diff options
95 files changed, 8132 insertions, 518 deletions
diff --git a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex index acfdb1a..2e43cec 100644 --- a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex +++ b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex @@ -263,7 +263,7 @@ werden können, muss auch = \int_{-1}^1 q(x)p(x)\,dx = -\sum_{i=0}^n q(x_i)p(x_i) +\sum_{i=0}^n A_iq(x_i)p(x_i) \] für jedes beliebige Polynom $q\in R_{n-1}$ gelten. Da man für $q$ die Interpolationspolynome $l_j(x)$ verwenden @@ -272,9 +272,11 @@ kann, den Grad $n-1$ haben, folgt 0 = \sum_{i=0}^n -l_j(x_i)p(x_i) +A_il_j(x_i)p(x_i) = -\sum_{i=0}^n \delta_{ij}p(x_i), +\sum_{i=0}^n A_i\delta_{ij}p(x_i) += +A_jp(x_j), \] die Stützstellen $x_i$ müssen also die Nullstellen des Polynoms $p(x)$ sein. diff --git a/buch/chapters/070-orthogonalitaet/orthogonal.tex b/buch/chapters/070-orthogonalitaet/orthogonal.tex index a84248a..677e865 100644 --- a/buch/chapters/070-orthogonalitaet/orthogonal.tex +++ b/buch/chapters/070-orthogonalitaet/orthogonal.tex @@ -842,14 +842,14 @@ bei geeigneter Normierung die {\em Hermite-Polynome}. % % Laguerre-Gewichtsfunktion % -\subsection{Laguerre-Gewichtsfunktion} +\subsubsection{Laguerre-Gewichtsfunktion} Ähnlich wie die Hermite-Gewichtsfunktion ist die {\em Laguerre-Gewichtsfunktion} \index{Laguerre-Gewichtsfunktion}% \[ w_{\text{Laguerre}}(x) = -w^{-x} +e^{-x} \] auf ganz $\mathbb{R}$ definiert, und sie geht für $x\to\infty$ wieder sehr rasch gegen $0$. diff --git a/buch/chapters/070-orthogonalitaet/rekursion.tex b/buch/chapters/070-orthogonalitaet/rekursion.tex index 5ec7fed..dc5531b 100644 --- a/buch/chapters/070-orthogonalitaet/rekursion.tex +++ b/buch/chapters/070-orthogonalitaet/rekursion.tex @@ -30,7 +30,7 @@ Skalarproduktes $\langle\,\;,\;\rangle_w$, wenn für alle $n$, $m$. \end{definition} -\subsection{Allgemeine Drei-Term-Rekursion für orthogonale Polynome} +\subsubsection{Allgemeine Drei-Term-Rekursion für orthogonale Polynome} Der folgende Satz besagt, dass $p_n$ eine Rekursionsbeziehung erfüllt. \begin{satz} @@ -55,7 +55,7 @@ C_{n+1} = \frac{A_{n+1}}{A_n}\frac{h_{n+1}}{h_n}. \end{equation} \end{satz} -\subsection{Multiplikationsoperator mit $x$} +\subsubsection{Multiplikationsoperator mit $x$} Man kann die Relation auch nach dem Produkt $xp_n(x)$ auflösen, dann wird sie \begin{equation} @@ -72,7 +72,7 @@ Die Multiplikation mit $x$ ist eine lineare Abbildung im Raum der Funktionen. Die Relation~\eqref{buch:orthogonal:eqn:multixrelation} besagt, dass diese Abbildung in der Basis der Polynome $p_k$ tridiagonale Form hat. -\subsection{Drei-Term-Rekursion für die Tschebyscheff-Polynome} +\subsubsection{Drei-Term-Rekursion für die Tschebyscheff-Polynome} Eine Relation der Form~\eqref{buch:orthogonal:eqn:multixrelation} wurde bereits in Abschnitt~\ref{buch:potenzen:tschebyscheff:rekursionsbeziehungen} @@ -80,12 +80,12 @@ hergeleitet. In der Form~\eqref{buch:orthogonal:eqn:rekursion} geschrieben lautet sie \[ -T_{n+1}(x) = 2x\,T_n(x)-T_{n-1}(x). +T_{n+1}(x) = 2x\,T_n(x)-T_{n-1}(x), \] also $A_n=2$, $B_n=0$ und $C_n=1$. -\subsection{Beweis von Satz~\ref{buch:orthogonal:satz:drei-term-rekursion}} +\subsubsection{Beweis von Satz~\ref{buch:orthogonal:satz:drei-term-rekursion}} Die Relation~\eqref{buch:orthogonal:eqn:multixrelation} zeigt auch, dass der Beweis die Koeffizienten $\langle xp_k,p_j\rangle_w$ berechnen muss. diff --git a/buch/chapters/070-orthogonalitaet/sturm.tex b/buch/chapters/070-orthogonalitaet/sturm.tex index c9c9cc6..35054ab 100644 --- a/buch/chapters/070-orthogonalitaet/sturm.tex +++ b/buch/chapters/070-orthogonalitaet/sturm.tex @@ -375,7 +375,7 @@ automatisch für diese Funktionenfamilien. \subsubsection{Trigonometrische Funktionen} Die trigonometrischen Funktionen sind Eigenfunktionen des Operators $d^2/dx^2$, also eines Sturm-Liouville-Operators mit $p(x)=1$, $q(x)=0$ -und $w(x)=0$. +und $w(x)=1$. Auf dem Intervall $(-\pi,\pi)$ können wir die Randbedingungen \bgroup \renewcommand{\arraycolsep}{2pt} diff --git a/buch/papers/common/addpapers.tex b/buch/papers/common/addpapers.tex index dd2b07a..eb353d7 100644 --- a/buch/papers/common/addpapers.tex +++ b/buch/papers/common/addpapers.tex @@ -3,7 +3,6 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\input{papers/000template/main.tex} \input{papers/lambertw/main.tex} \input{papers/fm/main.tex} \input{papers/parzyl/main.tex} diff --git a/buch/papers/common/paperlist b/buch/papers/common/paperlist index d4e5c20..f607279 100644 --- a/buch/papers/common/paperlist +++ b/buch/papers/common/paperlist @@ -1,4 +1,3 @@ -000template lambertw fm parzyl diff --git a/buch/papers/fm/anim/Makefile b/buch/papers/fm/anim/Makefile new file mode 100644 index 0000000..f4c7850 --- /dev/null +++ b/buch/papers/fm/anim/Makefile @@ -0,0 +1,12 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller +# +all: animation.pdf + +parts.tex: fm.m + octave fm.m + +animation.pdf: animation.tex parts.tex + pdflatex animation.tex diff --git a/buch/papers/fm/anim/animation.tex b/buch/papers/fm/anim/animation.tex new file mode 100644 index 0000000..4a6f428 --- /dev/null +++ b/buch/papers/fm/anim/animation.tex @@ -0,0 +1,85 @@ +% +% animation.tex +% +% (c) 2022 Prof Dr Andreas Müller, +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{nccmath} +\usepackage{mathtools} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\begin{document} + +\def\spektrum#1#2{ +\only<#1>{ + \begin{scope} + \color{red} + \input{#2} + \end{scope} +} +} + +\begin{frame} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\df{0.37} +\def\da{1} + +\draw[->,color=gray] (0,-0.1) -- (0,6.3) [right] coordinate[label={right:$a$}]; + +\foreach \a in {1,...,5}{ + \draw[color=gray!50] (-6,{(6-\a)*\da}) -- (6,{(6-\a)*\da}); +} +\draw[color=gray!50] (-6,{6*\da}) -- (6,{6*\da}); +\foreach \f in {-15,-10,-5,5,10,15}{ + \draw[color=gray!50] ({\f*\df},0) -- ({\f*\df},{6*\da}); +} + +\input{parts.tex} + +\draw[->] (-6.1,0) -- (6.9,0) coordinate[label={$f$}]; +\foreach \f in {-16,...,16}{ + \draw ({\f*\df},-0.05) -- ({\f*\df},0.05); +} +\foreach \f in {-15,-10,-5,5,10,15}{ + \node at ({\f*\df},-0.1) [below] {$\f f_m$}; + \draw ({\f*\df},-0.1) -- ({\f*\df},0.1); +} +\node at (0,-0.1) [below] {$0$}; + +\foreach \a in {1,...,5}{ + \node at (6,{(6-\a)*\da}) [right] {$-\a$}; +} +\node at (6,{6*\da}) [right] {$\phantom{-}0$}; + +\end{tikzpicture} +\end{center} +\end{frame} + +\end{document} diff --git a/buch/papers/fm/anim/fm.m b/buch/papers/fm/anim/fm.m new file mode 100644 index 0000000..9062818 --- /dev/null +++ b/buch/papers/fm/anim/fm.m @@ -0,0 +1,98 @@ +# +# fm.m -- animation frequenzspektrum +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global fc; +fc = 1e6; +global width; +width = 16; +global fm; +fm = 1000; +global gamma; +gamma = 2; +global resolution; +resolution = 300; + +function retval = spektrum(beta, fm) + global width; + global fc; + retval = zeros(2 * width + 1, 2); + center = width + 1; + for k = (0:width) + retval(center - k, 1) = fc - k * fm; + retval(center + k, 1) = fc + k * fm; + a = besselj(k, beta); + retval(center - k, 2) = a; + retval(center + k, 2) = a; + endfor +endfunction + +function drawspectrum(fn, spectrum, foffset, fscale, beta) + n = size(spectrum)(1,1); + for i = (1:n) + f = (spectrum(i, 1) - foffset)/fscale; + a = log10(spectrum(i, 2)) + 6; + if (a < 0) + a = 0; + end + fprintf(fn, "\\draw[line width=3.5pt] "); + fprintf(fn, "({%.2f*\\df},0) -- ({%.2f*\\df},{%.5f*\\da});\n", + f, f, abs(a)); + fprintf(fn, "\\node at ({-15*\\df},5.5) [right] {$\\beta = %.3f$};", beta); + endfor +endfunction + +function drawhull(fn, beta) + global resolution; + fprintf(fn, "\\begin{scope}\n"); + fprintf(fn, "\\clip ({-16.5*\\df},0) rectangle ({16.5*\\df},{6*\\da});\n"); + p = zeros(resolution, 2); + for k = (1:resolution) + nu = 16.5 * (k - 1) / resolution; + p(k,1) = nu; + y = log10(abs(besselj(nu, beta))) + 6; + p(k,2) = y; + end + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + -p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\end{scope}\n"); +endfunction + +function animation(betamin, betamax, steps) + global fm; + global fc; + global gamma; + fa = fopen("parts.tex", "w"); + for k = (1:steps) + % add entry to parts.tex + fprintf(fa, "\\spektrum{%d}{texfiles/a%04d.tex}\n", k, k); + % compute beta + x = (k - 1) / (steps - 1); + beta = betamin + (betamax - betamin) * (x ^ gamma); + % create a new file + name = sprintf("texfiles/a%04d.tex", k); + fn = fopen(name, "w"); + % write the hull + drawhull(fn, beta); + % compute and write the spectrum + spectrum = spektrum(beta, fm); + drawspectrum(fn, spectrum, fc, fm, beta); + fclose(fn); + endfor + fclose(fa); +endfunction + +animation(0.001,10.1,200) diff --git a/buch/papers/fresnel/Makefile b/buch/papers/fresnel/Makefile index c8aa073..11af3a7 100644 --- a/buch/papers/fresnel/Makefile +++ b/buch/papers/fresnel/Makefile @@ -1,9 +1,22 @@ # # Makefile -- make file for the paper fresnel # -# (c) 2020 Prof Dr Andreas Mueller +# (c) 2022 Prof Dr Andreas Mueller # +all: fresnelgraph.pdf eulerspirale.pdf pfad.pdf images: @echo "no images to be created in fresnel" +eulerpath.tex: eulerspirale.m + octave eulerspirale.m + +fresnelgraph.pdf: fresnelgraph.tex eulerpath.tex + pdflatex fresnelgraph.tex + +eulerspirale.pdf: eulerspirale.tex eulerpath.tex + pdflatex eulerspirale.tex + +pfad.pdf: pfad.tex + pdflatex pfad.tex + diff --git a/buch/papers/fresnel/eulerspirale.m b/buch/papers/fresnel/eulerspirale.m new file mode 100644 index 0000000..84e3696 --- /dev/null +++ b/buch/papers/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/buch/papers/fresnel/eulerspirale.pdf b/buch/papers/fresnel/eulerspirale.pdf Binary files differnew file mode 100644 index 0000000..4a85a50 --- /dev/null +++ b/buch/papers/fresnel/eulerspirale.pdf diff --git a/buch/papers/fresnel/eulerspirale.tex b/buch/papers/fresnel/eulerspirale.tex new file mode 100644 index 0000000..38ef756 --- /dev/null +++ b/buch/papers/fresnel/eulerspirale.tex @@ -0,0 +1,41 @@ +% +% eulerspirale.tex -- Darstellung der Eulerspirale +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} + +\def\s{8} + +\begin{scope}[scale=\s] +\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5); +\draw[color=darkgreen,line width=1.4pt] \fresnela; +\draw[color=darkgreen,line width=1.4pt] \fresnelb; +\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}]; +\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}]; +\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s}); +\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s}); +\node at (-0.5,0) [above left] {$\frac12$}; +\node at (0.5,0) [below right] {$\frac12$}; +\node at (0,-0.5) [below right] {$\frac12$}; +\node at (0,0.5) [above left] {$\frac12$}; +\end{scope} + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];; +\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/fresnelgraph.pdf b/buch/papers/fresnel/fresnelgraph.pdf Binary files differnew file mode 100644 index 0000000..9ccad56 --- /dev/null +++ b/buch/papers/fresnel/fresnelgraph.pdf diff --git a/buch/papers/fresnel/fresnelgraph.tex b/buch/papers/fresnel/fresnelgraph.tex new file mode 100644 index 0000000..20df951 --- /dev/null +++ b/buch/papers/fresnel/fresnelgraph.tex @@ -0,0 +1,46 @@ +% +% fresnelgraph.tex -- Graphs of the fresnel functions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} +\def\dx{1.3} +\def\dy{2.6} + +\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy}); +\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy}); + +\draw[color=blue,line width=1.4pt] \Splotright; +\draw[color=blue,line width=1.4pt] \Splotleft; + +\draw[color=red,line width=1.4pt] \Cplotright; +\draw[color=red,line width=1.4pt] \Cplotleft; + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}]; +\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}]; + +\foreach \x in {1,2,3,4,5}{ + \draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05); + \draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05); + \node at ({\x*\dx},-0.05) [below] {$\x$}; + \node at ({-\x*\dx},0.05) [above] {$-\x$}; +} +\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy}); +\node at (-0.05,{0.5*\dy}) [left] {$\frac12$}; +\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy}); +\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/main.tex b/buch/papers/fresnel/main.tex index bbaf7e6..e6ee3b5 100644 --- a/buch/papers/fresnel/main.tex +++ b/buch/papers/fresnel/main.tex @@ -3,29 +3,11 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:fresnel}} -\lhead{Thema} +\chapter{Fresnel-Integrale\label{chapter:fresnel}} +\lhead{Fresnel-Integrale} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Andreas Müller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} \input{papers/fresnel/teil0.tex} \input{papers/fresnel/teil1.tex} diff --git a/buch/papers/fresnel/pfad.pdf b/buch/papers/fresnel/pfad.pdf Binary files differnew file mode 100644 index 0000000..ff514cc --- /dev/null +++ b/buch/papers/fresnel/pfad.pdf diff --git a/buch/papers/fresnel/pfad.tex b/buch/papers/fresnel/pfad.tex new file mode 100644 index 0000000..5439a71 --- /dev/null +++ b/buch/papers/fresnel/pfad.tex @@ -0,0 +1,34 @@ +% +% pfad.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}]; +\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}]; + +\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0); +\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7); +\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0); + +\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$}; +\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$}; +\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$}; + +\node at (7,0) [below] {$R$}; +\node at (45:7) [above] {$Re^{i\pi/4}$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/references.bib b/buch/papers/fresnel/references.bib index 84cd3bc..58e9242 100644 --- a/buch/papers/fresnel/references.bib +++ b/buch/papers/fresnel/references.bib @@ -33,3 +33,14 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@online{fresnel:fresnelC, + url = { https://functions.wolfram.com/GammaBetaErf/FresnelC/introductions/FresnelIntegrals/ShowAll.html }, + title = { FresnelC }, + date = { 2022-05-13 } +} + +@online{fresnel:wikipedia, + url = { https://en.wikipedia.org/wiki/Fresnel_integral }, + title = { Fresnel Integral }, + date = { 2022-05-13 } +} diff --git a/buch/papers/fresnel/teil0.tex b/buch/papers/fresnel/teil0.tex index 5e9fdaf..253e2f3 100644 --- a/buch/papers/fresnel/teil0.tex +++ b/buch/papers/fresnel/teil0.tex @@ -1,22 +1,101 @@ % -% einleitung.tex -- Beispiel-File für die Einleitung +% teil0.tex -- Definition % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{fresnel:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{fresnel:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Definition\label{fresnel:section:teil0}} +\rhead{Definition} +Die Funktion $e^{x^2}$ hat bekanntermassen keine elementare Stammfunktion, +weshalb die Fehlerfunktion als Stammfunktion definiert wurde. +Die Funktionen $\cos x^2$ und $\sin x^2$ sind eng mit $e^{x^2}$ +verwandt, es ist daher nicht überraschend, dass sie ebenfalls +keine elementare Stammfunktionen haben. +Dies rechtfertigt die Definition der Fresnel-Integrale als neue spezielle +Funktionen. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\begin{definition} +Die Funktionen +\begin{align*} +C(x) &= \int_0^x \cos\biggl(\frac{\pi}2 t^2\biggr)\,dt +\\ +S(x) &= \int_0^x \sin\biggl(\frac{\pi}2 t^2\biggr)\,dt +\end{align*} +heissen die Fesnel-Integrale. +\end{definition} +Der Faktor $\frac{\pi}2$ ist einigermassen willkürlich, man könnte +daher noch allgemeiner die Funktionen +\begin{align*} +C_a(x) &= \int_0^x \cos(at^2)\,dt +\\ +S_a(x) &= \int_0^x \sin(at^2)\,dt +\end{align*} +definieren, so dass die Funktionen $C(x)$ und $S(x)$ der Fall +$a=\frac{\pi}2$ werden, also +\[ +\begin{aligned} +C(x) &= C_{\frac{\pi}2}(x), +& +S(x) &= S_{\frac{\pi}2}(x). +\end{aligned} +\] +Durch eine Substution $t=bs$ erhält man +\begin{align*} +C_a(x) +&= +\int_0^x \cos(at^2)\,dt += +b +\int_0^{\frac{x}b} \cos(ab^2s^2)\,ds += +b +C_{ab^2}\biggl(\frac{x}b\biggr) +\\ +S_a(x) +&= +\int_0^x \sin(at^2)\,dt += +b +\int_0^{\frac{x}b} \sin(ab^2s^2)\,ds += +b +S_{ab^2}\biggl(\frac{x}b\biggr). +\end{align*} +Indem man $ab^2=\frac{\pi}2$ setzt, also +\[ +b += +\sqrt{\frac{\pi}{2a}} +, +\] +kann man die Funktionen $C_a(x)$ und $S_a(x)$ durch $C(x)$ und $S(x)$ +ausdrücken: +\begin{align} +C_a(x) +&= +\sqrt{\frac{\pi}{2a}} +C\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr) +&&\text{und}& +S_a(x) +&= +\sqrt{\frac{\pi}{2a}} +S\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr). +\label{fresnel:equation:arg} +\end{align} +Im Folgenden werden wir meistens nur den Fall $a=1$, also die Funktionen +$C_1(x)$ und $S_1(x)$ betrachten, da in diesem Fall die Formeln einfacher +werden. +\begin{figure} +\centering +\includegraphics{papers/fresnel/fresnelgraph.pdf} +\caption{Graph der Funktionen $C(x)$ ({\color{red}rot}) +und $S(x)$ ({\color{blue}blau}) +\label{fresnel:figure:plot}} +\end{figure} +Die Abbildung~\ref{fresnel:figure:plot} zeigt die Graphen der +Funktion $C(x)$ und $S(x)$. diff --git a/buch/papers/fresnel/teil1.tex b/buch/papers/fresnel/teil1.tex index a2df138..a41ddb7 100644 --- a/buch/papers/fresnel/teil1.tex +++ b/buch/papers/fresnel/teil1.tex @@ -1,55 +1,202 @@ % -% teil1.tex -- Beispiel-File für das Paper +% teil1.tex -- Euler-Spirale % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 -\label{fresnel:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{fresnel:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +\section{Euler-Spirale +\label{fresnel:section:eulerspirale}} +\rhead{Euler-Spirale} +\begin{figure} +\centering +\includegraphics{papers/fresnel/eulerspirale.pdf} +\caption{Die Eulerspirale ist die Kurve mit der Parameterdarstellung +$x\mapsto (C(x),S(x))$, sie ist rot dargestellt. +Sie windet sich unendlich oft um die beiden Punkte $(\pm\frac12,\pm\frac12)$. +\label{fresnel:figure:eulerspirale}} +\end{figure} +Ein besseres Verständnis für die beiden Funktionen $C(x)$ und $S(x)$ +als die Darstellung~\ref{fresnel:figure:plot} ermöglicht die +Abbildung~\ref{fresnel:figure:eulerspirale}, die die beiden Funktionen +als die $x$- und $y$-Koordinaten der Parameterdarstellung einer Kurve +zeigt. +Sie heisst die {\em Euler-Spirale}. +Die Spirale scheint sich für $x\to\pm\infty$ um die Punkte +$(\pm\frac12,\pm\frac12)$ zu winden. -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? +\begin{figure} +\centering +\includegraphics{papers/fresnel/pfad.pdf} +\caption{Pfad zur Berechnung der Grenzwerte $C_1(\infty)$ und +$S_1(\infty)$ mit Hilfe des Cauchy-Integralsatzes +\label{fresnel:figure:pfad}} +\end{figure} -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. -Et harum quidem rerum facilis est et expedita distinctio -\ref{fresnel:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{fresnel:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\begin{satz} +Die Grenzwerte der Fresnel-Integrale für $x\to\pm\infty$ sind +\[ +\lim_{x\to\pm\infty} C(x) += +\lim_{x\to\pm\infty} S(x) += +\frac12. +\] +\end{satz} +\begin{proof}[Beweis] +Die komplexe Funktion +\( +f(z) = e^{-z^2} +\) +ist eine ganze Funktion, das Integral über einen geschlossenen +Pfad in der komplexen Ebene verschwindet daher. +Wir verwenden den Pfad in Abbildung~\ref{fresnel:figure:pfad} +bestehend aus den drei Segmenten $\gamma_1$ entlang der reellen +Achse von $0$ bis $R$, dem Kreisbogen $\gamma_2$ um $0$ mit Radius $R$ +und $\gamma_3$ mit der Parametrisierung $t\mapsto te^{i\pi/4}$. + +Das Teilintegral über $\gamma_1$ ist +\[ +\lim_{R\to\infty} +\int_{\gamma_1} e^{-z^2}\,dz += +\int_0^\infty e^{-t^2}\,dt += +\frac{\sqrt{\pi}}2. +\] +Das Integral über $\gamma_3$ ist +\begin{align*} +\lim_{R\to\infty} +\int_{\gamma_3} +e^{-z^2}\,dz +&= +-\int_0^\infty \exp(-t^2 e^{i\pi/2}) e^{i\pi/4}\,dt += +- +\int_0^\infty e^{-it^2}\,dt\, +e^{i\pi/4} +\\ +&= +-e^{i\pi/4}\int_0^\infty \cos t^2 - i \sin t^2\,dt +\\ +&= +-\frac{1}{\sqrt{2}}(1+i) +\bigl( +C_1(\infty) +-i +S_1(\infty) +\bigr) +\\ +&= +-\frac{1}{\sqrt{2}} +\bigl( +C_1(\infty)+S_1(\infty) ++ +i(C_1(\infty)-S_1(\infty)) +\bigr), +\end{align*} +wobei wir +\[ +C_1(\infty) = \lim_{R\to\infty} C_1(R) +\qquad\text{und}\qquad +S_1(\infty) = \lim_{R\to\infty} S_1(R) +\] +abgekürzt haben. +Das Integral über das Segment $\gamma_2$ lässt sich +mit der Parametrisierung +\( +\gamma_2(t) += +Re^{it} += +R(\cos t + i\sin t) +\) +wie folgt +abschätzen: +\begin{align*} +\biggl|\int_{\gamma_2} e^{-z^2} \,dz\biggr| +&= +\biggl| +\int_0^{\frac{\pi}4} +\exp(-R^2(\cos 2t + i\sin 2t)) iR e^{it}\,dt +\biggr| +\\ +&\le +R +\int_0^{\frac{\pi}4} +e^{-R^2\cos 2t} +\,dt +\le +R +\int_0^{\frac{\pi}4} +e^{-R^2(1-\frac{4}{\pi}t)} +\,dt. +\intertext{Dabei haben wir $\cos 2t\ge 1-\frac{4}\pi t$ verwendet. +Mit dieser Vereinfachung kann das Integral ausgewertet werden und +ergibt} +&= +Re^{-R^2} +\int_0^{\frac{\pi}4} +e^{R^2\frac{\pi}4t} +\,dt += +Re^{-R^2} +\biggl[ +\frac{4}{\pi R^2} +e^{R^2\frac{\pi}4t} +\biggr]_0^{\frac{\pi}4} += +\frac{4}{\pi R} +e^{-R^2}(e^{R^2}-1) += +\frac{4}{\pi R} +(1-e^{-R^2}) +\to 0 +\end{align*} +für $R\to \infty$. +Im Grenzwert $R\to \infty$ kann der Teil $\gamma_2$ des Pfades +vernachlässigt werden. + +Das Integral über den geschlossenen Pfad $\gamma$ verschwindet. +Da der Teil $\gamma_2$ keine Rolle spielt, müssen sich die +Integrale über $\gamma_1$ und $\gamma_3$ wegheben, also +\begin{align*} +0 += +\int_\gamma e^{-z^2}\,dz +&= +\int_{\gamma_1} e^{-z^2}\,dz ++ +\int_{\gamma_2} e^{-z^2}\,dz ++ +\int_{\gamma_3} e^{-z^2}\,dz +\\ +&\to +\frac{\sqrt{\pi}}2 +-\frac{1}{\sqrt{2}}(C_1(\infty)+S_1(\infty)) +-\frac{i}{\sqrt{2}}(C_1(\infty)-S_1(\infty)). +\end{align*} +Der Imaginärteil ist $C_1(\infty)-S_1(\infty)$, da er verschwinden +muss, folgt $C_1(\infty)=S_1(\infty)$. +Nach Multlikation mit $\sqrt{2}$ folgt aus der Tatsache, dass auch +der Realteil verschwinden muss +\[ +\frac{\sqrt{\pi}}{\sqrt{2}} = C_1(\infty)+S_1(\infty) +\qquad +\Rightarrow +\qquad +C_1(\infty) += +S_1(\infty) += +\frac{\sqrt{\pi}}{2\sqrt{2}}. +\] +Aus +\eqref{fresnel:equation:arg} +erhält man dann auch die Grenzwerte +\[ +C(\infty)=S(\infty)=\frac12. +\qedhere +\] +\end{proof} diff --git a/buch/papers/fresnel/teil2.tex b/buch/papers/fresnel/teil2.tex index 701c3ee..22d2a89 100644 --- a/buch/papers/fresnel/teil2.tex +++ b/buch/papers/fresnel/teil2.tex @@ -3,38 +3,22 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 -\label{fresnel:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Klothoide +\label{fresnel:section:klothoide}} +\rhead{Klothoide} +In diesem Abschnitt soll gezeigt werden, dass die Krümmung der +Euler-Spirale proportional zur vom Nullpunkt aus gemessenen Bogenlänge +ist. -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\begin{definition} +Eine ebene Kurve, deren Krümmung proportionale zur Kurvenlänge ist, +heisst {\em Klothoide}. +\end{definition} +Die Klothoide wird zum Beispiel im Strassenbau bei Autobahnkurven +angewendet. +Fährt man mit konstanter Geschwindigkeit mit entlang einer Klothoide, +muss man die Krümmung mit konstaner Geschwindigkeit ändern, +also das Lenkrad mit konstanter Geschwindigkeit drehen. +Dies ermöglicht eine ruhige Fahrweise. diff --git a/buch/papers/fresnel/teil3.tex b/buch/papers/fresnel/teil3.tex index d4f15f6..37e6bee 100644 --- a/buch/papers/fresnel/teil3.tex +++ b/buch/papers/fresnel/teil3.tex @@ -3,38 +3,110 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 -\label{fresnel:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Numerische Berechnung der Fresnel-Integrale +\label{fresnel:section:numerik}} +\rhead{Numerische Berechnung} +Die Fresnel-Integrale können mit verschiedenen Methoden effizient berechnet +werden. -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\subsection{Komplexe Fehlerfunktionen} +Es wurde schon darauf hingewiesen, dass der Integrand der Fresnel-Integrale +mit $e^{t^2}$ verwandt ist. +Tatsächlich kann gezeigt werden dass sich die Fresnel-Integrale mit +Hilfe der komplexen Fehlerfunktion als +\[ +\left. +\begin{matrix} +S_1(z) +\\ +C_1(z) +\end{matrix} +\; +\right\} += +\frac{1\pm i}4\biggl( +\operatorname{erf}\biggl(\frac{1+i}2\sqrt{\pi}z\biggr) +\mp +\operatorname{erf}\biggl(\frac{1-i}2\sqrt{\pi}z\biggr) +\biggr) +\] +ausdrücken lassen \cite{fresnel:fresnelC}. +Diese Darstellung ist jedoch für die numerische Berechnung nur +beschränkt nützlich, weil die meisten Bibliotheken für die Fehlerfunktion +diese nur für reelle Argument auszuwerten gestatten. + +\subsection{Als Lösung einer Differentialgleichung} +Da die Fresnel-Integrale die sehr einfachen Differentialgleichungen +\[ +C'(x) = \cos \biggl(\frac{\pi}2 x^2\biggr) +\qquad\text{und}\qquad +S'(x) = \sin \biggl(\frac{\pi}2 x^2\biggr) +\] +erfüllen, kann man eine Methode zur Lösung von Differentialgleichung +verwenden. +Die Abbildungen~\ref{fresnel:figure:plot} und \ref{fresnel:figure:eulerspirale} +wurden auf diese Weise erzeugt. + +\subsection{Taylor-Reihe integrieren} +Die Taylorreihen +\begin{align*} +\cos x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} x^{2k} +&&\text{und}& +\sin x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k+1} +\intertext{% +der trigonometrischen Funktionen werden durch Einsetzen von $x=t^2$ +zu} +\cos t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} t^{4k} +&&\text{und}& +\sin t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} t^{4k+2}. +\intertext{% +Die Fresnel-Integrale $C_1(x)$ und $S_1(x)$ können daher durch +termweise Integration mit Hilfe der Reihen} +C_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} \frac{x^{4k+1}}{4k+1} +&&\text{und}& +S_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} \frac{x^{4k+3}}{4k+3} +\end{align*} +berechnet werden. +Diese Reihen sind insbesondere für kleine Werte von $x$ sehr +schnell konvergent. + +\subsection{Hypergeometrische Reihen} +Aus der Reihenentwicklung kann jetzt auch eine Darstellung der +Fresnel-Integrale durch hypergeometrische Reihen gefunden werden +\cite{fresnel:fresnelC}. +Es ergibt sich +\begin{align*} +S(z) +&= +\frac{\pi z^3}{6} +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac34\\\frac32,\frac74\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr) +\\ +C(z) +&= +z +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac14\\\frac12,\frac54\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr). +\end{align*} diff --git a/buch/papers/laguerre/Makefile b/buch/papers/laguerre/Makefile index 606d7e1..0f0985a 100644 --- a/buch/papers/laguerre/Makefile +++ b/buch/papers/laguerre/Makefile @@ -4,6 +4,8 @@ # (c) 2020 Prof Dr Andreas Mueller # -images: - @echo "no images to be created in laguerre" +images: images/laguerre_polynomes.pdf + +images/laguerre_polynomes.pdf: scripts/laguerre_plot.py + python3 scripts/laguerre_plot.py diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index 1eb5034..12b0935 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -9,8 +9,7 @@ dependencies-laguerre = \ papers/laguerre/references.bib \ papers/laguerre/definition.tex \ papers/laguerre/eigenschaften.tex \ - papers/laguerre/quadratur.tex \ - papers/laguerre/transformation.tex \ - papers/laguerre/wasserstoff.tex + papers/laguerre/quadratur.tex \ + papers/laguerre/gamma.tex diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index 5f6d8bd..d111f6f 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -4,45 +4,154 @@ % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % \section{Definition -\label{laguerre:section:definition}} + \label{laguerre:section:definition}} \rhead{Definition} - +Die verallgemeinerte Laguerre-Differentialgleichung ist gegeben durch \begin{align} - x y''(x) + (1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:dgl} +x y''(x) + (\nu + 1 - x) y'(x) + n y(x) += +0 +, \quad +n \in \mathbb{N}_0 +, \quad +x \in \mathbb{R} +. +\label{laguerre:dgl} \end{align} - +Die klassische Laguerre-Diffentialgleichung erhält man, wenn $\nu = 0$. +Hier wird die verallgemeinerte Laguerre-Differentialgleichung verwendet, +weil die Lösung mit der selben Methode berechnet werden kann, +aber man zusätzlich die Lösung für den allgmeinen Fall erhält. +Zur Lösung der Gleichung \eqref{laguerre:dgl} verwenden wir einen +Potenzreihenansatz. +Da wir bereits wissen, dass die Lösung orthogonale Polynome sind, +erscheint dieser Ansatz sinnvoll. +Setzt man nun den Ansatz +\begin{align*} +y(x) + & = +\sum_{k=0}^\infty a_k x^k +\\ +y'(x) + & = +\sum_{k=1}^\infty k a_k x^{k-1} += +\sum_{k=0}^\infty (k+1) a_{k+1} x^k +\\ +y''(x) + & = +\sum_{k=2}^\infty k (k-1) a_k x^{k-2} += +\sum_{k=1}^\infty (k+1) k a_{k+1} x^{k-1} +\end{align*} +in die Differentialgleichung ein, erhält man: +\begin{align*} +\sum_{k=1}^\infty (k+1) k a_{k+1} x^k ++ +(\nu + 1)\sum_{k=0}^\infty (k+1) a_{k+1} x^k +- +\sum_{k=0}^\infty k a_k x^k ++ +n \sum_{k=0}^\infty a_k x^k + & = +0 \\ +\sum_{k=1}^\infty +\left[ (k+1) k a_{k+1} + (\nu + 1)(k+1) a_{k+1} - k a_k + n a_k \right] x^k + & = +0. +\end{align*} +Daraus lässt sich die Rekursionsbeziehung +\begin{align*} +a_{k+1} + & = +\frac{k-n}{(k+1) (k + \nu + 1)} a_k +\end{align*} +ableiten. +Für ein konstantes $n$ erhalten wir als Potenzreihenlösung ein Polynom vom Grad +$n$, +denn für $k=n$ wird $a_{n+1} = 0$ und damit auch $a_{n+2}=a_{n+3}=\ldots=0$. +Aus der Rekursionsbeziehung ist zudem ersichtlich, +dass $a_0 \neq 0$ beliebig gewählt werden kann. +Wählen wir nun $a_0 = 1$, dann folgt für die Koeffizienten $a_1, a_2, a_3$ +\begin{align*} +a_1 += +-\frac{n}{1 \cdot (\nu + 1)} +, & & +a_2 += +\frac{(n-1)n}{1 \cdot 2 \cdot (\nu + 1)(\nu + 2)} +, & & +a_3 += +-\frac{(n-2)(n-1)n}{1 \cdot 2 \cdot 3 \cdot (\nu + 1)(\nu + 2)(\nu + 3)} +\end{align*} +und allgemein +\begin{align*} +k + & \leq +n: + & +a_k + & = +(-1)^k \frac{n!}{(n-k)!} \frac{1}{k!(\nu + 1)_k} += +\frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} +\\ +k & >n: + & +a_k + & = +0. +\end{align*} +Somit erhalten wir für $\nu = 0$ die Laguerre-Polynome \begin{align} - L_n(x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n \\ - k - \end{pmatrix} - x^k - \label{laguerre:polynom} +L_n(x) += +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +\label{laguerre:polynom} \end{align} - +und mit $\nu \in \mathbb{R}$ die verallgemeinerten Laguerre-Polynome \begin{align} - x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:generell_dgl} +L_n^\nu(x) += +\sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. +\label{laguerre:allg_polynom} \end{align} -\begin{align} - L_n^\alpha (x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n + \alpha \\ - n - k - \end{pmatrix} - x^k - \label{laguerre:polynom} -\end{align} +\subsection{Analytische Fortsetzung} +Durch die analytische Fortsetzung erhalten wir zudem noch die zweite Lösung der +Differentialgleichung mit der Form +\begin{align*} +\Xi_n(x) += +L_n(x) \ln(x) + \sum_{k=1}^\infty d_k x^k +\end{align*} +Nach einigen mühsamen Rechnungen, +die den Rahmen dieses Kapitel sprengen würden, +erhalten wir +\begin{align*} +\Xi_n += +L_n(x) \ln(x) ++ +\sum_{k=1}^n \frac{(-1)^k}{k!} \binom{n}{k} +(\alpha_{n-k} - \alpha_n - 2 \alpha_k)x^k ++ +(-1)^n \sum_{k=1}^\infty \frac{(k-1)!n!}{((n+k)!)^2} x^{n+k}, +\end{align*} +wobei $\alpha_0 = 0$ und $\alpha_k =\sum_{i=1}^k i^{-1}$, +$\forall k \in \mathbb{N}$. +Die Laguerre-Polynome von Grad $0$ bis $7$ sind in +Abbildung~\ref{laguerre:fig:polyeval} dargestellt. +\begin{figure} +\centering +\includegraphics[width=0.7\textwidth]{% + papers/laguerre/images/laguerre_polynomes.pdf% +} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\label{laguerre:fig:polyeval} +\end{figure} + +% https://www.math.kit.edu/iana1/lehre/hm3phys2012w/media/laguerre.pdf +% http://www.physics.okayama-u.ac.jp/jeschke_homepage/E4/kapitel4.pdf diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex index b7597e5..b0cc3a3 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -4,5 +4,116 @@ % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % \section{Eigenschaften -\label{laguerre:section:eigenschaften}} -\rhead{Eigenschaften}
\ No newline at end of file + \label{laguerre:section:eigenschaften}} +{ +\large \color{red} +TODO: +Evtl. nur Orthogonalität hier behandeln, da nur diese für die Gauss-Quadratur +benötigt wird. +} + +Die Laguerre-Polynome besitzen einige interessante Eigenschaften +\rhead{Eigenschaften} + +\subsection{Orthogonalität + \label{laguerre:subsection:orthogonal}} +Im Abschnitt~\ref{laguerre:section:definition} haben wir behauptet, +dass die Laguerre-Polynome orthogonale Polynome sind. +Zu dieser Behauptung möchten wir nun einen Beweis liefern. +Wenn wir die Laguerre\--Differentialgleichung in ein +Sturm\--Liouville\--Problem umwandeln können, haben wir bewiesen, dass es sich +bei +den Laguerre\--Polynomen um orthogonale Polynome handelt (siehe +Abschnitt~\ref{buch:integrale:subsection:sturm-liouville-problem}). +Der Sturm-Liouville-Operator hat die Form +\begin{align} +S += +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right). +\label{laguerre:slop} +\end{align} +Aus der Beziehung +\begin{align} +S + & = +\Lambda +\nonumber +\\ +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right) + & = +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx} +\label{laguerre:sl-lag} +\end{align} +lässt sich sofort erkennen, dass $q(x) = 0$. +Ausserdem ist ersichtlich, dass $p(x)$ die Differentialgleichung +\begin{align*} +x \frac{dp}{dx} += +-(\nu + 1 - x) p, +\end{align*} +erfüllen muss. +Durch Separation erhalten wir dann +\begin{align*} +\int \frac{dp}{p} + & = +-\int \frac{\nu + 1 - x}{x}dx +\\ +\log p + & = +-\log \nu + 1 - x + C +\\ +p(x) + & = +-C x^{\nu + 1} e^{-x} +\end{align*} +Eingefügt in Gleichung~\eqref{laguerre:sl-lag} erhalten wir +\begin{align*} +\frac{C}{w(x)} +\left( +x^{\nu+1} e^{-x} \frac{d^2}{dx^2} + +(\nu + 1 - x) x^{\nu} e^{-x} \frac{d}{dx} +\right) += +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx}. +\end{align*} +Mittels Koeffizientenvergleich kann nun abgelesen werden, dass $w(x) = x^\nu +e^{-x}$ und $C=1$ mit $\nu > -1$. +Die Gewichtsfunktion $w(x)$ wächst für $x\rightarrow-\infty$ sehr schnell an, +deshalb ist die Laguerre-Gewichtsfunktion nur geeignet für den +Definitionsbereich $(0, \infty)$. +Bleibt nur noch sicherzustellen, dass die Randbedingungen, +\begin{align} +k_0 y(0) + h_0 p(0)y'(0) + & = +0 +\label{laguerre:sllag_randa} +\\ +k_\infty y(\infty) + h_\infty p(\infty) y'(\infty) + & = +0 +\label{laguerre:sllag_randb} +\end{align} +mit $|k_i|^2 + |h_i|^2 \neq 0,\,\forall i \in \{0, \infty\}$, erfüllt sind. +Am linken Rand (Gleichung~\eqref{laguerre:sllag_randa}) kann $y(0) = 1$, $k_0 = +0$ und $h_0 = 1$ verwendet werden, +was auch die Laguerre-Polynome ergeben haben. +Für den rechten Rand ist die Bedingung (Gleichung~\eqref{laguerre:sllag_randb}) +\begin{align*} +\lim_{x \rightarrow \infty} p(x) y'(x) + & = +\lim_{x \rightarrow \infty} -x^{\nu + 1} e^{-x} y'(x) += +0 +\end{align*} +für beliebige Polynomlösungen erfüllt für $k_\infty=0$ und $h_\infty=1$. +Damit können wir schlussfolgern, dass die Laguerre-Polynome orthogonal +bezüglich des Skalarproduktes auf dem Intervall $(0, \infty)$ mit der Laguerre\--Gewichtsfunktion +$w(x)=x^\nu e^{-x}$ sind. + + +\subsection{Rodrigues-Formel} + +\subsection{Drei-Terme Rekursion} + +\subsection{Beziehung mit der Hypergeometrischen Funktion} + diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex new file mode 100644 index 0000000..e3838b0 --- /dev/null +++ b/buch/papers/laguerre/gamma.tex @@ -0,0 +1,76 @@ +% +% gamma.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Anwendung: Berechnung der Gamma-Funktion + \label{laguerre:section:quad-gamma}} +Die Gauss-Laguerre-Quadratur kann nun verwendet werden, +um exponentiell abfallende Funktionen im Definitionsbereich $(0, \infty)$ zu +berechnen. +Dabei bietet sich z.B. die Gamma-Funkion bestens an, wie wir in den folgenden +Abschnitten sehen werden. + +\subsection{Gamma-Funktion} +Die Gamma-Funktion ist eine Erweiterung der Fakultät auf die reale und komplexe +Zahlenmenge. +Die Definition~\ref{buch:rekursion:def:gamma} beschreibt die Gamma-Funktion als +Integral der Form +\begin{align} +\Gamma(z) + & = +\int_0^\infty t^{z-1} e^{-t} dt +, +\quad +\text{wobei Realteil von $z$ grösser als $0$} +, +\label{laguerre:gamma} +\end{align} +welches alle Eigenschaften erfüllt, um mit der Gauss-Laguerre-Quadratur +berechnet zu werden. + +\subsubsection{Funktionalgleichung} +Die Funktionalgleichung besagt +\begin{align} +z \Gamma(z) = \Gamma(z+1). +\label{laguerre:gamma_funktional} +\end{align} +Mittels dieser Gleichung kann der Wert an einer bestimmten, +geeigneten Stelle evaluiert werden und dann zurückverschoben werden, +um das gewünschte Resultat zu erhalten. + +\subsection{Berechnung mittels Gauss-Laguerre-Quadratur} + +Fehlerterm: +\begin{align*} +R_n += +(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z-2n-1} +\end{align*} + +\subsubsection{Finden der optimalen Berechnungsstelle} +Nun stellt sich die Frage, +ob die Approximation mittels Gauss-Laguerre-Quadratur verbessert werden kann, +wenn man das Problem an einer geeigneten Stelle evaluiert und +dann zurückverschiebt mit der Funktionalgleichung. +Dazu wollen wir den Fehlerterm in +Gleichung~\eqref{laguerre:lagurre:lag_error} anpassen und dann minimieren. +Zunächst wollen wir dies nur für $z\in \mathbb{R}$ und $0<z<1$ definieren. +Zudem nehmen wir an, dass die optimale Stelle $x^* \in \mathbb{R}$, $z < x^*$ +ist. +Dann fügen wir einen Verschiebungsterm um $m$ Stellen ein, daraus folgt +\begin{align*} +R_n += +\frac{(z - 2n)_{2n}}{(z - m)_m} \frac{(n!)^2}{(2n)!} \xi^{z + m - 2n - 1} +. +\end{align*} + +{ +\large \color{red} +TODO: +Geeignete Minimierung für Fehler finden, so dass sie mit den emprisich +bestimmen optimalen Punkten übereinstimmen. +} + +\subsection{Resultate} diff --git a/buch/papers/laguerre/images/laguerre_polynomes.pdf b/buch/papers/laguerre/images/laguerre_polynomes.pdf Binary files differnew file mode 100644 index 0000000..3976bc7 --- /dev/null +++ b/buch/papers/laguerre/images/laguerre_polynomes.pdf diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex index 1fe0f8b..00e3b43 100644 --- a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -8,13 +8,14 @@ \begin{refsection} \chapterauthor{Patrik Müller} -Hier kommt eine Einleitung. +{\large \color{red} TODO: Einleitung} \input{papers/laguerre/definition} \input{papers/laguerre/eigenschaften} \input{papers/laguerre/quadratur} -\input{papers/laguerre/transformation} -\input{papers/laguerre/wasserstoff} +\input{papers/laguerre/gamma} +% \input{papers/laguerre/transformation} +% \input{papers/laguerre/wasserstoff} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/laguerre/packages.tex b/buch/papers/laguerre/packages.tex index ab55228..4ebc172 100644 --- a/buch/papers/laguerre/packages.tex +++ b/buch/papers/laguerre/packages.tex @@ -7,4 +7,3 @@ % if your paper needs special packages, add package commands as in the % following example \usepackage{derivative} - diff --git a/buch/papers/laguerre/quadratur.tex b/buch/papers/laguerre/quadratur.tex index 8ab1af5..60fad7f 100644 --- a/buch/papers/laguerre/quadratur.tex +++ b/buch/papers/laguerre/quadratur.tex @@ -3,27 +3,77 @@ % % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % -\section{Gauss-Laguerre Quadratur -\label{laguerre:section:quadratur}} +\section{Gauss-Quadratur + \label{laguerre:section:quadratur}} + {\large \color{red} TODO: Einleitung und kurze Beschreibung Gauss-Quadratur} +\begin{align} +\int_a^b f(x) w(x) +\approx +\sum_{i=1}^N f(x_i) A_i +\label{laguerre:gaussquadratur} +\end{align} +\subsection{Gauss-Laguerre-Quadratur +\label{laguerre:subsection:gausslag-quadratur}} +Die Gauss-Quadratur kann auch auf Skalarprodukte mit Gewichtsfunktionen +ausgeweitet werden. +In unserem Falle möchten wir die Gauss Quadratur auf die Laguerre-Polynome +$L_n$ ausweiten. +Diese sind orthogonal im Intervall $(0, \infty)$ bezüglich +der Gewichtsfunktion $e^{-x}$. +Gleichung~\eqref{laguerre:laguerrequadratur} lässt sich wiefolgt umformulieren: \begin{align} - \int_a^b f(x) w(x) - \approx - \sum_{i=1}^N f(x_i) A_i - \label{laguerre:gaussquadratur} +\int_{0}^{\infty} f(x) e^{-x} dx +\approx +\sum_{i=1}^{N} f(x_i) A_i +\label{laguerre:laguerrequadratur} \end{align} +\subsubsection{Stützstellen und Gewichte} +Nach der Definition der Gauss-Quadratur müssen als Stützstellen die Nullstellen +des verwendeten Polynoms genommen werden. +Das heisst für das Laguerre-Polynom $L_n$ müssen dessen Nullstellen $x_i$ und +als Gewichte $A_i$ werden die Integrale $l_i(x)e^{-x}$ verwendet werden. +Dabei sind +\begin{align*} +l_i(x_j) += +\delta_{ij} += +\begin{cases} +1 & i=j \\ +0 & \text{sonst.} +\end{cases} +\end{align*} +Laut \cite{abramowitz+stegun} sind die Gewichte also \begin{align} - \int_{0}^{\infty} f(x) e^{-x} dx - \approx - \sum_{i=1}^{N} f(x_i) A_i - \label{laguerre:laguerrequadratur} +A_i += +\frac{x_i}{(n + 1)^2 \left[ L_{n + 1}(x_i)\right]^2} +. +\label{laguerre:quadratur_gewichte} \end{align} +\subsubsection{Fehlerterm} +Der Fehlerterm $R_n$ folgt direkt aus der Approximation +\begin{align*} +\int_0^{\infty} f(x) e^{-x} dx += +\sum_{i=1}^n f(x_i) A_i + R_n +\end{align*} +un \cite{abramowitz+stegun} gibt in als \begin{align} - A_i - = - \frac{x_i}{(n + 1)^2 \left[ L_{n + 1}(x_i)\right]^2} - \label{laguerre:quadratur_gewichte} +R_n += +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +,\quad +0 < \xi < \infty +\label{lagurre:lag_error} \end{align} +an. +{ +\large \color{red} +TODO: +Noch mehr Text / bessere Beschreibungen in allen Abschnitten +} diff --git a/buch/papers/laguerre/references.bib b/buch/papers/laguerre/references.bib index caf270f..6956ade 100644 --- a/buch/papers/laguerre/references.bib +++ b/buch/papers/laguerre/references.bib @@ -4,32 +4,19 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{laguerre:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{laguerre:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{laguerre:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} -} - +@book{abramowitz+stegun, + added-at = {2008-06-25T06:25:58.000+0200}, + address = {New York}, + author = {Abramowitz, Milton and Stegun, Irene A.}, + biburl = {https://www.bibsonomy.org/bibtex/223ec744709b3a776a1af0a3fd65cd09f/a_olympia}, + description = {BibTeX - Wikipedia, the free encyclopedia}, + edition = {ninth Dover printing, tenth GPO printing}, + interhash = {d4914a420f489f7c5129ed01ec3cf80c}, + intrahash = {23ec744709b3a776a1af0a3fd65cd09f}, + keywords = {Handbook}, + publisher = {Dover}, + pages = {890}, + timestamp = {2008-06-25T06:25:58.000+0200}, + title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, + year = 1972 +}
\ No newline at end of file diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb new file mode 100644 index 0000000..44f3abd --- /dev/null +++ b/buch/papers/laguerre/scripts/gamma_approx.ipynb @@ -0,0 +1,395 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Gauss-Laguerre Quadratur für die Gamma-Funktion\n", + "\n", + "$$\n", + " \\Gamma(z)\n", + " = \n", + " \\int_0^\\infty t^{z-1}e^{-t}dt\n", + "$$\n", + "\n", + "$$\n", + " \\int_0^\\infty f(x) e^{-x} dx \n", + " \\approx \n", + " \\sum_{i=1}^{N} f(x_i) w_i\n", + " \\qquad\\text{ wobei }\n", + " w_i = \\frac{x_i}{(n+1)^2 [L_{n+1}(x_i)]^2}\n", + "$$\n", + "und $x_i$ sind Nullstellen des Laguerre Polynoms $L_n(x)$\n", + "\n", + "Der Fehler ist gegeben als\n", + "\n", + "$$\n", + " E \n", + " =\n", + " \\frac{(n!)^2}{(2n)!} f^{(2n)}(\\xi) \n", + " = \n", + " \\frac{(-2n + z)_{2n}}{(z-m)_m} \\frac{(n!)^2}{(2n)!} \\xi^{z + m - 2n - 1}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from cmath import exp, pi, sin, sqrt\n", + "import scipy.special\n", + "\n", + "EPSILON = 1e-07\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "lanczos_p = [\n", + " 676.5203681218851,\n", + " -1259.1392167224028,\n", + " 771.32342877765313,\n", + " -176.61502916214059,\n", + " 12.507343278686905,\n", + " -0.13857109526572012,\n", + " 9.9843695780195716e-6,\n", + " 1.5056327351493116e-7,\n", + "]\n", + "\n", + "\n", + "def drop_imag(z):\n", + " if abs(z.imag) <= EPSILON:\n", + " z = z.real\n", + " return z\n", + "\n", + "\n", + "def lanczos_gamma(z):\n", + " z = complex(z)\n", + " if z.real < 0.5:\n", + " y = pi / (sin(pi * z) * lanczos_gamma(1 - z)) # Reflection formula\n", + " else:\n", + " z -= 1\n", + " x = 0.99999999999980993\n", + " for (i, pval) in enumerate(lanczos_p):\n", + " x += pval / (z + i + 1)\n", + " t = z + len(lanczos_p) - 0.5\n", + " y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x\n", + " return drop_imag(y)\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "zeros, weights = np.polynomial.laguerre.laggauss(8)\n", + "# zeros = np.array(\n", + "# [\n", + "# 1.70279632305101000e-1,\n", + "# 9.03701776799379912e-1,\n", + "# 2.25108662986613069e0,\n", + "# 4.26670017028765879e0,\n", + "# 7.04590540239346570e0,\n", + "# 1.07585160101809952e1,\n", + "# 1.57406786412780046e1,\n", + "# 2.28631317368892641e1,\n", + "# ]\n", + "# )\n", + "\n", + "# weights = np.array(\n", + "# [\n", + "# 3.69188589341637530e-1,\n", + "# 4.18786780814342956e-1,\n", + "# 1.75794986637171806e-1,\n", + "# 3.33434922612156515e-2,\n", + "# 2.79453623522567252e-3,\n", + "# 9.07650877335821310e-5,\n", + "# 8.48574671627253154e-7,\n", + "# 1.04800117487151038e-9,\n", + "# ]\n", + "# )\n", + "\n", + "\n", + "def pochhammer(z, n):\n", + " return np.prod(z + np.arange(n))\n", + "\n", + "\n", + "def find_shift(z, target):\n", + " factor = 1.0\n", + " steps = int(np.floor(target - np.real(z)))\n", + " zs = z + steps\n", + " if steps > 0:\n", + " factor = 1 / pochhammer(z, steps)\n", + " elif steps < 0:\n", + " factor = pochhammer(zs, -steps)\n", + " return zs, factor\n", + "\n", + "\n", + "def laguerre_gamma(z, x, w, target=11):\n", + " # res = 0.0\n", + " z = complex(z)\n", + " if z.real < 1e-3:\n", + " res = pi / (\n", + " sin(pi * z) * laguerre_gamma(1 - z, x, w, target)\n", + " ) # Reflection formula\n", + " else:\n", + " z_shifted, correction_factor = find_shift(z, target)\n", + " res = np.sum(x ** (z_shifted - 1) * w)\n", + " res *= correction_factor\n", + " res = drop_imag(res)\n", + " return res\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def eval_laguerre(x, target=12):\n", + " return np.array([laguerre_gamma(xi, zeros, weights, target) for xi in x])\n", + "\n", + "\n", + "def eval_lanczos(x):\n", + " return np.array([lanczos_gamma(xi) for xi in x])\n", + "\n", + "\n", + "def eval_mean_laguerre(x, targets):\n", + " return np.mean([eval_laguerre(x, target) for target in targets], 0)\n", + "\n", + "\n", + "def calc_rel_error(x, y):\n", + " return (y - x) / x\n", + "\n", + "\n", + "def evaluate(x, target=12):\n", + " lanczos_gammas = eval_lanczos(x)\n", + " laguerre_gammas = eval_laguerre(x, target)\n", + " rel_error = calc_rel_error(lanczos_gammas, laguerre_gammas)\n", + " return rel_error\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Test with real values" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Empirische Tests zeigen:\n", + "- $n=4 \\Rightarrow m=6$\n", + "- $n=5 \\Rightarrow m=7$ oder $m=8$\n", + "- $n=6 \\Rightarrow m=9$\n", + "- $n=7 \\Rightarrow m=10$\n", + "- $n=8 \\Rightarrow m=11$ oder $m=12$\n", + "- $n=9 \\Rightarrow m=13$\n", + "- $n=10 \\Rightarrow m=14$\n", + "- $n=11 \\Rightarrow m=15$ oder $m=16$\n", + "- $n=12 \\Rightarrow m=17$\n", + "- $n=13 \\Rightarrow m=18 \\Rightarrow $ Beginnt numerisch instabil zu werden \n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "zeros, weights = np.polynomial.laguerre.laggauss(12)\n", + "targets = np.arange(16, 21)\n", + "mean_targets = ((16, 17),)\n", + "x = np.linspace(EPSILON, 1 - EPSILON, 101)\n", + "_, axs = plt.subplots(\n", + " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", + ")\n", + "\n", + "lanczos = eval_lanczos(x)\n", + "for mean_target in mean_targets:\n", + " vals = eval_mean_laguerre(x, mean_target)\n", + " rel_error_mean = calc_rel_error(lanczos, vals)\n", + " axs[0].plot(x, rel_error_mean, label=mean_target)\n", + " axs[1].semilogy(x, np.abs(rel_error_mean), label=mean_target)\n", + "\n", + "mins = []\n", + "maxs = []\n", + "for target in targets:\n", + " rel_error = evaluate(x, target)\n", + " mins.append(np.min(np.abs(rel_error[(0.1 <= x) & (x <= 0.9)])))\n", + " maxs.append(np.max(np.abs(rel_error)))\n", + " axs[0].plot(x, rel_error, label=target)\n", + " axs[1].semilogy(x, np.abs(rel_error), label=target)\n", + "# axs[0].set_ylim(*(np.array([-1, 1]) * 3.5e-8))\n", + "\n", + "axs[0].set_xlim(x[0], x[-1])\n", + "axs[1].set_ylim(np.min(mins), 1.04*np.max(maxs))\n", + "for ax in axs:\n", + " ax.legend()\n", + " ax.grid(which=\"both\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "targets = (16, 17)\n", + "xmax = 15\n", + "x = np.linspace(-xmax + EPSILON, xmax - EPSILON, 1000)\n", + "\n", + "mean_lag = eval_mean_laguerre(x, targets)\n", + "lanczos = eval_lanczos(x)\n", + "rel_error = calc_rel_error(lanczos, mean_lag)\n", + "rel_error_simple = evaluate(x, targets[-1])\n", + "# rel_error = evaluate(x, target)\n", + "\n", + "_, axs = plt.subplots(\n", + " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", + ")\n", + "axs[0].plot(x, rel_error, label=targets)\n", + "axs[1].semilogy(x, np.abs(rel_error), label=targets)\n", + "axs[0].plot(x, rel_error_simple, label=targets[-1])\n", + "axs[1].semilogy(x, np.abs(rel_error_simple), label=targets[-1])\n", + "axs[0].set_xlim(x[0], x[-1])\n", + "# axs[0].set_ylim(*(np.array([-1, 1]) * 4.2e-8))\n", + "# axs[1].set_ylim(1e-10, 5e-8)\n", + "for ax in axs:\n", + " ax.legend()\n", + "\n", + "x2 = np.linspace(-5 + EPSILON, 5, 4001)\n", + "_, ax = plt.subplots(constrained_layout=True, figsize=(8, 6))\n", + "ax.plot(x2, eval_mean_laguerre(x2, targets))\n", + "ax.set_xlim(x2[0], x2[-1])\n", + "ax.set_ylim(-7.5, 25)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Test with complex values" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "targets = (16, 17)\n", + "vals = np.linspace(-5 + EPSILON, 5, 100)\n", + "x, y = np.meshgrid(vals, vals)\n", + "mesh = x + 1j * y\n", + "input = mesh.flatten()\n", + "\n", + "mean_lag = eval_mean_laguerre(input, targets).reshape(mesh.shape)\n", + "lanczos = eval_lanczos(input).reshape(mesh.shape)\n", + "rel_error = np.abs(calc_rel_error(lanczos, mean_lag))\n", + "\n", + "lag = eval_laguerre(input, targets[-1]).reshape(mesh.shape)\n", + "rel_error_simple = np.abs(calc_rel_error(lanczos, lag))\n", + "# rel_error = evaluate(x, target)\n", + "\n", + "fig, axs = plt.subplots(\n", + " 2,\n", + " 2,\n", + " sharex=True,\n", + " sharey=True,\n", + " clear=True,\n", + " constrained_layout=True,\n", + " figsize=(12, 10),\n", + ")\n", + "_c = axs[0, 1].pcolormesh(x, y, np.log10(np.abs(lanczos - mean_lag)), shading=\"gouraud\")\n", + "_c = axs[0, 0].pcolormesh(x, y, np.log10(np.abs(lanczos - lag)), shading=\"gouraud\")\n", + "fig.colorbar(_c, ax=axs[0, :])\n", + "_c = axs[1, 1].pcolormesh(x, y, np.log10(rel_error), shading=\"gouraud\")\n", + "_c = axs[1, 0].pcolormesh(x, y, np.log10(rel_error_simple), shading=\"gouraud\")\n", + "fig.colorbar(_c, ax=axs[1, :])\n", + "_ = axs[0, 0].set_title(\"Absolute Error\")\n", + "_ = axs[1, 0].set_title(\"Relative Error\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "z = 0.5\n", + "ns = [4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12] # np.arange(4, 13)\n", + "ms = np.arange(6, 18)\n", + "xi = np.logspace(0, 2, 201)[:, None]\n", + "lanczos = eval_lanczos([z])[0]\n", + "\n", + "_, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(12, 8))\n", + "ax.grid(1)\n", + "for n, m in zip(ns, ms):\n", + " zeros, weights = np.polynomial.laguerre.laggauss(n)\n", + " c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n", + " e = np.abs(\n", + " scipy.special.poch(z - 2 * n, 2 * n)\n", + " / scipy.special.poch(z - m, m)\n", + " * c\n", + " * xi ** (z - 2 * n + m - 1)\n", + " )\n", + " ez = np.sum(\n", + " scipy.special.poch(z - 2 * n, 2 * n)\n", + " / scipy.special.poch(z - m, m)\n", + " * c\n", + " * zeros[:, None] ** (z - 2 * n + m - 1),\n", + " 0,\n", + " )\n", + " lag = eval_laguerre([z], m)[0]\n", + " err = np.abs(lanczos - lag)\n", + " # print(m+z,ez)\n", + " # for zi,ezi in zip(z[0], ez):\n", + " # print(f\"{m+zi}: {ezi}\")\n", + " # ax.semilogy(xi, e, color=color)\n", + " lines = ax.loglog(xi, e, label=str(n))\n", + " ax.axhline(err, color=lines[0].get_color())\n", + " # ax.set_xticks(np.arange(xi[-1] + 1))\n", + " # ax.set_ylim(1e-8, 1e5)\n", + "_ = ax.legend()\n", + "# _ = ax.legend([f\"z={zi}\" for zi in z[0]])\n", + "# _ = [ax.axvline(x) for x in zeros]\n" + ] + } + ], + "metadata": { + "interpreter": { + "hash": "767d51c1340bd893661ea55ea3124f6de3c7a262a8b4abca0554b478b1e2ff90" + }, + "kernelspec": { + "display_name": "Python 3.8.10 64-bit", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py new file mode 100644 index 0000000..b9088d0 --- /dev/null +++ b/buch/papers/laguerre/scripts/laguerre_plot.py @@ -0,0 +1,100 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Some plots for Laguerre Polynomials.""" + +import os +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np +import scipy.special as ss + + +def get_ticks(start, end, step=1): + ticks = np.arange(start, end, step) + return ticks[ticks != 0] + + +N = 1000 +step = 5 +t = np.linspace(-1.05, 10.5, N)[:, None] +root = str(Path(__file__).parent) +img_path = f"{root}/../images" +os.makedirs(img_path, exist_ok=True) + + +# fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7)) +# ax = fig.add_subplot(axes_class=AxesZero) +fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) +for n in np.arange(0, 8): + k = np.arange(0, n + 1)[None] + L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) + ax.plot(t, L, label=f"n={n}") + +ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True) +ax.set_xticks(get_ticks(0, t[-1], step)) +ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0])) +ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large") + +ylim = 13 +ax.set_yticks(np.arange(-ylim, ylim), minor=True) +ax.set_yticks(np.arange(-step * (ylim // step), ylim, step)) +ax.set_ylim(-ylim, ylim) +ax.set_ylabel(r"$y$", y=0.95, labelpad=-18, rotation=0, fontsize="large") + +ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large") + +# set the x-spine +ax.spines[["left", "bottom"]].set_position("zero") +ax.spines[["right", "top"]].set_visible(False) +ax.xaxis.set_ticks_position("bottom") +hlx = 0.4 +dx = t[-1, 0] - t[0, 0] +dy = 2 * ylim +hly = dy / dx * hlx +dps = fig.dpi_scale_trans.inverted() +bbox = ax.get_window_extent().transformed(dps) +width, height = bbox.width, bbox.height + +# manual arrowhead width and length +hw = 1.0 / 60.0 * dy +hl = 1.0 / 30.0 * dx +lw = 0.5 # axis line width +ohg = 0.0 # arrow overhang + +# compute matching arrowhead length and width +yhw = hw / dy * dx * height / width +yhl = hl / dx * dy * width / height + +# draw x and y axis +ax.arrow( + t[-1, 0] - hl, + 0, + hl, + 0.0, + fc="k", + ec="k", + lw=lw, + head_width=hw, + head_length=hl, + overhang=ohg, + length_includes_head=True, + clip_on=False, +) + +ax.arrow( + 0, + ylim - yhl, + 0.0, + yhl, + fc="k", + ec="k", + lw=lw, + head_width=yhw, + head_length=yhl, + overhang=ohg, + length_includes_head=True, + clip_on=False, +) + +fig.savefig(f"{img_path}/laguerre_polynomes.pdf") diff --git a/buch/papers/laguerre/transformation.tex b/buch/papers/laguerre/transformation.tex deleted file mode 100644 index 4de86b6..0000000 --- a/buch/papers/laguerre/transformation.tex +++ /dev/null @@ -1,31 +0,0 @@ -% -% transformation.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Laguerre Transformation -\label{laguerre:section:transformation}} -\begin{align} - L \left\{ f(x) \right\} - = - \tilde{f}_\alpha(n) - = - \int_0^\infty e^{-x} x^\alpha L_n^\alpha(x) f(x) dx - \label{laguerre:transformation} -\end{align} - -\begin{align} - L^{-1} \left\{ \tilde{f}_\alpha(n) \right\} - = - f(x) - = - \sum_{n=0}^{\infty} - \begin{pmatrix} - n + \alpha \\ - n - \end{pmatrix}^{-1} - \frac{1}{\Gamma(\alpha + 1)} - \tilde{f}_\alpha(n) - L_n^\alpha(x) - \label{laguerre:inverse_transformation} -\end{align}
\ No newline at end of file diff --git a/buch/papers/laguerre/wasserstoff.tex b/buch/papers/laguerre/wasserstoff.tex deleted file mode 100644 index caaa6af..0000000 --- a/buch/papers/laguerre/wasserstoff.tex +++ /dev/null @@ -1,29 +0,0 @@ -% -% wasserstoff.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Radialer Schwingungsanteil eines Wasserstoffatoms -\label{laguerre:section:radial_h_atom}} - -\begin{align} - \nonumber - - \frac{\hbar^2}{2m} - & - \left( - \frac{1}{r^2} \pdv{}{r} - \left( r^2 \pdv{}{r} \right) - + - \frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} - \left( \sin \vartheta \pdv{}{\vartheta} \right) - + - \frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} - \right) - u(r, \vartheta, \varphi) - \\ - & - - \frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) - = - E u(r, \vartheta, \varphi) - \label{laguerre:pdg_h_atom} -\end{align} diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile new file mode 100644 index 0000000..c9dcacc --- /dev/null +++ b/buch/papers/nav/images/Makefile @@ -0,0 +1,108 @@ +# +# Makefile to build images +# +# (c) 2022 +# +all: dreiecke3d + +dreieck.pdf: dreieck.tex dreieckdata.tex macros.tex + pdflatex dreieck.tex + +dreieckdata.tex: pk.m + octave pk.m + +DREIECKE = \ + dreieck1.pdf \ + dreieck2.pdf \ + dreieck3.pdf \ + dreieck4.pdf \ + dreieck5.pdf \ + dreieck6.pdf \ + dreieck7.pdf + +dreiecke: $(DREIECKE) + +dreieck1.pdf: dreieck1.tex dreieckdata.tex macros.tex + pdflatex dreieck1.tex + +dreieck2.pdf: dreieck2.tex dreieckdata.tex macros.tex + pdflatex dreieck2.tex + +dreieck3.pdf: dreieck3.tex dreieckdata.tex macros.tex + pdflatex dreieck3.tex + +dreieck4.pdf: dreieck4.tex dreieckdata.tex macros.tex + pdflatex dreieck4.tex + +dreieck5.pdf: dreieck5.tex dreieckdata.tex macros.tex + pdflatex dreieck5.tex + +dreieck6.pdf: dreieck6.tex dreieckdata.tex macros.tex + pdflatex dreieck6.tex + +dreieck7.pdf: dreieck7.tex dreieckdata.tex macros.tex + pdflatex dreieck7.tex + +DREIECKE3D = \ + dreieck3d1.pdf \ + dreieck3d2.pdf \ + dreieck3d3.pdf \ + dreieck3d4.pdf \ + dreieck3d5.pdf \ + dreieck3d6.pdf \ + dreieck3d7.pdf + +dreiecke3d: $(DREIECKE3D) + +POVRAYOPTIONS = -W1080 -H1080 +#POVRAYOPTIONS = -W480 -H480 + +dreieck3d1.png: dreieck3d1.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d1.png dreieck3d1.pov +dreieck3d1.jpg: dreieck3d1.png + convert dreieck3d1.png -density 300 -units PixelsPerInch dreieck3d1.jpg +dreieck3d1.pdf: dreieck3d1.tex dreieck3d1.jpg + pdflatex dreieck3d1.tex + +dreieck3d2.png: dreieck3d2.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d2.png dreieck3d2.pov +dreieck3d2.jpg: dreieck3d2.png + convert dreieck3d2.png -density 300 -units PixelsPerInch dreieck3d2.jpg +dreieck3d2.pdf: dreieck3d2.tex dreieck3d2.jpg + pdflatex dreieck3d2.tex + +dreieck3d3.png: dreieck3d3.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d3.png dreieck3d3.pov +dreieck3d3.jpg: dreieck3d3.png + convert dreieck3d3.png -density 300 -units PixelsPerInch dreieck3d3.jpg +dreieck3d3.pdf: dreieck3d3.tex dreieck3d3.jpg + pdflatex dreieck3d3.tex + +dreieck3d4.png: dreieck3d4.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d4.png dreieck3d4.pov +dreieck3d4.jpg: dreieck3d4.png + convert dreieck3d4.png -density 300 -units PixelsPerInch dreieck3d4.jpg +dreieck3d4.pdf: dreieck3d4.tex dreieck3d4.jpg + pdflatex dreieck3d4.tex + +dreieck3d5.png: dreieck3d5.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d5.png dreieck3d5.pov +dreieck3d5.jpg: dreieck3d5.png + convert dreieck3d5.png -density 300 -units PixelsPerInch dreieck3d5.jpg +dreieck3d5.pdf: dreieck3d5.tex dreieck3d5.jpg + pdflatex dreieck3d5.tex + +dreieck3d6.png: dreieck3d6.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d6.png dreieck3d6.pov +dreieck3d6.jpg: dreieck3d6.png + convert dreieck3d6.png -density 300 -units PixelsPerInch dreieck3d6.jpg +dreieck3d6.pdf: dreieck3d6.tex dreieck3d6.jpg + pdflatex dreieck3d6.tex + +dreieck3d7.png: dreieck3d7.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d7.png dreieck3d7.pov +dreieck3d7.jpg: dreieck3d7.png + convert dreieck3d7.png -density 300 -units PixelsPerInch dreieck3d7.jpg +dreieck3d7.pdf: dreieck3d7.tex dreieck3d7.jpg + pdflatex dreieck3d7.tex + diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc new file mode 100644 index 0000000..33d9384 --- /dev/null +++ b/buch/papers/nav/images/common.inc @@ -0,0 +1,149 @@ +// +// common.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.034; + +#declare A = vnormalize(< 0, 1, 0>); +#declare B = vnormalize(< 1, 2, -8>); +#declare C = vnormalize(< 5, 1, 0>); +#declare P = vnormalize(< 5, -1, -7>); + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + +light_source { + <10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro grosskreis(normale, staerke) +union { + #declare v1 = vcross(normale, <normale.x, normale.z, normale.y>); + #declare v1 = vnormalize(v1); + #declare v2 = vnormalize(vcross(v1, normale)); + #declare phisteps = 100; + #declare phistep = pi / phisteps; + #declare phi = 0; + #declare p1 = v1; + #while (phi < 2 * pi - phistep/2) + sphere { p1, staerke } + #declare phi = phi + phistep; + #declare p2 = v1 * cos(phi) + v2 * sin(phi); + cylinder { p1, p2, staerke } + #declare p1 = p2; + #end +} +#end + +#macro seite(p, q, staerke) + #declare n = vcross(p, q); + intersection { + grosskreis(n, staerke) + plane { -vcross(n, q) * vdot(vcross(n, q), p), 0 } + plane { -vcross(n, p) * vdot(vcross(n, p), q), 0 } + } +#end + +#macro winkel(w, p, q, staerke) + #declare n = vnormalize(w); + #declare pp = vnormalize(p - vdot(n, p) * n); + #declare qq = vnormalize(q - vdot(n, q) * n); + intersection { + sphere { <0, 0, 0>, 1 + staerke } + cone { <0, 0, 0>, 0, 1.2 * vnormalize(w), 0.4 } + plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 } + plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 } + } +#end + +#macro punkt(p, staerke) + sphere { p, 1.5 * staerke } +#end + +#declare fett = 0.015; +#declare fine = 0.010; + +#declare dreieckfarbe = rgb<0.6,0.6,0.6>; +#declare rot = rgb<0.8,0.2,0.2>; +#declare gruen = rgb<0,0.6,0>; +#declare blau = rgb<0.2,0.2,0.8>; + +sphere { + <0, 0, 0>, 1 + pigment { + color rgb<0.8,0.8,0.8> + } +} + +//union { +// sphere { A, 0.02 } +// sphere { B, 0.02 } +// sphere { C, 0.02 } +// sphere { P, 0.02 } +// pigment { +// color Red +// } +//} + +//union { +// winkel(A, B, C) +// winkel(B, P, C) +// seite(B, C, 0.01) +// seite(B, P, 0.01) +// pigment { +// color rgb<0,0.6,0> +// } +//} diff --git a/buch/papers/nav/images/dreieck.tex b/buch/papers/nav/images/dreieck.tex new file mode 100644 index 0000000..55f6a81 --- /dev/null +++ b/buch/papers/nav/images/dreieck.tex @@ -0,0 +1,68 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\def\punktbeschriftung{ + \node at (A) [above] {$A$}; + \node at (B) [left] {$B$}; + \node at (C) [right] {$C$}; + \node at (P) [below] {$P$}; +} + +\winkelKappa{gray} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\winkelOmega{gray} +\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\punktbeschriftung + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck1.pdf b/buch/papers/nav/images/dreieck1.pdf Binary files differnew file mode 100644 index 0000000..5bdf23d --- /dev/null +++ b/buch/papers/nav/images/dreieck1.pdf diff --git a/buch/papers/nav/images/dreieck1.tex b/buch/papers/nav/images/dreieck1.tex new file mode 100644 index 0000000..436314c --- /dev/null +++ b/buch/papers/nav/images/dreieck1.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck2.pdf b/buch/papers/nav/images/dreieck2.pdf Binary files differnew file mode 100644 index 0000000..a872b25 --- /dev/null +++ b/buch/papers/nav/images/dreieck2.pdf diff --git a/buch/papers/nav/images/dreieck2.tex b/buch/papers/nav/images/dreieck2.tex new file mode 100644 index 0000000..99aabb7 --- /dev/null +++ b/buch/papers/nav/images/dreieck2.tex @@ -0,0 +1,59 @@ +% +% dreieck2.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck3.pdf b/buch/papers/nav/images/dreieck3.pdf Binary files differnew file mode 100644 index 0000000..65070c6 --- /dev/null +++ b/buch/papers/nav/images/dreieck3.pdf diff --git a/buch/papers/nav/images/dreieck3.tex b/buch/papers/nav/images/dreieck3.tex new file mode 100644 index 0000000..0cf5363 --- /dev/null +++ b/buch/papers/nav/images/dreieck3.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d1.pov b/buch/papers/nav/images/dreieck3d1.pov new file mode 100644 index 0000000..8afe60e --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.pov @@ -0,0 +1,58 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fine) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, A, fine) + pigment { + color gruen + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, A, B, fine) + pigment { + color blau + } + finish { + specular 0.95 + metallic + } +} diff --git a/buch/papers/nav/images/dreieck3d1.tex b/buch/papers/nav/images/dreieck3d1.tex new file mode 100644 index 0000000..799b21a --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.tex @@ -0,0 +1,53 @@ +% +% dreieck3d1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +\node at (-2.5,-0.5) {$\beta$}; +\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d2.pov b/buch/papers/nav/images/dreieck3d2.pov new file mode 100644 index 0000000..c23a54c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.pov @@ -0,0 +1,26 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d2.tex b/buch/papers/nav/images/dreieck3d2.tex new file mode 100644 index 0000000..0f6e10c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.tex @@ -0,0 +1,53 @@ +% +% dreieck3d2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d3.pov b/buch/papers/nav/images/dreieck3d3.pov new file mode 100644 index 0000000..f2496b5 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fine) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d3.tex b/buch/papers/nav/images/dreieck3d3.tex new file mode 100644 index 0000000..a047b1b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.tex @@ -0,0 +1,53 @@ +% +% dreieck3d3.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d4.pov b/buch/papers/nav/images/dreieck3d4.pov new file mode 100644 index 0000000..bddcf7c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fine) + seite(A, C, fine) + punkt(A, fine) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, P, fine) + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d4.tex b/buch/papers/nav/images/dreieck3d4.tex new file mode 100644 index 0000000..d49fb66 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.tex @@ -0,0 +1,54 @@ +% +% dreieck3d4.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d4.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.3,-1.5) {$\beta_1$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d5.pov b/buch/papers/nav/images/dreieck3d5.pov new file mode 100644 index 0000000..32fc9e6 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.pov @@ -0,0 +1,26 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fine) + seite(A, C, fine) + punkt(A, fine) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d5.tex b/buch/papers/nav/images/dreieck3d5.tex new file mode 100644 index 0000000..8011b37 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.tex @@ -0,0 +1,53 @@ +% +% dreieck3d5.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d5.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d6.pov b/buch/papers/nav/images/dreieck3d6.pov new file mode 100644 index 0000000..7611950 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(A, C, fett) + seite(B, P, fett) + seite(C, P, fett) + seite(A, P, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, A, P, fine) + pigment { + color rgb<0.6,0.2,0.6> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d6.tex b/buch/papers/nav/images/dreieck3d6.tex new file mode 100644 index 0000000..bbca2ca --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.tex @@ -0,0 +1,55 @@ +% +% dreieck3d6.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d6.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.4,-0.6) {$\kappa$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d7.pov b/buch/papers/nav/images/dreieck3d7.pov new file mode 100644 index 0000000..fa48f5b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.pov @@ -0,0 +1,39 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, C, fett) + seite(A, P, fett) + seite(C, P, fett) + + seite(A, B, fine) + seite(B, C, fine) + seite(B, P, fine) + punkt(A, fett) + punkt(C, fett) + punkt(P, fett) + punkt(B, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fine) + pigment { + color rgb<0.4,0.4,1> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d7.tex b/buch/papers/nav/images/dreieck3d7.tex new file mode 100644 index 0000000..4027a8b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.tex @@ -0,0 +1,55 @@ +% +% dreieck3d7.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d7.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (0.8,3.1) {$\omega$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck4.pdf b/buch/papers/nav/images/dreieck4.pdf Binary files differnew file mode 100644 index 0000000..4871a1e --- /dev/null +++ b/buch/papers/nav/images/dreieck4.pdf diff --git a/buch/papers/nav/images/dreieck4.tex b/buch/papers/nav/images/dreieck4.tex new file mode 100644 index 0000000..19a7d12 --- /dev/null +++ b/buch/papers/nav/images/dreieck4.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck5.pdf b/buch/papers/nav/images/dreieck5.pdf Binary files differnew file mode 100644 index 0000000..cf686e0 --- /dev/null +++ b/buch/papers/nav/images/dreieck5.pdf diff --git a/buch/papers/nav/images/dreieck5.tex b/buch/papers/nav/images/dreieck5.tex new file mode 100644 index 0000000..d1117d1 --- /dev/null +++ b/buch/papers/nav/images/dreieck5.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +%\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck6.pdf b/buch/papers/nav/images/dreieck6.pdf Binary files differnew file mode 100644 index 0000000..7efd673 --- /dev/null +++ b/buch/papers/nav/images/dreieck6.pdf diff --git a/buch/papers/nav/images/dreieck6.tex b/buch/papers/nav/images/dreieck6.tex new file mode 100644 index 0000000..87db1c2 --- /dev/null +++ b/buch/papers/nav/images/dreieck6.tex @@ -0,0 +1,64 @@ +% +% dreieck6.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +%\seiteA{gray} + +\seiteL{black} +\seitePB{black} +\seitePC{black} + +\draw[color=black,line width=1.4pt] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +%\draw[color=gray] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck7.pdf b/buch/papers/nav/images/dreieck7.pdf Binary files differnew file mode 100644 index 0000000..aa83e28 --- /dev/null +++ b/buch/papers/nav/images/dreieck7.pdf diff --git a/buch/papers/nav/images/dreieck7.tex b/buch/papers/nav/images/dreieck7.tex new file mode 100644 index 0000000..f084708 --- /dev/null +++ b/buch/papers/nav/images/dreieck7.tex @@ -0,0 +1,64 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{gray} + +\seiteL{black} +\seitePB{gray} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +\draw[color=gray] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{gray}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node[color=gray] at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieckdata.tex b/buch/papers/nav/images/dreieckdata.tex new file mode 100644 index 0000000..c0fb720 --- /dev/null +++ b/buch/papers/nav/images/dreieckdata.tex @@ -0,0 +1,16 @@ +\coordinate (P) at (0.0000,0.0000); +\coordinate (A) at (1.0000,8.0000); +\coordinate (B) at (-3.0000,3.0000); +\coordinate (C) at (4.0000,4.0000); +\def\kanteAB{(1.0000,8.0000) arc (114.77514:167.90524:7.1589)} +\def\kanteBA{(-3.0000,3.0000) arc (167.90524:114.77514:7.1589)} +\def\kanteAC{(1.0000,8.0000) arc (63.43495:10.30485:5.5902)} +\def\kanteCA{(4.0000,4.0000) arc (10.30485:63.43495:5.5902)} +\def\kanteAP{(1.0000,8.0000) arc (146.30993:199.44003:9.0139)} +\def\kantePA{(0.0000,0.0000) arc (199.44003:146.30993:9.0139)} +\def\kanteBC{(-3.0000,3.0000) arc (-95.90614:-67.83365:14.5774)} +\def\kanteCB{(4.0000,4.0000) arc (-67.83365:-95.90614:14.5774)} +\def\kanteBP{(-3.0000,3.0000) arc (-161.56505:-108.43495:4.7434)} +\def\kantePB{(0.0000,0.0000) arc (-108.43495:-161.56505:4.7434)} +\def\kanteCP{(4.0000,4.0000) arc (-30.96376:-59.03624:11.6619)} +\def\kantePC{(0.0000,0.0000) arc (-59.03624:-30.96376:11.6619)} diff --git a/buch/papers/nav/images/macros.tex b/buch/papers/nav/images/macros.tex new file mode 100644 index 0000000..69a620d --- /dev/null +++ b/buch/papers/nav/images/macros.tex @@ -0,0 +1,54 @@ +\def\winkelAlpha#1{ + \begin{scope} + \clip (A) circle[radius=1.1]; + \fill[color=#1!20] \kanteAB -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(222:0.82)$) {$\alpha$}; +} + +\def\winkelOmega#1{ + \begin{scope} + \clip (A) circle[radius=0.7]; + \fill[color=#1!20] \kanteAP -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(285:0.50)$) {$\omega$}; +} + +\def\winkelGamma#1{ + \begin{scope} + \clip (C) circle[radius=1.0]; + \fill[color=#1!20] \kanteCA -- \kanteBC -- cycle; + \end{scope} + \node[color=#1] at ($(C)+(155:0.60)$) {$\gamma$}; +} + +\def\winkelKappa#1{ + \begin{scope} + \clip (B) circle[radius=1.2]; + \fill[color=#1!20] \kanteBP -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(15:1.00)$) {$\kappa$}; +} + +\def\winkelBeta#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBC -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(35:0.40)$) {$\beta$}; +} + +\def\winkelBetaEins#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBP -- \kanteCB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(330:0.60)$) {$\beta_1$}; +} + +\def\seiteC#1{ \node[color=#1] at (-1.9,5.9) {$c$}; } +\def\seiteB#1{ \node[color=#1] at (3.2,6.5) {$b$}; } +\def\seiteL#1{ \node[color=#1] at (-0.2,4.5) {$l$}; } +\def\seiteA#1{ \node[color=#1] at (2,3) {$a$}; } +\def\seitePB#1{ \node[color=#1] at (-2.1,1) {$p_b$}; } +\def\seitePC#1{ \node[color=#1] at (2.5,1.5) {$p_c$}; } diff --git a/buch/papers/nav/images/pk.m b/buch/papers/nav/images/pk.m new file mode 100644 index 0000000..6e89e9a --- /dev/null +++ b/buch/papers/nav/images/pk.m @@ -0,0 +1,55 @@ +# +# pk.m -- Punkte und Kanten für sphärisches Dreieck +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +A = [ 1, 8 ]; +B = [ -3, 3 ]; +C = [ 4, 4 ]; +P = [ 0, 0 ]; + +global fn; +fn = fopen("dreieckdata.tex", "w"); + +fprintf(fn, "\\coordinate (P) at (%.4f,%.4f);\n", P(1,1), P(1,2)); +fprintf(fn, "\\coordinate (A) at (%.4f,%.4f);\n", A(1,1), A(1,2)); +fprintf(fn, "\\coordinate (B) at (%.4f,%.4f);\n", B(1,1), B(1,2)); +fprintf(fn, "\\coordinate (C) at (%.4f,%.4f);\n", C(1,1), C(1,2)); + +function retval = seite(A, B, l, nameA, nameB) + global fn; + d = fliplr(B - A); + d(1, 2) = -d(1, 2); + # Zentrum + C = 0.5 * (A + B) + l * d; + # Radius: + r = hypot(C(1,1)-A(1,1), C(1,2)-A(1,2)) + # Winkel von + winkelvon = atan2(A(1,2)-C(1,2),A(1,1)-C(1,1)); + # Winkel bis + winkelbis = atan2(B(1,2)-C(1,2),B(1,1)-C(1,1)); + if (abs(winkelvon - winkelbis) > pi) + if (winkelbis < winkelvon) + winkelbis = winkelbis + 2 * pi + else + winkelvon = winkelvon + 2 * pi + end + end + # Kurve + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameA, nameB, + A(1,1), A(1,2), winkelvon * 180 / pi, winkelbis * 180 / pi, r); + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameB, nameA, + B(1,1), B(1,2), winkelbis * 180 / pi, winkelvon * 180 / pi, r); +endfunction + +seite(A, B, -1, "A", "B"); +seite(A, C, 1, "A", "C"); +seite(A, P, -1, "A", "P"); +seite(B, C, -2, "B", "C"); +seite(B, P, -1, "B", "P"); +seite(C, P, 2, "C", "P"); + +fclose(fn); diff --git a/buch/papers/zeta/Makefile.inc b/buch/papers/zeta/Makefile.inc index 11c7697..14babe2 100644 --- a/buch/papers/zeta/Makefile.inc +++ b/buch/papers/zeta/Makefile.inc @@ -7,8 +7,7 @@ dependencies-zeta = \ papers/zeta/packages.tex \ papers/zeta/main.tex \ papers/zeta/references.bib \ - papers/zeta/teil0.tex \ - papers/zeta/teil1.tex \ - papers/zeta/teil2.tex \ - papers/zeta/teil3.tex + papers/zeta/einleitung.tex \ + papers/zeta/analytic_continuation.tex \ + papers/zeta/zeta_gamma.tex \ diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex new file mode 100644 index 0000000..bb95b92 --- /dev/null +++ b/buch/papers/zeta/analytic_continuation.tex @@ -0,0 +1,264 @@ +\section{Analytische Fortsetzung} \label{zeta:section:analytische_fortsetzung} +\rhead{Analytische Fortsetzung} + +%TODO missing Text + +\subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0} +Zuerst definieren die Dirichletsche Etafunktion als +\begin{equation}\label{zeta:equation:eta} + \eta(s) + = + \sum_{n=1}^{\infty} + \frac{(-1)^{n-1}}{n^s}, +\end{equation} +wobei die Reihe bis auf die alternierenden Vorzeichen die selbe wie in der Zetafunktion ist. +Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) > 0$, da dann die einzelnen Glieder monoton fallend sind. + +Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion mit der Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. +Die folgenden Schritte zeigen, wie man dazu kommt: +\begin{align} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{1}{n^s} \label{zeta:align1} + \\ + \frac{1}{2^{s-1}} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{2}{(2n)^s} \label{zeta:align2} + \\ + \left(1 - \frac{1}{2^{s-1}} \right) + \zeta(s) + &= + \frac{1}{1^s} + \underbrace{-\frac{2}{2^s} + \frac{1}{2^s}}_{-\frac{1}{2^s}} + + \frac{1}{3^s} + \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}} + \ldots + && \text{\eqref{zeta:align1}} - \text{\eqref{zeta:align2}} + \\ + &= \eta(s) + \\ + \zeta(s) + &= + \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s). +\end{align} + +\subsection{Fortsetzung auf ganz $\mathbb{C}$} \label{zeta:subsection:auf_ganz} +Für die Fortsetzung auf den Rest von $\mathbb{C}$, verwenden wir den Zusammenhang von Gamma- und Zetafunktion aus \ref{zeta:section:zusammenhang_mit_gammafunktion}. +Wir beginnen damit, die Gammafunktion für den halben Funktionswert zu berechnen als +\begin{equation} + \Gamma \left( \frac{s}{2} \right) + = + \int_0^{\infty} t^{\frac{s}{2}-1} e^{-t} dt. +\end{equation} +Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten +\begin{align} + \Gamma \left( \frac{s}{2} \right) + &= + \int_0^{\infty} + (\pi n^2)^{\frac{s}{2}} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + dx + && \text{Division durch } (\pi n^2)^{\frac{s}{2}} + \\ + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} + &= + \int_0^{\infty} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + dx + && \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} + \\ + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + &= + \int_0^{\infty} + x^{\frac{s}{2}-1} + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + dx. \label{zeta:equation:integral1} +\end{align} +Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. +%TODO Wieso folgendes -> aus Fourier Signal +Es gilt +\begin{equation}\label{zeta:equation:psi} + \psi(x) + = + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}. +\end{equation} + +Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als +\begin{equation}\label{zeta:equation:integral2} + \int_0^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx, +\end{equation} +wobei wir uns nun auf den ersten Teil konzentrieren werden. +Dabei setzen wir das Wissen aus \eqref{zeta:equation:psi} ein und erhalten +\begin{align} + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \left( + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}. + \right) + dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + + \frac{1}{2} + \left( + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \right) + dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + dx + + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + dx. \label{zeta:equation:integral3} +\end{align} +Dabei kann das zweite Integral gelöst werden als +\begin{equation} + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + dx + = + \frac{1}{s(s-1)}. +\end{equation} +Das erste Integral aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. +Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. +Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. +Dies ergibt +\begin{align} + \int_{\infty}^{1} + {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{-du}{u^2} + &= + \int_{1}^{\infty} + {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{du}{u^2} + \\ + &= + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx, +\end{align} +wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. +Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind. +Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten +\begin{equation} + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + = + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx + + + \frac{1}{s(s-1)}. +\end{equation} +Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um schlussendlich +\begin{align} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + \nonumber + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + \left( + x^{-\frac{s}{2}-\frac{1}{2}} + + + x^{\frac{s}{2}-1} + \right) + \psi(x) + dx + \\ + &= + \frac{-1}{s(1-s)} + + + \int_{1}^{\infty} + \left( + x^{\frac{1-s}{2}} + + + x^{\frac{s}{2}} + \right) + \frac{\psi(x)}{x} + dx, +\end{align} +zu erhalten. +Wenn wir dieses Resultat genau anschauen, erkennen wir dass sich nichts verändert wenn $s$ mit $1-s$ ersetzt wird. +Somit haben wir die analytische Fortsetzung gefunden als +\begin{equation}\label{zeta:equation:functional} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + = + \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}} + \zeta(1-s). +\end{equation} + diff --git a/buch/papers/zeta/einleitung.tex b/buch/papers/zeta/einleitung.tex new file mode 100644 index 0000000..3b70531 --- /dev/null +++ b/buch/papers/zeta/einleitung.tex @@ -0,0 +1,11 @@ +\section{Einleitung} \label{zeta:section:einleitung} +\rhead{Einleitung} + +Die Riemannsche Zetafunktion ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als +\begin{equation}\label{zeta:equation1} + \zeta(s) + = + \sum_{n=1}^{\infty} + \frac{1}{n^s}. +\end{equation} + diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex index 1d9e059..e0ea8e1 100644 --- a/buch/papers/zeta/main.tex +++ b/buch/papers/zeta/main.tex @@ -3,34 +3,16 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:zeta}} -\lhead{Thema} +\chapter{Riemannsche Zetafunktion\label{chapter:zeta}} +\lhead{Riemannsche Zetafunktion} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Raphael Unterer} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +%TODO Einleitung -\input{papers/zeta/teil0.tex} -\input{papers/zeta/teil1.tex} -\input{papers/zeta/teil2.tex} -\input{papers/zeta/teil3.tex} +\input{papers/zeta/einleitung.tex} +\input{papers/zeta/zeta_gamma.tex} +\input{papers/zeta/analytic_continuation.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/zeta/teil0.tex b/buch/papers/zeta/teil0.tex deleted file mode 100644 index 56c0b1b..0000000 --- a/buch/papers/zeta/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{zeta:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{zeta:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/zeta/teil1.tex b/buch/papers/zeta/teil1.tex deleted file mode 100644 index 4017ee8..0000000 --- a/buch/papers/zeta/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{zeta:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{zeta:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{zeta:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{zeta:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/zeta/teil2.tex b/buch/papers/zeta/teil2.tex deleted file mode 100644 index 9e8a96e..0000000 --- a/buch/papers/zeta/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{zeta:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/teil3.tex b/buch/papers/zeta/teil3.tex deleted file mode 100644 index 6610cc3..0000000 --- a/buch/papers/zeta/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{zeta:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex new file mode 100644 index 0000000..59c8744 --- /dev/null +++ b/buch/papers/zeta/zeta_gamma.tex @@ -0,0 +1,53 @@ +\section{Zusammenhang mit Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion} +\rhead{Zusammenhang mit Gammafunktion} + +Dieser Abschnitt stellt die Verbindung zwischen der Gamma- und der Zetafunktion her. + +%TODO ref Gamma +Wenn in der Gammafunkion die Integrationsvariable $t$ substituieren mit $t = nu$ und $dt = n du$, dann können wir die Gleichung umstellen und erhalten den Zusammenhang mit der Zetafunktion +\begin{align} + \Gamma(s) + &= + \int_0^{\infty} t^{s-1} e^{-t} dt + \\ + &= + \int_0^{\infty} n^{s\cancel{-1}}u^{s-1} e^{-nu} \cancel{n}du + && + \text{Division durch }n^s + \\ + \frac{\Gamma(s)}{n^s} + &= + \int_0^{\infty} u^{s-1} e^{-nu}du + && + \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} + \\ + \Gamma(s) \zeta(s) + &= + \int_0^{\infty} u^{s-1} + \sum_{n=1}^{\infty}e^{-nu} + du. + \label{zeta:equation:zeta_gamma1} +\end{align} +Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhalten +\begin{align} + \sum_{n=1}^{\infty}e^{-u^n} + &= + \sum_{n=0}^{\infty}e^{-u^n} + - + 1 + \\ + &= + \frac{1}{1 - e^{-u}} - 1 + \\ + &= + \frac{1}{e^u - 1}. +\end{align} +Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir +\begin{equation}\label{zeta:equation:zeta_gamma_final} + \zeta(s) + = + \frac{1}{\Gamma(s)} + \int_0^{\infty} + \frac{u^{s-1}}{e^u -1} + du. +\end{equation} 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/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) + -- 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(-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) + -- (-0.4295,-0.5199) + -- (-0.4276,-0.5101) + -- (-0.4270,-0.5001) + -- (-0.4279,-0.4902) + -- (-0.4301,-0.4804) + -- (-0.4336,-0.4711) + -- (-0.4383,-0.4623) + -- (-0.4443,-0.4542) + -- (-0.4512,-0.4471) + -- (-0.4591,-0.4410) + -- (-0.4678,-0.4360) + -- (-0.4771,-0.4323) + -- (-0.4868,-0.4299) + -- (-0.4967,-0.4289) + -- (-0.5067,-0.4293) + -- (-0.5165,-0.4311) + -- (-0.5260,-0.4343) + -- (-0.5350,-0.4387) + -- (-0.5432,-0.4444) + -- (-0.5505,-0.4512) + -- (-0.5568,-0.4590) + -- (-0.5619,-0.4676) + -- (-0.5658,-0.4768) + -- (-0.5683,-0.4864) + -- (-0.5694,-0.4964) + -- (-0.5690,-0.5064) + -- (-0.5672,-0.5162) + -- (-0.5641,-0.5257) + -- (-0.5595,-0.5346) + -- (-0.5538,-0.5427) + -- (-0.5469,-0.5500) + -- (-0.5391,-0.5562) + -- (-0.5304,-0.5611) + -- (-0.5211,-0.5648) + -- (-0.5114,-0.5670) + -- (-0.5014,-0.5678) + -- (-0.4914,-0.5671) + -- (-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 |
