides/fresnel/integrale.tex new file mode 100644 index 0000000..906aec1 --- /dev/null +++ b/vorlesungen/slides/fresnel/integrale.tex @@ -0,0 +1,119 @@ +% +% integrale.tex -- Definition der Fresnel Integrale +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\input{../slides/fresnel/eulerpath.tex} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Fresnel-Integrale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Fresnel-Integrale: +\begin{align*} +\color{red}S(t) +&= +\int_0^t \sin\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau +\\ +\color{blue}C(t) +&= +\int_0^t \cos\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau +\end{align*} +\uncover<3->{% +Können nicht in geschlossener Form ausgewertet werden. +} +\end{block} +\uncover<4->{% +\begin{block}{Euler-Spirale} +\[ +\gamma_a(t) += +\begin{pmatrix} +C_a(t)\\S_a(t) +\end{pmatrix} += +\begin{pmatrix} +\displaystyle +\int_0^t \cos (a\tau^2)\,d\tau\\[8pt] +\displaystyle +\int_0^t \sin (a\tau^2)\,d\tau +\end{pmatrix} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\ifthenelse{\boolean{presentation}}{ +\only<2-4>{% +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=1] +\def\dx{0.6} +\def\dy{1.5} + +\begin{scope} + \draw[color=gray!50] (0,{0.5*\dy}) -- (3,{0.5*\dy}); + \draw[color=gray!50] (0,{-0.5*\dy}) -- (-3,{-0.5*\dy}); + \draw[->] (-3,0) -- (3.3,0) coordinate[label={$t$}]; + \draw[->] (0,-1.5) -- (0,1.5) coordinate[label={left:$S(t)$}]; + \draw (-0.1,{0.5*\dy}) -- (0.1,{0.5*\dy}); + \node at (-0.1,{0.5*\dy}) [left] {$\frac12$}; + \draw (-0.1,{-0.5*\dy}) -- (0.1,{-0.5*\dy}); + \node at (0.1,{-0.5*\dy}) [right] {$-\frac12$}; + \draw[color=red,line width=1.4pt] \Splotright; + \draw[color=red,line width=1.4pt] \Splotleft; +\end{scope} + +\begin{scope}[yshift=-3.4cm] + \draw[color=gray!50] (0,{0.5*\dy}) -- (3,{0.5*\dy}); + \draw[color=gray!50] (0,{-0.5*\dy}) -- (-3,{-0.5*\dy}); + \draw[->] (-3,0) -- (3.3,0) coordinate[label={$t$}]; + \draw[->] (0,-1.5) -- (0,1.5) coordinate[label={left:$C(t)$}]; + \draw (-0.1,{0.5*\dy}) -- (0.1,{0.5*\dy}); + \node at (-0.1,{0.5*\dy}) [left] {$\frac12$}; + \draw (-0.1,{-0.5*\dy}) -- (0.1,{-0.5*\dy}); + \node at (0.1,{-0.5*\dy}) [right] {$-\frac12$}; + \draw[color=blue,line width=1.4pt] \Cplotright; + \draw[color=blue,line width=1.4pt] \Cplotleft; +\end{scope} + +\end{tikzpicture} +\end{center} +}}{} +\uncover<5->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=3.5] + +\draw[color=gray!50] (-0.5,-0.5) rectangle (0.5,0.5); + +\draw[->] (-0.8,0) -- (0.9,0) coordinate[label={$\color{blue}C(t)$}]; +\draw[->] (0,-0.8) -- (0,0.9) coordinate[label={right:$\color{red}S(t)$}]; + +\draw[color=darkgreen,line width=1.0pt] \fresnela; +\draw[color=darkgreen,line width=1.0pt] \fresnelb; + +\fill[color=orange] (0.5,0.5) circle[radius=0.02]; +\fill[color=orange] (-0.5,-0.5) circle[radius=0.02]; + +\draw (0.5,-0.02) -- (0.5,0.02); +\node at (0.5,-0.02) [below right] {$\frac12$}; + +\draw (-0.5,-0.02) -- (-0.5,0.02); +\node at (-0.5,0.02) [above left] {$-\frac12$}; + +\draw (-0.01,0.5) -- (0.02,0.5); +\node at (-0.02,0.5) [above left] {$\frac12$}; + +\draw (-0.02,-0.5) -- (0.02,-0.5); +\node at (0.02,-0.5) [below right] {$-\frac12$}; + +\end{tikzpicture} +\end{center} +} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/fresnel/klothoide.tex b/vorlesungen/slides/fresnel/klothoide.tex new file mode 100644 index 0000000..bf43644 --- /dev/null +++ b/vorlesungen/slides/fresnel/klothoide.tex @@ -0,0 +1,68 @@ +% +% klothoide.tex -- Klothoide +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Klothoide} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Krümmung der Euler-Spirale} +\begin{align*} +\frac{d}{dt}\gamma_1(t) +&= +\dot{\gamma}_1(t) += +\begin{pmatrix} +\cos t^2\\ +\sin t^2 +\end{pmatrix} +\intertext{\uncover<2->{Bogenlänge:}} +\uncover<2->{ +|\dot{\gamma}_1(t)| +&= +\sqrt{\cos^2 t^2 + \sin^2 t^2} += +1 +} +\intertext{\uncover<3->{Polarwinkel:}} +\uncover<3->{ +\varphi&=t^2 +\intertext{\uncover<4->{Krümmung:}} +\uncover<4->{ +\frac{d\varphi}{dt} +&= +2t +} +} +\end{align*} +\uncover<5->{% +$\Rightarrow$ Krümmung ist proportional zur Bogenlänge +} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Definition} +Eine Kurve, deren Krümmung proportional zur Bogenlänge ist, heisst +{\em Klothoide} +\end{block}} +\uncover<7->{% +\begin{block}{Anwendung} +\begin{itemize} +\item<8-> +Strassenbau: Um mit konstanter Geschwindigkeit auf einer +Klothoide zu fahren, muss man das Lenkrad mit konstanter Geschwindigkeit +drehen +\item<9-> +Apfel + Sparschäler +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/fresnel/kruemmung.tex b/vorlesungen/slides/fresnel/kruemmung.tex new file mode 100644 index 0000000..06f6b9b --- /dev/null +++ b/vorlesungen/slides/fresnel/kruemmung.tex @@ -0,0 +1,91 @@ +% +% kruemmung.tex -- Kruemmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Krümmung einer Kurve} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Krümmungsradius} +Bogen und Radius: +\[ +s=r\cdot\Delta\varphi +\uncover<2->{ +\quad +\Rightarrow +\quad +r += +\frac{s}{\Delta\varphi} +} +\] +\end{block} +\vspace*{-12pt} +\uncover<3->{ +\begin{block}{Krümmung} +Je grösser der Krümmungsradius, desto kleiner die Krümmung: +\[ +\kappa = \frac{1}{r} +\] +\end{block}} +\vspace*{-12pt} +\uncover<5->{% +\begin{block}{Definition} +Änderungsgeschwindigkeit des Polarwinkels der Tangente +\[ +\kappa += +\frac{1}{r} +\uncover<6->{= +\frac{\Delta\varphi}{s}} +\uncover<7->{= +\frac{d\varphi}{dt}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\begin{scope} +\clip (-1,-1) rectangle (4,4); + +\def\r{3} +\def\winkel{30} + +\fill[color=blue!20] (0,0) -- (0:\r) arc (0:\winkel:\r) -- cycle; +\node[color=blue] at ({0.5*\winkel}:{0.5*\r}) {$\Delta\varphi$}; + +\draw[line width=0.3pt] (0,0) circle[radius=\r]; + +\draw[->] (0,0) -- (0:\r); +\draw[->] (0,0) -- (\winkel:\r); + +\uncover<4->{ +\draw[->] (0:\r) -- ($(0:\r)+(90:0.7*\r)$); +\draw[->] (\winkel:\r) -- ($(\winkel:\r)+({90+\winkel}:0.7*\r)$); +} + +\draw[color=red,line width=1.4pt] (0:\r) arc (0:\winkel:\r); +\node[color=red] at ({0.5*\winkel}:\r) [left] {$s$}; +\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{center} +\uncover<4->{% +Für $\varphi$ kann man auch den Polarwinkel des Tangentialvektors nehmen +} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/fresnel/numerik.tex b/vorlesungen/slides/fresnel/numerik.tex new file mode 100644 index 0000000..0bd4d5a --- /dev/null +++ b/vorlesungen/slides/fresnel/numerik.tex @@ -0,0 +1,124 @@ +% +% numerik.tex -- numerische Berechnung der Fresnel Integrale +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Numerik} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\sin t^{\uncover<2->{\color<2>{red}2}} +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{ +\ifthenelse{\boolean{presentation}}{\only<1>{2k+1}}{} +\only<2->{\color<2>{red}4k+2} +} +}{ +(2k+1)! +} +\\ +%\int \sin t^2\,dt +\uncover<4->{ +S_1(t) +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{4k+3}}{(2k+1)!(4n+3)} +} +\\ +\cos t^{\uncover<3->{\color<3>{red}2}} +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{ +\ifthenelse{\boolean{presentation}}{\only<-2>{2k}}{} +\only<3->{\color<3>{red}4k}} +}{ +(2k)! +} +\\ +%\int \sin t^2\,dt +\uncover<5->{ +C_1(t) +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{4k+1}}{(2k)!(4k+1)} +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{ +\begin{block}{Differentialgleichung} +\[ +\dot{\gamma}_1(t) += +\begin{pmatrix} +\cos t^2\\ \sin t^2 +\end{pmatrix} +\uncover<7->{ +\; +\to +\; +\gamma_1(t) += +\begin{pmatrix} +C_1(t)\\S_1(t) +\end{pmatrix} +} +\] +\end{block}} +\uncover<8->{% +\begin{block}{Hypergeometrische Reihen} +\begin{align*} +\uncover<9->{% +S(t) +&= +\frac{\pi z^3}{6} +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac34\\\frac32,\frac74\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr) +} +\\ +\uncover<10->{ +C(t) +&= +z +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac14\\\frac12,\frac54\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr)} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<11->{% +\begin{block}{Komplexe Fehlerfunktion} +\[ +\left. +\begin{matrix} +S(z)\\ +C(z) +\end{matrix} +\right\} += +\frac{1\pm i}{4} +\left( +\operatorname{erf}\biggl({\frac{1+i}2}\sqrt{\pi}z\biggr) +\mp i +\operatorname{erf}\biggl({\frac{1-i}2}\sqrt{\pi}z\biggr) +\right) +\] +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/fresnel/test.tex b/vorlesungen/slides/fresnel/test.tex deleted file mode 100644 index 6c2f25b..0000000 --- a/vorlesungen/slides/fresnel/test.tex +++ /dev/null @@ -1,19 +0,0 @@ -% -% template.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Template für Klothoide} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\end{column} -\begin{column}{0.48\textwidth} -\end{column} -\end{columns} -\end{frame} -\egroup diff --git a/vorlesungen/slides/hermite/Makefile.inc b/vorlesungen/slides/hermite/Makefile.inc index 5c55467..58c21f2 100644 --- a/vorlesungen/slides/hermite/Makefile.inc +++ b/vorlesungen/slides/hermite/Makefile.inc @@ -4,4 +4,9 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapterhermite = \ + ../slides/hermite/normalintegrale.tex \ + ../slides/hermite/normalhermite.tex \ + ../slides/hermite/hermiteentwicklung.tex \ + ../slides/hermite/loesung.tex \ + ../slides/hermite/skalarprodukt.tex \ ../slides/hermite/test.tex diff --git a/vorlesungen/slides/hermite/hermiteentwicklung.tex b/vorlesungen/slides/hermite/hermiteentwicklung.tex new file mode 100644 index 0000000..5f6e1c9 --- /dev/null +++ b/vorlesungen/slides/hermite/hermiteentwicklung.tex @@ -0,0 +1,72 @@ +% +% hermiteentwicklung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beliebige Polynome} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Polynom} +\[ +P(x) += +p_0 + p_1x + p_2x^2 + \dots + p_nx^n +\] +\uncover<2->{% +als Linearkombination von Hermite-Polynome schreiben: +\begin{align*} +P(x) +&= +a_0H_0(x)% + a_1H_1(x) ++ \dots + a_nH_n(x) +\\ +&= +a_0\cdot 1 +\\ +&\quad + a_1\cdot 2x +\\ +&\quad + a_2\cdot(4x^2-2) +\\ +&\quad + a_3\cdot(8x^3-12x) +\\ +&\quad + a_4\cdot(16x^4-48x^2+12) +\\ +&\quad\;\;\vdots +\\ +&\quad + a_n(2^nx^n + \dots) +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Koeffizientenvergleich} +führt auf ein Gleichungssystem +\begin{center} +\begin{tabular}{|>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}c<{$}|>{$}c<{$}|} +\hline +a_0&a_1&a_2&a_3&a_4&\dots&\\ +\hline + 1& 0& 0& 0& 0&\dots&p_0\\ + 0& 2& 0& 0& 0&\dots&p_1\\ +-2& 0& 4& 0& 0&\dots&p_2\\ + 0&-12& 0& 8& 0&\dots&p_3\\ +12& 0&-48& 0& 16&\dots&p_4\\ +\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ +\hline +\end{tabular} +\end{center} +\uncover<4->{% +Dreiecksmatrix}\uncover<5->{, Diagonalelement +$\ne 0$} +\uncover<6->{$\Rightarrow$ +$\exists$ eindeutige Lösung} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/loesung.tex b/vorlesungen/slides/hermite/loesung.tex new file mode 100644 index 0000000..68ee32e --- /dev/null +++ b/vorlesungen/slides/hermite/loesung.tex @@ -0,0 +1,65 @@ +% +% loesung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frage} +Für welche Polynome $P(t)$ kann man eine Stammfunktion +\[ +\int +P(t)e^{-\frac{t^2}2} +\,dt +\] +in geschlossener Form angeben? +\end{block} +\uncover<2->{% +\begin{block}{``Hermite-Antwort''} +\[ +\int H_n(x)e^{-x^2}\,dx +\] +kann genau für $n>0$ in geschlossener Form angegeben werden. +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Allgemein} +\begin{align*} +\int P(x)e^{-x^2}\,dx +&\uncover<4->{= +\int \sum_{k=0}^n a_kH_k(x)e^{-x^2}\,dx} +\\ +\uncover<5->{ +&= +\sum_{k=0}^n +a_k +\int +H_k(x)e^{-x^2}\,dx +} +\\ +\uncover<6->{ +&= +a_0\operatorname{erf}(x) + C +} +\\ +\uncover<6->{ +&\hspace*{2mm} + \sum_{k=1}^n a_k\int H_k(x)e^{-x^2}\,dx +} +\end{align*} +\end{block}} +\uncover<7->{% +\begin{theorem} +Das Integral von $P(x)e^{-x^2}$ ist genau dann elementar darstellbar, wenn +$a_0=0$ +\end{theorem}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/normalhermite.tex b/vorlesungen/slides/hermite/normalhermite.tex new file mode 100644 index 0000000..98721dc --- /dev/null +++ b/vorlesungen/slides/hermite/normalhermite.tex @@ -0,0 +1,103 @@ +% +% normalhermite.tex -- integrability of hermite polynomials +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hermite-Polynome} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition (Rodrigues-Formel)} +\[ +H_n(x) += +(-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Orthogonalität} +$H_n(x)$ sind orthogonale Polynome bezüglich $w(x)=e^{-x^2}$, d.~h. +\begin{align*} +\langle H_n,H_m\rangle_w +&= +\int H_n(x)H_m(x)e^{-x^2}\,dx +\\ +&= +\biggl\{ +\renewcommand{\arraycolsep}{1pt} +\begin{array}{l@{\quad}l} +1&\text{falls $n=m$}\\ +0&\text{sonst} +\end{array} +\biggr\} += +\delta_{mn} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Rekursion: Auf-/Absteigeoperatoren} +Rekursionsformel: +\[ +H_n(x) += +2x\cdot H_{n-1}(x) - H_{n-1}'(x) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Stammfunktion} +\begin{align*} +\uncover<4->{ +\int H_n(x) e^{-x^2}\,dx} +&\uncover<5->{= +\int \bigl({\color{red}2x}H_{n-1}(x)} +\\ +\uncover<5->{ +&\qquad -H_{n-1}'(x)\bigr) e^{-x^2}\,dx +} +\\ +\uncover<6->{ +{\color{gray}((e^{-x^2})'=-2x)} +&= +{\color{red}-}\int {\color{red}(e^{-x^2})'} H_{n-1}(x)\,dx +} +\\ +\uncover<6->{ +&\qquad +- +\int H_{n-1}'(x) e^{-x^2}\,dx +} +\\ +\uncover<7->{ +\text{\color{gray}(Produktregel)} +&= +\int (e^{-x^2}H_{n-1}(x))'\,dx +} +\\ +\uncover<8->{ +\text{\color{gray}(Ableitung)} +&= +e^{-x^2}H_{n-1}(x) +} +\end{align*} +\uncover<9->{% +ausser für $n=0$: +\[ +\int +H_0(x)e^{-x^2}\,dx += +\int +e^{-x^2}\,dx +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/normalintegrale.tex b/vorlesungen/slides/hermite/normalintegrale.tex new file mode 100644 index 0000000..32333cd --- /dev/null +++ b/vorlesungen/slides/hermite/normalintegrale.tex @@ -0,0 +1,57 @@ +% +% normalintegrale.tex -- +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Integranden $P(t)e^{-t^2}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frage} +Für welche Polynome $P(t)$ kann man eine Stammfunktion +\[ +\int +P(t)e^{-t^2} +\,dt +\] +in geschlossener Form angeben? +\end{block} +\uncover<4->{% +\begin{block}{Allgemeine Antwort} +Satz von Liouville und +Risch- Algorithmus können entscheiden, ob es eine elementare Stammfunktion gibt +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Negativbeispiel} +$P(t) = 1$, das Normalverteilungsintegral +\[ +F(x) += +\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2}\,dt +\] +ist nicht elementar darstellbar. +\end{block}} +\uncover<3->{% +\begin{block}{Positivbeispiel} +$P(t)=t$. Wegen +\begin{align*} +\frac{d}{dx}e^{-x^2} +&= +-xe^{-x^2} +\intertext{ist} +\int te^{-t^2}\,dt +&= +-e^{-x^2}+C +\end{align*} +elementar darstellbar. +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/skalarprodukt.tex b/vorlesungen/slides/hermite/skalarprodukt.tex new file mode 100644 index 0000000..a51e9f6 --- /dev/null +++ b/vorlesungen/slides/hermite/skalarprodukt.tex @@ -0,0 +1,82 @@ +% +% skalarprodukt.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Skalarprodukt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Orthogonale Zerlegung} +Orthogonale $H_k$ normalisieren: +\[ +\tilde{H}_k(x) = \frac{1}{\|H_k\|_w} H_k(x) +\] +mit Gewichtsfunktion $w(x)=e^{-x^2}$ +\end{block} +\uncover<2->{% +\begin{block}{``Hermite''-Analyse} +\begin{align*} +P(x) +&= +\sum_{k=1}^\infty a_k H_k(x) += +\sum_{k=1}^\infty \tilde{a}_k \tilde{H}_k(x) +\\ +\uncover<3->{ +\tilde{a}_k +&= +\| H_k\|_w\, a_k +} +\\ +\uncover<4->{ +a_k +&= +\frac{1}{\|H_k\|} +\langle \tilde{H}_k, P\rangle_w +}\uncover<5->{= +\frac{1}{\|H_k\|^2} +\langle H_k, P\rangle_w +} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Integrationsproblem} +Bedingung: +\begin{align*} +a_0=0 +\uncover<7->{% +\qquad\Leftrightarrow\qquad +\langle H_0,P\rangle_w +&= +0} +\\ +\uncover<8->{% +\int_{-\infty}^\infty +P(t) w(t) \,dt +}\uncover<9->{% += +\int_{-\infty}^\infty +P(t) e^{-t^2} \,dt +&= +0} +\end{align*} +\end{block}} +\uncover<10->{% +\begin{theorem} +Das Integral von $P(t)e^{-t^2}$ ist in geschlossener Form darstellbar +genau dann, wenn +\[ +\int_{-\infty}^\infty P(t)e^{-t^2}\,dt = 0 +\] +\end{theorem}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 6aa09f8..ca4ccc9 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,4 +3,8 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{0/intro.tex} +\folie{hermite/normalintegrale.tex} +\folie{hermite/normalhermite.tex} +\folie{hermite/hermiteentwicklung.tex} +\folie{hermite/loesung.tex} +\folie{hermite/skalarprodukt.tex} diff --git a/vorlesungen/stream/countdown.html b/vorlesungen/stream/countdown.html index d8ec82e..e9d7d6e 100644 --- a/vorlesungen/stream/countdown.html +++ b/vorlesungen/stream/countdown.html @@ -25,7 +25,7 @@ function checkfor(d) { console.log("time string: " + ds); let start = new Date(ds).getTime(); console.log("now: " + now); - if ((start > now) && ((start-now) < 3600*1000)) { + if ((start > now) && ((start-now) < 3300*1000)) { deadline = start; console.log("set deadline to: " + ds); } else { |
