Merge branch 'master' of github.com:HeadAndToes/SeminarSpezielleFunktionen

86 files changed, 7794 insertions, 332 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
new 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
new 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
new 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
new 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
new 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$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck2.pdf b/buch/papers/nav/images/dreieck2.pdf
new file mode 100644
index 0000000..a872b25
--- /dev/null
+++ b/buch/papers/nav/images/dreieck2.pdf
diff --git a/buch/papers/nav/images/dreieck2.tex b/buch/papers/nav/images/dreieck2.tex
new file mode 100644
index 0000000..99aabb7
--- /dev/null
+++ b/buch/papers/nav/images/dreieck2.tex
@@ -0,0 +1,59 @@
+%
+% dreieck2.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+%\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+\seiteC{black}
+\seiteB{black}
+%\seiteA{black}
+
+%\seiteL{gray}
+\seitePB{gray}
+\seitePC{gray}
+
+\draw[line width=1.4pt] \kanteAB;
+\draw[line width=1.4pt] \kanteAC;
+%\draw[color=gray] \kanteAP;
+\draw[line width=1.4pt] \kanteBC;
+\draw[color=gray] \kanteBP;
+\draw[color=gray] \kanteCP;
+
+\punkt{(A)}{black};
+\punkt{(B)}{black};
+\punkt{(C)}{black};
+\punkt{(P)}{gray};
+
+\node at (A) [above] {$A$};
+\node at (B) [left] {$B$};
+\node at (C) [right] {$C$};
+\node[color=gray] at (P) [below] {$P$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck3.pdf b/buch/papers/nav/images/dreieck3.pdf
new file mode 100644
index 0000000..65070c6
--- /dev/null
+++ b/buch/papers/nav/images/dreieck3.pdf
diff --git a/buch/papers/nav/images/dreieck3.tex b/buch/papers/nav/images/dreieck3.tex
new file mode 100644
index 0000000..0cf5363
--- /dev/null
+++ b/buch/papers/nav/images/dreieck3.tex
@@ -0,0 +1,59 @@
+%
+% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+\seiteC{black}
+\seiteB{black}
+%\seiteA{black}
+
+%\seiteL{gray}
+\seitePB{gray}
+\seitePC{gray}
+
+\draw[line width=1.4pt] \kanteAB;
+\draw[line width=1.4pt] \kanteAC;
+%\draw[color=gray] \kanteAP;
+\draw[line width=1.4pt] \kanteBC;
+\draw[color=gray] \kanteBP;
+\draw[color=gray] \kanteCP;
+
+\punkt{(A)}{black};
+\punkt{(B)}{black};
+\punkt{(C)}{black};
+\punkt{(P)}{gray};
+
+\node at (A) [above] {$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$};
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck4.pdf b/buch/papers/nav/images/dreieck4.pdf
new file mode 100644
index 0000000..4871a1e
--- /dev/null
+++ b/buch/papers/nav/images/dreieck4.pdf
diff --git a/buch/papers/nav/images/dreieck4.tex b/buch/papers/nav/images/dreieck4.tex
new file mode 100644
index 0000000..19a7d12
--- /dev/null
+++ b/buch/papers/nav/images/dreieck4.tex
@@ -0,0 +1,64 @@
+%
+% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+%\winkelKappa{gray}
+
+%\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+%\winkelOmega{gray}
+\winkelBetaEins{brown}
+
+%\seiteC{gray}
+%\seiteB{gray}
+%\seiteL{gray}
+
+\seiteA{black}
+\seitePB{black}
+\seitePC{black}
+
+\draw[color=gray] \kanteAB;
+\draw[color=gray] \kanteAC;
+%\draw[color=gray] \kanteAP;
+\draw[color=black,line width=1.4pt] \kanteBC;
+\draw[color=black,line width=1.4pt] \kanteBP;
+\draw[color=black,line width=1.4pt] \kanteCP;
+
+\punkt{(A)}{gray};
+\punkt{(B)}{black};
+\punkt{(C)}{black};
+\punkt{(P)}{black};
+
+\node[color=gray] at (A) [above] {$A$};
+\node at (B) [left] {$B$};
+\node at (C) [right] {$C$};
+\node at (P) [below] {$P$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck5.pdf b/buch/papers/nav/images/dreieck5.pdf
new file mode 100644
index 0000000..cf686e0
--- /dev/null
+++ b/buch/papers/nav/images/dreieck5.pdf
diff --git a/buch/papers/nav/images/dreieck5.tex b/buch/papers/nav/images/dreieck5.tex
new file mode 100644
index 0000000..d1117d1
--- /dev/null
+++ b/buch/papers/nav/images/dreieck5.tex
@@ -0,0 +1,64 @@
+%
+% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+%\winkelKappa{gray}
+
+%\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+%\winkelOmega{gray}
+%\winkelBetaEins{brown}
+
+%\seiteC{gray}
+%\seiteB{gray}
+%\seiteL{gray}
+
+%\seiteA{black}
+\seitePB{black}
+\seitePC{black}
+
+\draw[color=gray] \kanteAB;
+\draw[color=gray] \kanteAC;
+%\draw[color=gray] \kanteAP;
+\draw[color=black,line width=1.4pt] \kanteBC;
+\draw[color=black,line width=1.4pt] \kanteBP;
+\draw[color=black,line width=1.4pt] \kanteCP;
+
+\punkt{(A)}{gray};
+\punkt{(B)}{black};
+\punkt{(C)}{black};
+\punkt{(P)}{black};
+
+\node[color=gray] at (A) [above] {$A$};
+\node at (B) [left] {$B$};
+\node at (C) [right] {$C$};
+\node at (P) [below] {$P$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck6.pdf b/buch/papers/nav/images/dreieck6.pdf
new file mode 100644
index 0000000..7efd673
--- /dev/null
+++ b/buch/papers/nav/images/dreieck6.pdf
diff --git a/buch/papers/nav/images/dreieck6.tex b/buch/papers/nav/images/dreieck6.tex
new file mode 100644
index 0000000..87db1c2
--- /dev/null
+++ b/buch/papers/nav/images/dreieck6.tex
@@ -0,0 +1,64 @@
+%
+% dreieck6.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+\winkelKappa{gray}
+
+%\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+%\winkelOmega{gray}
+%\winkelBetaEins{brown}
+
+\seiteC{black}
+\seiteB{black}
+%\seiteA{gray}
+
+\seiteL{black}
+\seitePB{black}
+\seitePC{black}
+
+\draw[color=black,line width=1.4pt] \kanteAB;
+\draw[color=black,line width=1.4pt] \kanteAC;
+\draw[color=black,line width=1.4pt] \kanteAP;
+%\draw[color=gray] \kanteBC;
+\draw[color=black,line width=1.4pt] \kanteBP;
+\draw[color=black,line width=1.4pt] \kanteCP;
+
+\punkt{(A)}{black};
+\punkt{(B)}{black};
+\punkt{(C)}{black};
+\punkt{(P)}{black};
+
+\node at (A) [above] {$A$};
+\node at (B) [left] {$B$};
+\node at (C) [right] {$C$};
+\node at (P) [below] {$P$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/nav/images/dreieck7.pdf b/buch/papers/nav/images/dreieck7.pdf
new file mode 100644
index 0000000..aa83e28
--- /dev/null
+++ b/buch/papers/nav/images/dreieck7.pdf
diff --git a/buch/papers/nav/images/dreieck7.tex b/buch/papers/nav/images/dreieck7.tex
new file mode 100644
index 0000000..f084708
--- /dev/null
+++ b/buch/papers/nav/images/dreieck7.tex
@@ -0,0 +1,64 @@
+%
+% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\skala{1}
+
+\def\punkt#1#2{
+	\fill[color=#2] #1 circle[radius=0.08];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{dreieckdata.tex}
+\input{macros.tex}
+
+%\winkelKappa{gray}
+
+%\winkelAlpha{red}
+%\winkelGamma{blue}
+%\winkelBeta{darkgreen}
+
+\winkelOmega{gray}
+%\winkelBetaEins{brown}
+
+\seiteC{black}
+\seiteB{black}
+\seiteA{gray}
+
+\seiteL{black}
+\seitePB{gray}
+\seitePC{black}
+
+\draw[color=gray] \kanteAB;
+\draw[color=black,line width=1.4pt] \kanteAC;
+\draw[color=black,line width=1.4pt] \kanteAP;
+\draw[color=gray] \kanteBC;
+\draw[color=gray] \kanteBP;
+\draw[line width=1.4pt] \kanteCP;
+
+\punkt{(A)}{black};
+\punkt{(B)}{gray};
+\punkt{(C)}{black};
+\punkt{(P)}{black};
+
+\node at (A) [above] {$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/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
new 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
new file mode 100644
index 0000000..f413852
--- /dev/null
+++ b/vorlesungen/slides/fresnel/apfel.png
diff --git a/vorlesungen/slides/fresnel/apfel.tex b/vorlesungen/slides/fresnel/apfel.tex
new file mode 100644
index 0000000..090c3d5
--- /dev/null
+++ b/vorlesungen/slides/fresnel/apfel.tex
@@ -0,0 +1,32 @@
+%
+% apfel.tex -- Apfelschale als Klothoide
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\input{../slides/fresnel/eulerpath.tex}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Apfelschale}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+\clip(-1,-1) rectangle (7,6);
+\uncover<2->{
+\node at (3.1,2.2) [rotate=-3]
+	{\includegraphics[width=9.4cm]{../slides/fresnel/apfel.png}};
+}
+\end{scope}
+\draw[color=gray!50] (0,0) rectangle (4,4);
+\draw[->] (-0.5,0) -- (7.5,0) coordinate[label={$C(t)$}];
+\draw[->] (0,-0.5) -- (0,6.0) coordinate[label={left:$S(t)$}];
+\uncover<3->{
+\begin{scope}[scale=8]
+\draw[color=red,opacity=0.5,line width=1.4pt] \fresnela;
+\end{scope}
+}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/fresnel/chapter.tex b/vorlesungen/slides/fresnel/chapter.tex
index dc5d031..916a3a9 100644
--- a/vorlesungen/slides/fresnel/chapter.tex
+++ b/vorlesungen/slides/fresnel/chapter.tex
@@ -3,4 +3,8 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
-\folie{fresnel/test.tex}
+\folie{fresnel/integrale.tex}
+\folie{fresnel/kruemmung.tex}
+\folie{fresnel/klothoide.tex}
+\folie{fresnel/numerik.tex}
+\folie{fresnel/apfel.tex}
diff --git a/vorlesungen/slides/fresnel/eulerpath.tex b/vorlesungen/slides/fresnel/eulerpath.tex
new file mode 100644
index 0000000..ecd0b2b
--- /dev/null
+++ b/vorlesungen/slides/fresnel/eulerpath.tex
@@ -0,0 +1,4012 @@
+\def\fresnela{ (0,0)
+	-- (0.0100,0.0000)
+	-- (0.0200,0.0000)
+	-- (0.0300,0.0000)
+	-- (0.0400,0.0000)
+	-- (0.0500,0.0001)
+	-- (0.0600,0.0001)
+	-- (0.0700,0.0002)
+	-- (0.0800,0.0003)
+	-- (0.0900,0.0004)
+	-- (0.1000,0.0005)
+	-- (0.1100,0.0007)
+	-- (0.1200,0.0009)
+	-- (0.1300,0.0012)
+	-- (0.1400,0.0014)
+	-- (0.1500,0.0018)
+	-- (0.1600,0.0021)
+	-- (0.1700,0.0026)
+	-- (0.1800,0.0031)
+	-- (0.1899,0.0036)
+	-- (0.1999,0.0042)
+	-- (0.2099,0.0048)
+	-- (0.2199,0.0056)
+	-- (0.2298,0.0064)
+	-- (0.2398,0.0072)
+	-- (0.2498,0.0082)
+	-- (0.2597,0.0092)
+	-- (0.2696,0.0103)
+	-- (0.2796,0.0115)
+	-- (0.2895,0.0128)
+	-- (0.2994,0.0141)
+	-- (0.3093,0.0156)
+	-- (0.3192,0.0171)
+	-- (0.3290,0.0188)
+	-- (0.3389,0.0205)
+	-- (0.3487,0.0224)
+	-- (0.3585,0.0244)
+	-- (0.3683,0.0264)
+	-- (0.3780,0.0286)
+	-- (0.3878,0.0309)
+	-- (0.3975,0.0334)
+	-- (0.4072,0.0359)
+	-- (0.4168,0.0386)
+	-- (0.4264,0.0414)
+	-- (0.4359,0.0443)
+	-- (0.4455,0.0474)
+	-- (0.4549,0.0506)
+	-- (0.4644,0.0539)
+	-- (0.4738,0.0574)
+	-- (0.4831,0.0610)
+	-- (0.4923,0.0647)
+	-- (0.5016,0.0686)
+	-- (0.5107,0.0727)
+	-- (0.5198,0.0769)
+	-- (0.5288,0.0812)
+	-- (0.5377,0.0857)
+	-- (0.5466,0.0904)
+	-- (0.5553,0.0952)
+	-- (0.5640,0.1001)
+	-- (0.5726,0.1053)
+	-- (0.5811,0.1105)
+	-- (0.5895,0.1160)
+	-- (0.5978,0.1216)
+	-- (0.6059,0.1273)
+	-- (0.6140,0.1333)
+	-- (0.6219,0.1393)
+	-- (0.6298,0.1456)
+	-- (0.6374,0.1520)
+	-- (0.6450,0.1585)
+	-- (0.6524,0.1653)
+	-- (0.6597,0.1721)
+	-- (0.6668,0.1792)
+	-- (0.6737,0.1864)
+	-- (0.6805,0.1937)
+	-- (0.6871,0.2012)
+	-- (0.6935,0.2089)
+	-- (0.6998,0.2167)
+	-- (0.7058,0.2246)
+	-- (0.7117,0.2327)
+	-- (0.7174,0.2410)
+	-- (0.7228,0.2493)
+	-- (0.7281,0.2579)
+	-- (0.7331,0.2665)
+	-- (0.7379,0.2753)
+	-- (0.7425,0.2841)
+	-- (0.7469,0.2932)
+	-- (0.7510,0.3023)
+	-- (0.7548,0.3115)
+	-- (0.7584,0.3208)
+	-- (0.7617,0.3303)
+	-- (0.7648,0.3398)
+	-- (0.7676,0.3494)
+	-- (0.7702,0.3590)
+	-- (0.7724,0.3688)
+	-- (0.7744,0.3786)
+	-- (0.7760,0.3885)
+	-- (0.7774,0.3984)
+	-- (0.7785,0.4083)
+	-- (0.7793,0.4183)
+	-- (0.7797,0.4283)
+	-- (0.7799,0.4383)
+	-- (0.7797,0.4483)
+	-- (0.7793,0.4582)
+	-- (0.7785,0.4682)
+	-- (0.7774,0.4782)
+	-- (0.7759,0.4880)
+	-- (0.7741,0.4979)
+	-- (0.7721,0.5077)
+	-- (0.7696,0.5174)
+	-- (0.7669,0.5270)
+	-- (0.7638,0.5365)
+	-- (0.7604,0.5459)
+	-- (0.7567,0.5552)
+	-- (0.7526,0.5643)
+	-- (0.7482,0.5733)
+	-- (0.7436,0.5821)
+	-- (0.7385,0.5908)
+	-- (0.7332,0.5993)
+	-- (0.7276,0.6075)
+	-- (0.7217,0.6156)
+	-- (0.7154,0.6234)
+	-- (0.7089,0.6310)
+	-- (0.7021,0.6383)
+	-- (0.6950,0.6454)
+	-- (0.6877,0.6522)
+	-- (0.6801,0.6587)
+	-- (0.6722,0.6648)
+	-- (0.6641,0.6707)
+	-- (0.6558,0.6763)
+	-- (0.6473,0.6815)
+	-- (0.6386,0.6863)
+	-- (0.6296,0.6908)
+	-- (0.6205,0.6950)
+	-- (0.6112,0.6987)
+	-- (0.6018,0.7021)
+	-- (0.5923,0.7050)
+	-- (0.5826,0.7076)
+	-- (0.5728,0.7097)
+	-- (0.5630,0.7114)
+	-- (0.5531,0.7127)
+	-- (0.5431,0.7135)
+	-- (0.5331,0.7139)
+	-- (0.5231,0.7139)
+	-- (0.5131,0.7134)
+	-- (0.5032,0.7125)
+	-- (0.4933,0.7111)
+	-- (0.4834,0.7093)
+	-- (0.4737,0.7070)
+	-- (0.4641,0.7043)
+	-- (0.4546,0.7011)
+	-- (0.4453,0.6975)
+	-- (0.4361,0.6935)
+	-- (0.4272,0.6890)
+	-- (0.4185,0.6841)
+	-- (0.4100,0.6788)
+	-- (0.4018,0.6731)
+	-- (0.3939,0.6670)
+	-- (0.3862,0.6605)
+	-- (0.3790,0.6536)
+	-- (0.3720,0.6464)
+	-- (0.3655,0.6389)
+	-- (0.3593,0.6310)
+	-- (0.3535,0.6229)
+	-- (0.3482,0.6144)
+	-- (0.3433,0.6057)
+	-- (0.3388,0.5968)
+	-- (0.3348,0.5876)
+	-- (0.3313,0.5782)
+	-- (0.3283,0.5687)
+	-- (0.3258,0.5590)
+	-- (0.3238,0.5492)
+	-- (0.3224,0.5393)
+	-- (0.3214,0.5293)
+	-- (0.3211,0.5194)
+	-- (0.3212,0.5094)
+	-- (0.3219,0.4994)
+	-- (0.3232,0.4895)
+	-- (0.3250,0.4796)
+	-- (0.3273,0.4699)
+	-- (0.3302,0.4603)
+	-- (0.3336,0.4509)
+	-- (0.3376,0.4418)
+	-- (0.3420,0.4328)
+	-- (0.3470,0.4241)
+	-- (0.3524,0.4157)
+	-- (0.3584,0.4077)
+	-- (0.3648,0.4000)
+	-- (0.3716,0.3927)
+	-- (0.3788,0.3858)
+	-- (0.3865,0.3793)
+	-- (0.3945,0.3733)
+	-- (0.4028,0.3678)
+	-- (0.4115,0.3629)
+	-- (0.4204,0.3584)
+	-- (0.4296,0.3545)
+	-- (0.4391,0.3511)
+	-- (0.4487,0.3484)
+	-- (0.4584,0.3462)
+	-- (0.4683,0.3447)
+	-- (0.4783,0.3437)
+	-- (0.4883,0.3434)
+	-- (0.4982,0.3437)
+	-- (0.5082,0.3447)
+	-- (0.5181,0.3462)
+	-- (0.5278,0.3484)
+	-- (0.5374,0.3513)
+	-- (0.5468,0.3547)
+	-- (0.5560,0.3587)
+	-- (0.5648,0.3633)
+	-- (0.5734,0.3685)
+	-- (0.5816,0.3743)
+	-- (0.5894,0.3805)
+	-- (0.5967,0.3873)
+	-- (0.6036,0.3945)
+	-- (0.6100,0.4022)
+	-- (0.6159,0.4103)
+	-- (0.6212,0.4188)
+	-- (0.6259,0.4276)
+	-- (0.6300,0.4367)
+	-- (0.6335,0.4461)
+	-- (0.6363,0.4557)
+	-- (0.6384,0.4655)
+	-- (0.6399,0.4754)
+	-- (0.6407,0.4853)
+	-- (0.6408,0.4953)
+	-- (0.6401,0.5053)
+	-- (0.6388,0.5152)
+	-- (0.6368,0.5250)
+	-- (0.6340,0.5346)
+	-- (0.6306,0.5440)
+	-- (0.6266,0.5532)
+	-- (0.6218,0.5620)
+	-- (0.6165,0.5704)
+	-- (0.6105,0.5784)
+	-- (0.6040,0.5860)
+	-- (0.5970,0.5931)
+	-- (0.5894,0.5996)
+	-- (0.5814,0.6056)
+	-- (0.5729,0.6110)
+	-- (0.5641,0.6157)
+	-- (0.5550,0.6197)
+	-- (0.5455,0.6230)
+	-- (0.5359,0.6256)
+	-- (0.5261,0.6275)
+	-- (0.5161,0.6286)
+	-- (0.5061,0.6289)
+	-- (0.4961,0.6285)
+	-- (0.4862,0.6273)
+	-- (0.4764,0.6254)
+	-- (0.4668,0.6226)
+	-- (0.4574,0.6192)
+	-- (0.4483,0.6150)
+	-- (0.4396,0.6101)
+	-- (0.4313,0.6045)
+	-- (0.4235,0.5983)
+	-- (0.4161,0.5915)
+	-- (0.4094,0.5842)
+	-- (0.4033,0.5763)
+	-- (0.3978,0.5679)
+	-- (0.3930,0.5591)
+	-- (0.3889,0.5500)
+	-- (0.3856,0.5406)
+	-- (0.3831,0.5309)
+	-- (0.3814,0.5210)
+	-- (0.3805,0.5111)
+	-- (0.3805,0.5011)
+	-- (0.3812,0.4911)
+	-- (0.3828,0.4812)
+	-- (0.3853,0.4715)
+	-- (0.3885,0.4621)
+	-- (0.3925,0.4529)
+	-- (0.3973,0.4441)
+	-- (0.4028,0.4358)
+	-- (0.4090,0.4279)
+	-- (0.4158,0.4207)
+	-- (0.4233,0.4140)
+	-- (0.4313,0.4080)
+	-- (0.4397,0.4027)
+	-- (0.4487,0.3982)
+	-- (0.4579,0.3944)
+	-- (0.4675,0.3915)
+	-- (0.4773,0.3895)
+	-- (0.4872,0.3883)
+	-- (0.4972,0.3880)
+	-- (0.5072,0.3886)
+	-- (0.5171,0.3900)
+	-- (0.5268,0.3924)
+	-- (0.5362,0.3956)
+	-- (0.5454,0.3996)
+	-- (0.5541,0.4045)
+	-- (0.5624,0.4101)
+	-- (0.5701,0.4165)
+	-- (0.5772,0.4235)
+	-- (0.5836,0.4312)
+	-- (0.5893,0.4394)
+	-- (0.5942,0.4481)
+	-- (0.5983,0.4572)
+	-- (0.6015,0.4667)
+	-- (0.6038,0.4764)
+	-- (0.6053,0.4863)
+	-- (0.6057,0.4963)
+	-- (0.6052,0.5063)
+	-- (0.6038,0.5162)
+	-- (0.6015,0.5259)
+	-- (0.5982,0.5354)
+	-- (0.5941,0.5445)
+	-- (0.5891,0.5531)
+	-- (0.5833,0.5613)
+	-- (0.5767,0.5688)
+	-- (0.5695,0.5757)
+	-- (0.5616,0.5818)
+	-- (0.5531,0.5872)
+	-- (0.5442,0.5917)
+	-- (0.5349,0.5952)
+	-- (0.5253,0.5979)
+	-- (0.5154,0.5996)
+	-- (0.5054,0.6003)
+	-- (0.4954,0.6001)
+	-- (0.4855,0.5988)
+	-- (0.4758,0.5966)
+	-- (0.4663,0.5933)
+	-- (0.4572,0.5892)
+	-- (0.4486,0.5842)
+	-- (0.4405,0.5783)
+	-- (0.4331,0.5716)
+	-- (0.4263,0.5642)
+	-- (0.4204,0.5562)
+	-- (0.4153,0.5476)
+	-- (0.4111,0.5385)
+	-- (0.4079,0.5290)
+	-- (0.4057,0.5193)
+	-- (0.4045,0.5094)
+	-- (0.4043,0.4994)
+	-- (0.4052,0.4894)
+	-- (0.4071,0.4796)
+	-- (0.4100,0.4700)
+	-- (0.4139,0.4608)
+	-- (0.4188,0.4521)
+	-- (0.4246,0.4439)
+	-- (0.4311,0.4364)
+	-- (0.4385,0.4296)
+	-- (0.4465,0.4237)
+	-- (0.4551,0.4186)
+	-- (0.4643,0.4145)
+	-- (0.4738,0.4114)
+	-- (0.4835,0.4094)
+	-- (0.4935,0.4084)
+	-- (0.5035,0.4085)
+	-- (0.5134,0.4097)
+	-- (0.5231,0.4119)
+	-- (0.5326,0.4152)
+	-- (0.5416,0.4196)
+	-- (0.5501,0.4249)
+	-- (0.5579,0.4311)
+	-- (0.5650,0.4381)
+	-- (0.5713,0.4459)
+	-- (0.5767,0.4543)
+	-- (0.5811,0.4633)
+	-- (0.5845,0.4727)
+	-- (0.5868,0.4824)
+	-- (0.5880,0.4923)
+	-- (0.5880,0.5023)
+	-- (0.5869,0.5122)
+	-- (0.5848,0.5220)
+	-- (0.5815,0.5314)
+	-- (0.5771,0.5404)
+	-- (0.5718,0.5489)
+	-- (0.5655,0.5567)
+	-- (0.5584,0.5637)
+	-- (0.5505,0.5698)
+	-- (0.5419,0.5750)
+	-- (0.5329,0.5791)
+	-- (0.5233,0.5822)
+	-- (0.5135,0.5841)
+	-- (0.5036,0.5849)
+	-- (0.4936,0.5845)
+	-- (0.4837,0.5830)
+	-- (0.4741,0.5803)
+	-- (0.4649,0.5764)
+	-- (0.4562,0.5715)
+	-- (0.4481,0.5656)
+	-- (0.4408,0.5588)
+	-- (0.4343,0.5512)
+	-- (0.4289,0.5428)
+	-- (0.4244,0.5338)
+	-- (0.4211,0.5244)
+	-- (0.4189,0.5147)
+	-- (0.4180,0.5047)
+	-- (0.4182,0.4947)
+	-- (0.4197,0.4848)
+	-- (0.4223,0.4752)
+	-- (0.4261,0.4660)
+	-- (0.4311,0.4573)
+	-- (0.4370,0.4492)
+	-- (0.4439,0.4420)
+	-- (0.4516,0.4357)
+	-- (0.4601,0.4303)
+	-- (0.4691,0.4261)
+	-- (0.4786,0.4230)
+	-- (0.4885,0.4211)
+	-- (0.4984,0.4205)
+	-- (0.5084,0.4211)
+	-- (0.5182,0.4230)
+	-- (0.5277,0.4261)
+	-- (0.5368,0.4304)
+	-- (0.5452,0.4358)
+	-- (0.5528,0.4422)
+	-- (0.5596,0.4495)
+	-- (0.5654,0.4576)
+	-- (0.5701,0.4665)
+	-- (0.5737,0.4758)
+	-- (0.5760,0.4855)
+	-- (0.5771,0.4955)
+	-- (0.5768,0.5054)
+	-- (0.5753,0.5153)
+	-- (0.5725,0.5249)
+	-- (0.5684,0.5341)
+	-- (0.5633,0.5426)
+	-- (0.5570,0.5504)
+	-- (0.5498,0.5573)
+	-- (0.5417,0.5632)
+	-- (0.5329,0.5680)
+	-- (0.5236,0.5716)
+	-- (0.5139,0.5739)
+	-- (0.5040,0.5749)
+	-- (0.4940,0.5746)
+	-- (0.4841,0.5730)
+	-- (0.4746,0.5700)
+	-- (0.4655,0.5658)
+	-- (0.4571,0.5604)
+	-- (0.4494,0.5540)
+	-- (0.4428,0.5466)
+	-- (0.4371,0.5383)
+	-- (0.4327,0.5294)
+	-- (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\fresnelb{ (0,0)
+	-- (-0.0100,-0.0000)
+	-- (-0.0200,-0.0000)
+	-- (-0.0300,-0.0000)
+	-- (-0.0400,-0.0000)
+	-- (-0.0500,-0.0001)
+	-- (-0.0600,-0.0001)
+	-- (-0.0700,-0.0002)
+	-- (-0.0800,-0.0003)
+	-- (-0.0900,-0.0004)
+	-- (-0.1000,-0.0005)
+	-- (-0.1100,-0.0007)
+	-- (-0.1200,-0.0009)
+	-- (-0.1300,-0.0012)
+	-- (-0.1400,-0.0014)
+	-- (-0.1500,-0.0018)
+	-- (-0.1600,-0.0021)
+	-- (-0.1700,-0.0026)
+	-- (-0.1800,-0.0031)
+	-- (-0.1899,-0.0036)
+	-- (-0.1999,-0.0042)
+	-- (-0.2099,-0.0048)
+	-- (-0.2199,-0.0056)
+	-- (-0.2298,-0.0064)
+	-- (-0.2398,-0.0072)
+	-- (-0.2498,-0.0082)
+	-- (-0.2597,-0.0092)
+	-- (-0.2696,-0.0103)
+	-- (-0.2796,-0.0115)
+	-- (-0.2895,-0.0128)
+	-- (-0.2994,-0.0141)
+	-- (-0.3093,-0.0156)
+	-- (-0.3192,-0.0171)
+	-- (-0.3290,-0.0188)
+	-- (-0.3389,-0.0205)
+	-- (-0.3487,-0.0224)
+	-- (-0.3585,-0.0244)
+	-- (-0.3683,-0.0264)
+	-- (-0.3780,-0.0286)
+	-- (-0.3878,-0.0309)
+	-- (-0.3975,-0.0334)
+	-- (-0.4072,-0.0359)
+	-- (-0.4168,-0.0386)
+	-- (-0.4264,-0.0414)
+	-- (-0.4359,-0.0443)
+	-- (-0.4455,-0.0474)
+	-- (-0.4549,-0.0506)
+	-- (-0.4644,-0.0539)
+	-- (-0.4738,-0.0574)
+	-- (-0.4831,-0.0610)
+	-- (-0.4923,-0.0647)
+	-- (-0.5016,-0.0686)
+	-- (-0.5107,-0.0727)
+	-- (-0.5198,-0.0769)
+	-- (-0.5288,-0.0812)
+	-- (-0.5377,-0.0857)
+	-- (-0.5466,-0.0904)
+	-- (-0.5553,-0.0952)
+	-- (-0.5640,-0.1001)
+	-- (-0.5726,-0.1053)
+	-- (-0.5811,-0.1105)
+	-- (-0.5895,-0.1160)
+	-- (-0.5978,-0.1216)
+	-- (-0.6059,-0.1273)
+	-- (-0.6140,-0.1333)
+	-- (-0.6219,-0.1393)
+	-- (-0.6298,-0.1456)
+	-- (-0.6374,-0.1520)
+	-- (-0.6450,-0.1585)
+	-- (-0.6524,-0.1653)
+	-- (-0.6597,-0.1721)
+	-- (-0.6668,-0.1792)
+	-- (-0.6737,-0.1864)
+	-- (-0.6805,-0.1937)
+	-- (-0.6871,-0.2012)
+	-- (-0.6935,-0.2089)
+	-- (-0.6998,-0.2167)
+	-- (-0.7058,-0.2246)
+	-- (-0.7117,-0.2327)
+	-- (-0.7174,-0.2410)
+	-- (-0.7228,-0.2493)
+	-- (-0.7281,-0.2579)
+	-- (-0.7331,-0.2665)
+	-- (-0.7379,-0.2753)
+	-- (-0.7425,-0.2841)
+	-- (-0.7469,-0.2932)
+	-- (-0.7510,-0.3023)
+	-- (-0.7548,-0.3115)
+	-- (-0.7584,-0.3208)
+	-- (-0.7617,-0.3303)
+	-- (-0.7648,-0.3398)
+	-- (-0.7676,-0.3494)
+	-- (-0.7702,-0.3590)
+	-- (-0.7724,-0.3688)
+	-- (-0.7744,-0.3786)
+	-- (-0.7760,-0.3885)
+	-- (-0.7774,-0.3984)
+	-- (-0.7785,-0.4083)
+	-- (-0.7793,-0.4183)
+	-- (-0.7797,-0.4283)
+	-- (-0.7799,-0.4383)
+	-- (-0.7797,-0.4483)
+	-- (-0.7793,-0.4582)
+	-- (-0.7785,-0.4682)
+	-- (-0.7774,-0.4782)
+	-- (-0.7759,-0.4880)
+	-- (-0.7741,-0.4979)
+	-- (-0.7721,-0.5077)
+	-- (-0.7696,-0.5174)
+	-- (-0.7669,-0.5270)
+	-- (-0.7638,-0.5365)
+	-- (-0.7604,-0.5459)
+	-- (-0.7567,-0.5552)
+	-- (-0.7526,-0.5643)
+	-- (-0.7482,-0.5733)
+	-- (-0.7436,-0.5821)
+	-- (-0.7385,-0.5908)
+	-- (-0.7332,-0.5993)
+	-- (-0.7276,-0.6075)
+	-- (-0.7217,-0.6156)
+	-- (-0.7154,-0.6234)
+	-- (-0.7089,-0.6310)
+	-- (-0.7021,-0.6383)
+	-- (-0.6950,-0.6454)
+	-- (-0.6877,-0.6522)
+	-- (-0.6801,-0.6587)
+	-- (-0.6722,-0.6648)
+	-- (-0.6641,-0.6707)
+	-- (-0.6558,-0.6763)
+	-- (-0.6473,-0.6815)
+	-- (-0.6386,-0.6863)
+	-- (-0.6296,-0.6908)
+	-- (-0.6205,-0.6950)
+	-- (-0.6112,-0.6987)
+	-- (-0.6018,-0.7021)
+	-- (-0.5923,-0.7050)
+	-- (-0.5826,-0.7076)
+	-- (-0.5728,-0.7097)
+	-- (-0.5630,-0.7114)
+	-- (-0.5531,-0.7127)
+	-- (-0.5431,-0.7135)
+	-- (-0.5331,-0.7139)
+	-- (-0.5231,-0.7139)
+	-- (-0.5131,-0.7134)
+	-- (-0.5032,-0.7125)
+	-- (-0.4933,-0.7111)
+	-- (-0.4834,-0.7093)
+	-- (-0.4737,-0.7070)
+	-- (-0.4641,-0.7043)
+	-- (-0.4546,-0.7011)
+	-- (-0.4453,-0.6975)
+	-- (-0.4361,-0.6935)
+	-- (-0.4272,-0.6890)
+	-- (-0.4185,-0.6841)
+	-- (-0.4100,-0.6788)
+	-- (-0.4018,-0.6731)
+	-- (-0.3939,-0.6670)
+	-- (-0.3862,-0.6605)
+	-- (-0.3790,-0.6536)
+	-- (-0.3720,-0.6464)
+	-- (-0.3655,-0.6389)
+	-- (-0.3593,-0.6310)
+	-- (-0.3535,-0.6229)
+	-- (-0.3482,-0.6144)
+	-- (-0.3433,-0.6057)
+	-- (-0.3388,-0.5968)
+	-- (-0.3348,-0.5876)
+	-- (-0.3313,-0.5782)
+	-- (-0.3283,-0.5687)
+	-- (-0.3258,-0.5590)
+	-- (-0.3238,-0.5492)
+	-- (-0.3224,-0.5393)
+	-- (-0.3214,-0.5293)
+	-- (-0.3211,-0.5194)
+	-- (-0.3212,-0.5094)
+	-- (-0.3219,-0.4994)
+	-- (-0.3232,-0.4895)
+	-- (-0.3250,-0.4796)
+	-- (-0.3273,-0.4699)
+	-- (-0.3302,-0.4603)
+	-- (-0.3336,-0.4509)
+	-- (-0.3376,-0.4418)
+	-- (-0.3420,-0.4328)
+	-- (-0.3470,-0.4241)
+	-- (-0.3524,-0.4157)
+	-- (-0.3584,-0.4077)
+	-- (-0.3648,-0.4000)
+	-- (-0.3716,-0.3927)
+	-- (-0.3788,-0.3858)
+	-- (-0.3865,-0.3793)
+	-- (-0.3945,-0.3733)
+	-- (-0.4028,-0.3678)
+	-- (-0.4115,-0.3629)
+	-- (-0.4204,-0.3584)
+	-- (-0.4296,-0.3545)
+	-- (-0.4391,-0.3511)
+	-- (-0.4487,-0.3484)
+	-- (-0.4584,-0.3462)
+	-- (-0.4683,-0.3447)
+	-- (-0.4783,-0.3437)
+	-- (-0.4883,-0.3434)
+	-- (-0.4982,-0.3437)
+	-- (-0.5082,-0.3447)
+	-- (-0.5181,-0.3462)
+	-- (-0.5278,-0.3484)
+	-- (-0.5374,-0.3513)
+	-- (-0.5468,-0.3547)
+	-- (-0.5560,-0.3587)
+	-- (-0.5648,-0.3633)
+	-- (-0.5734,-0.3685)
+	-- (-0.5816,-0.3743)
+	-- (-0.5894,-0.3805)
+	-- (-0.5967,-0.3873)
+	-- (-0.6036,-0.3945)
+	-- (-0.6100,-0.4022)
+	-- (-0.6159,-0.4103)
+	-- (-0.6212,-0.4188)
+	-- (-0.6259,-0.4276)
+	-- (-0.6300,-0.4367)
+	-- (-0.6335,-0.4461)
+	-- (-0.6363,-0.4557)
+	-- (-0.6384,-0.4655)
+	-- (-0.6399,-0.4754)
+	-- (-0.6407,-0.4853)
+	-- (-0.6408,-0.4953)
+	-- (-0.6401,-0.5053)
+	-- (-0.6388,-0.5152)
+	-- (-0.6368,-0.5250)
+	-- (-0.6340,-0.5346)
+	-- (-0.6306,-0.5440)
+	-- (-0.6266,-0.5532)
+	-- (-0.6218,-0.5620)
+	-- (-0.6165,-0.5704)
+	-- (-0.6105,-0.5784)
+	-- (-0.6040,-0.5860)
+	-- (-0.5970,-0.5931)
+	-- (-0.5894,-0.5996)
+	-- (-0.5814,-0.6056)
+	-- (-0.5729,-0.6110)
+	-- (-0.5641,-0.6157)
+	-- (-0.5550,-0.6197)
+	-- (-0.5455,-0.6230)
+	-- (-0.5359,-0.6256)
+	-- (-0.5261,-0.6275)
+	-- (-0.5161,-0.6286)
+	-- (-0.5061,-0.6289)
+	-- (-0.4961,-0.6285)
+	-- (-0.4862,-0.6273)
+	-- (-0.4764,-0.6254)
+	-- (-0.4668,-0.6226)
+	-- (-0.4574,-0.6192)
+	-- (-0.4483,-0.6150)
+	-- (-0.4396,-0.6101)
+	-- (-0.4313,-0.6045)
+	-- (-0.4235,-0.5983)
+	-- (-0.4161,-0.5915)
+	-- (-0.4094,-0.5842)
+	-- (-0.4033,-0.5763)
+	-- (-0.3978,-0.5679)
+	-- (-0.3930,-0.5591)
+	-- (-0.3889,-0.5500)
+	-- (-0.3856,-0.5406)
+	-- (-0.3831,-0.5309)
+	-- (-0.3814,-0.5210)
+	-- (-0.3805,-0.5111)
+	-- (-0.3805,-0.5011)
+	-- (-0.3812,-0.4911)
+	-- (-0.3828,-0.4812)
+	-- (-0.3853,-0.4715)
+	-- (-0.3885,-0.4621)
+	-- (-0.3925,-0.4529)
+	-- (-0.3973,-0.4441)
+	-- (-0.4028,-0.4358)
+	-- (-0.4090,-0.4279)
+	-- (-0.4158,-0.4207)
+	-- (-0.4233,-0.4140)
+	-- (-0.4313,-0.4080)
+	-- (-0.4397,-0.4027)
+	-- (-0.4487,-0.3982)
+	-- (-0.4579,-0.3944)
+	-- (-0.4675,-0.3915)
+	-- (-0.4773,-0.3895)
+	-- (-0.4872,-0.3883)
+	-- (-0.4972,-0.3880)
+	-- (-0.5072,-0.3886)
+	-- (-0.5171,-0.3900)
+	-- (-0.5268,-0.3924)
+	-- (-0.5362,-0.3956)
+	-- (-0.5454,-0.3996)
+	-- (-0.5541,-0.4045)
+	-- (-0.5624,-0.4101)
+	-- (-0.5701,-0.4165)
+	-- (-0.5772,-0.4235)
+	-- (-0.5836,-0.4312)
+	-- (-0.5893,-0.4394)
+	-- (-0.5942,-0.4481)
+	-- (-0.5983,-0.4572)
+	-- (-0.6015,-0.4667)
+	-- (-0.6038,-0.4764)
+	-- (-0.6053,-0.4863)
+	-- (-0.6057,-0.4963)
+	-- (-0.6052,-0.5063)
+	-- (-0.6038,-0.5162)
+	-- (-0.6015,-0.5259)
+	-- (-0.5982,-0.5354)
+	-- (-0.5941,-0.5445)
+	-- (-0.5891,-0.5531)
+	-- (-0.5833,-0.5613)
+	-- (-0.5767,-0.5688)
+	-- (-0.5695,-0.5757)
+	-- (-0.5616,-0.5818)
+	-- (-0.5531,-0.5872)
+	-- (-0.5442,-0.5917)
+	-- (-0.5349,-0.5952)
+	-- (-0.5253,-0.5979)
+	-- (-0.5154,-0.5996)
+	-- (-0.5054,-0.6003)
+	-- (-0.4954,-0.6001)
+	-- (-0.4855,-0.5988)
+	-- (-0.4758,-0.5966)
+	-- (-0.4663,-0.5933)
+	-- (-0.4572,-0.5892)
+	-- (-0.4486,-0.5842)
+	-- (-0.4405,-0.5783)
+	-- (-0.4331,-0.5716)
+	-- (-0.4263,-0.5642)
+	-- (-0.4204,-0.5562)
+	-- (-0.4153,-0.5476)
+	-- (-0.4111,-0.5385)
+	-- (-0.4079,-0.5290)
+	-- (-0.4057,-0.5193)
+	-- (-0.4045,-0.5094)
+	-- (-0.4043,-0.4994)
+	-- (-0.4052,-0.4894)
+	-- (-0.4071,-0.4796)
+	-- (-0.4100,-0.4700)
+	-- (-0.4139,-0.4608)
+	-- (-0.4188,-0.4521)
+	-- (-0.4246,-0.4439)
+	-- (-0.4311,-0.4364)
+	-- (-0.4385,-0.4296)
+	-- (-0.4465,-0.4237)
+	-- (-0.4551,-0.4186)
+	-- (-0.4643,-0.4145)
+	-- (-0.4738,-0.4114)
+	-- (-0.4835,-0.4094)
+	-- (-0.4935,-0.4084)
+	-- (-0.5035,-0.4085)
+	-- (-0.5134,-0.4097)
+	-- (-0.5231,-0.4119)
+	-- (-0.5326,-0.4152)
+	-- (-0.5416,-0.4196)
+	-- (-0.5501,-0.4249)
+	-- (-0.5579,-0.4311)
+	-- (-0.5650,-0.4381)
+	-- (-0.5713,-0.4459)
+	-- (-0.5767,-0.4543)
+	-- (-0.5811,-0.4633)
+	-- (-0.5845,-0.4727)
+	-- (-0.5868,-0.4824)
+	-- (-0.5880,-0.4923)
+	-- (-0.5880,-0.5023)
+	-- (-0.5869,-0.5122)
+	-- (-0.5848,-0.5220)
+	-- (-0.5815,-0.5314)
+	-- (-0.5771,-0.5404)
+	-- (-0.5718,-0.5489)
+	-- (-0.5655,-0.5567)
+	-- (-0.5584,-0.5637)
+	-- (-0.5505,-0.5698)
+	-- (-0.5419,-0.5750)
+	-- (-0.5329,-0.5791)
+	-- (-0.5233,-0.5822)
+	-- (-0.5135,-0.5841)
+	-- (-0.5036,-0.5849)
+	-- (-0.4936,-0.5845)
+	-- (-0.4837,-0.5830)
+	-- (-0.4741,-0.5803)
+	-- (-0.4649,-0.5764)
+	-- (-0.4562,-0.5715)
+	-- (-0.4481,-0.5656)
+	-- (-0.4408,-0.5588)
+	-- (-0.4343,-0.5512)
+	-- (-0.4289,-0.5428)
+	-- (-0.4244,-0.5338)
+	-- (-0.4211,-0.5244)
+	-- (-0.4189,-0.5147)
+	-- (-0.4180,-0.5047)
+	-- (-0.4182,-0.4947)
+	-- (-0.4197,-0.4848)
+	-- (-0.4223,-0.4752)
+	-- (-0.4261,-0.4660)
+	-- (-0.4311,-0.4573)
+	-- (-0.4370,-0.4492)
+	-- (-0.4439,-0.4420)
+	-- (-0.4516,-0.4357)
+	-- (-0.4601,-0.4303)
+	-- (-0.4691,-0.4261)
+	-- (-0.4786,-0.4230)
+	-- (-0.4885,-0.4211)
+	-- (-0.4984,-0.4205)
+	-- (-0.5084,-0.4211)
+	-- (-0.5182,-0.4230)
+	-- (-0.5277,-0.4261)
+	-- (-0.5368,-0.4304)
+	-- (-0.5452,-0.4358)
+	-- (-0.5528,-0.4422)
+	-- (-0.5596,-0.4495)
+	-- (-0.5654,-0.4576)
+	-- (-0.5701,-0.4665)
+	-- (-0.5737,-0.4758)
+	-- (-0.5760,-0.4855)
+	-- (-0.5771,-0.4955)
+	-- (-0.5768,-0.5054)
+	-- (-0.5753,-0.5153)
+	-- (-0.5725,-0.5249)
+	-- (-0.5684,-0.5341)
+	-- (-0.5633,-0.5426)
+	-- (-0.5570,-0.5504)
+	-- (-0.5498,-0.5573)
+	-- (-0.5417,-0.5632)
+	-- (-0.5329,-0.5680)
+	-- (-0.5236,-0.5716)
+	-- (-0.5139,-0.5739)
+	-- (-0.5040,-0.5749)
+	-- (-0.4940,-0.5746)
+	-- (-0.4841,-0.5730)
+	-- (-0.4746,-0.5700)
+	-- (-0.4655,-0.5658)
+	-- (-0.4571,-0.5604)
+	-- (-0.4494,-0.5540)
+	-- (-0.4428,-0.5466)
+	-- (-0.4371,-0.5383)
+	-- (-0.4327,-0.5294)
+	-- (-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