Merge branch 'master' of https://github.com/AndreasFMueller/SeminarSpezielleFunktionen

16 files changed, 591 insertions, 0 deletions

diff --git a/buch/papers/fresnel/images/Makefile b/buch/papers/fresnel/images/Makefile
new file mode 100644
index 0000000..eb7dc57
--- /dev/null
+++ b/buch/papers/fresnel/images/Makefile
@@ -0,0 +1,38 @@
+#
+# Makefile
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	schale.pdf							\
+	fresnelgraph.pdf						\
+	eulerspirale.pdf						\
+	pfad.pdf							\
+	apfel.pdf							\
+	kruemmung.pdf
+
+schale.png:	schale.pov
+	povray +A0.1 -W1920 -H1080 -Oschale.png schale.pov
+
+schale.jpg:	schale.png Makefile
+	convert -extract 1240x1080+340 schale.png -density 300 -units PixelsPerInch schale.jpg
+
+schale.pdf:	schale.tex schale.jpg
+	pdflatex schale.tex
+
+eulerpath.tex:	eulerspirale.m
+	octave eulerspirale.m
+
+fresnelgraph.pdf:	fresnelgraph.tex eulerpath.tex
+	pdflatex fresnelgraph.tex
+
+eulerspirale.pdf:	eulerspirale.tex eulerpath.tex
+	pdflatex eulerspirale.tex
+
+pfad.pdf:	pfad.tex
+	pdflatex pfad.tex
+
+apfel.pdf:	apfel.tex apfel.jpg eulerpath.tex
+	pdflatex apfel.tex
+
+kruemmung.pdf:	kruemmung.tex
+	pdflatex kruemmung.tex
diff --git a/buch/papers/fresnel/images/apfel.jpg b/buch/papers/fresnel/images/apfel.jpg
new file mode 100644
index 0000000..76e48e7
--- /dev/null
+++ b/buch/papers/fresnel/images/apfel.jpg
diff --git a/buch/papers/fresnel/images/apfel.pdf b/buch/papers/fresnel/images/apfel.pdf
new file mode 100644
index 0000000..69e5092
--- /dev/null
+++ b/buch/papers/fresnel/images/apfel.pdf
diff --git a/buch/papers/fresnel/images/apfel.tex b/buch/papers/fresnel/images/apfel.tex
new file mode 100644
index 0000000..754886b
--- /dev/null
+++ b/buch/papers/fresnel/images/apfel.tex
@@ -0,0 +1,49 @@
+%
+% apfel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{7}
+\def\hoehe{4}
+
+\input{eulerpath.tex}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\begin{scope}
+\clip(-0.6,-0.6) rectangle (7,6);
+\node at (3.1,2.2) [rotate=-3] {\includegraphics[width=9.4cm]{apfel.jpg}};
+\end{scope}
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\draw[color=gray!50] (0,0) rectangle (4,4);
+\draw[->] (-0.5,0) -- (7.5,0) coordinate[label={$C(t)$}];
+\draw[->] (0,-0.5) -- (0,6.0) coordinate[label={left:$S(t)$}];
+\begin{scope}[scale=8]
+\draw[color=red,opacity=0.5,line width=1.4pt] \fresnela;
+\end{scope}
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/fresnel/images/eulerspirale.m b/buch/papers/fresnel/images/eulerspirale.m
new file mode 100644
index 0000000..84e3696
--- /dev/null
+++ b/buch/papers/fresnel/images/eulerspirale.m
@@ -0,0 +1,61 @@
+#
+# eulerspirale.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue
+#
+global n;
+n = 1000;
+global tmax;
+tmax = 10;
+global N;
+N = round(n*5/tmax);
+
+function retval = f(x, t)
+	x = pi * t^2 / 2;
+	retval = [ cos(x); sin(x) ];
+endfunction
+
+x0 = [ 0; 0 ];
+t = tmax * (0:n) / n;
+
+c = lsode(@f, x0, t);
+
+fn = fopen("eulerpath.tex", "w");
+
+fprintf(fn, "\\def\\fresnela{ (0,0)");
+for i = (2:n)
+	fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2));
+end
+fprintf(fn, "\n}\n\n");
+
+fprintf(fn, "\\def\\fresnelb{ (0,0)");
+for i = (2:n)
+	fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2));
+end
+fprintf(fn, "\n}\n\n");
+
+fprintf(fn, "\\def\\Cplotright{ (0,0)");
+for i = (2:N)
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,1));
+end
+fprintf(fn, "\n}\n\n");
+
+fprintf(fn, "\\def\\Cplotleft{ (0,0)");
+for i = (2:N)
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,1));
+end
+fprintf(fn, "\n}\n\n");
+
+fprintf(fn, "\\def\\Splotright{ (0,0)");
+for i = (2:N)
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,2));
+end
+fprintf(fn, "\n}\n\n");
+
+fprintf(fn, "\\def\\Splotleft{ (0,0)");
+for i = (2:N)
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,2));
+end
+fprintf(fn, "\n}\n\n");
+
+fclose(fn);
diff --git a/buch/papers/fresnel/images/eulerspirale.pdf b/buch/papers/fresnel/images/eulerspirale.pdf
new file mode 100644
index 0000000..db74e4b
--- /dev/null
+++ b/buch/papers/fresnel/images/eulerspirale.pdf
diff --git a/buch/papers/fresnel/images/eulerspirale.tex b/buch/papers/fresnel/images/eulerspirale.tex
new file mode 100644
index 0000000..38ef756
--- /dev/null
+++ b/buch/papers/fresnel/images/eulerspirale.tex
@@ -0,0 +1,41 @@
+%
+% eulerspirale.tex -- Darstellung der Eulerspirale
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{eulerpath.tex}
+
+\def\s{8}
+
+\begin{scope}[scale=\s]
+\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5);
+\draw[color=darkgreen,line width=1.4pt] \fresnela;
+\draw[color=darkgreen,line width=1.4pt] \fresnelb;
+\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}];
+\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}];
+\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s});
+\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s});
+\node at (-0.5,0) [above left] {$\frac12$};
+\node at (0.5,0) [below right] {$\frac12$};
+\node at (0,-0.5) [below right] {$\frac12$};
+\node at (0,0.5) [above left] {$\frac12$};
+\end{scope}
+
+\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];;
+\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];;
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/fresnel/images/fresnelgraph.pdf b/buch/papers/fresnel/images/fresnelgraph.pdf
new file mode 100644
index 0000000..c658901
--- /dev/null
+++ b/buch/papers/fresnel/images/fresnelgraph.pdf
diff --git a/buch/papers/fresnel/images/fresnelgraph.tex b/buch/papers/fresnel/images/fresnelgraph.tex
new file mode 100644
index 0000000..20df951
--- /dev/null
+++ b/buch/papers/fresnel/images/fresnelgraph.tex
@@ -0,0 +1,46 @@
+%
+% fresnelgraph.tex -- Graphs of the fresnel functions
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{eulerpath.tex}
+\def\dx{1.3}
+\def\dy{2.6}
+
+\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy});
+\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy});
+
+\draw[color=blue,line width=1.4pt] \Splotright;
+\draw[color=blue,line width=1.4pt] \Splotleft;
+
+\draw[color=red,line width=1.4pt] \Cplotright;
+\draw[color=red,line width=1.4pt] \Cplotleft;
+
+\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}];
+\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}];
+
+\foreach \x in {1,2,3,4,5}{
+	\draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05);
+	\draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05);
+	\node at ({\x*\dx},-0.05) [below] {$\x$};
+	\node at ({-\x*\dx},0.05) [above] {$-\x$};
+}
+\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy});
+\node at (-0.05,{0.5*\dy}) [left] {$\frac12$};
+\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy});
+\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/fresnel/images/kruemmung.pdf b/buch/papers/fresnel/images/kruemmung.pdf
new file mode 100644
index 0000000..1180116
--- /dev/null
+++ b/buch/papers/fresnel/images/kruemmung.pdf
diff --git a/buch/papers/fresnel/images/kruemmung.tex b/buch/papers/fresnel/images/kruemmung.tex
new file mode 100644
index 0000000..af0a1a9
--- /dev/null
+++ b/buch/papers/fresnel/images/kruemmung.tex
@@ -0,0 +1,51 @@
+%
+% kruemmung.tex -- Krümmung einer ebenen Kurve
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}
+\clip (-1,-1) rectangle (4,4);
+
+\def\r{3}
+\def\winkel{30}
+
+\fill[color=blue!20] (0,0) -- (0:{0.6*\r}) arc (0:\winkel:{0.6*\r}) -- cycle;
+\fill[color=blue!20] (\winkel:\r)
+	-- ($(\winkel:\r)+(0,{0.6*\r})$) arc (90:{90+\winkel}:{0.6*\r}) -- cycle;
+\node[color=blue] at ({0.5*\winkel}:{0.45*\r}) {$\Delta\varphi$};
+
+\node[color=blue] at ($(\winkel:\r)+({90+0.5*\winkel}:{0.45*\r})$)
+	{$\Delta\varphi$};
+
+\draw[line width=0.3pt] (0,0) circle[radius=\r];
+
+\draw[->] (0,0) -- (0:\r);
+\draw[->] (0,0) -- (\winkel:\r);
+
+\draw[->] (0:\r) -- ($(0:\r)+(90:0.7*\r)$);
+\draw[->] (\winkel:\r) -- ($(\winkel:\r)+({90+\winkel}:0.7*\r)$);
+\draw[->,color=gray] (\winkel:\r) -- ($(\winkel:\r)+(0,0.7*\r)$);
+
+\draw[color=red,line width=1.4pt] (0:\r) arc (0:\winkel:\r);
+\node[color=red] at ({0.5*\winkel}:\r) [left] {$\Delta s$};
+\fill[color=red] (0:\r) circle[radius=0.05];
+\fill[color=red] (\winkel:\r) circle[radius=0.05];
+
+\node at (\winkel:{0.5*\r}) [above] {$r$};
+\node at (0:{0.5*\r}) [below] {$r$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/fresnel/images/pfad.pdf b/buch/papers/fresnel/images/pfad.pdf
new file mode 100644
index 0000000..df3c7af
--- /dev/null
+++ b/buch/papers/fresnel/images/pfad.pdf
diff --git a/buch/papers/fresnel/images/pfad.tex b/buch/papers/fresnel/images/pfad.tex
new file mode 100644
index 0000000..680cd78
--- /dev/null
+++ b/buch/papers/fresnel/images/pfad.tex
@@ -0,0 +1,37 @@
+%
+% pfad.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\fill[color=gray!40] (0,0) -- (2,0) arc (0:45:2) -- cycle;
+\node at (22.5:1.4) {$\displaystyle\frac{\pi}4$};
+
+\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}];
+\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}];
+
+\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0);
+\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7);
+\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0);
+
+\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$};
+\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$};
+\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$};
+
+\node at (7,0) [below] {$R$};
+\node at (45:7) [above] {$Re^{i\pi/4}$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/fresnel/images/schale.pdf b/buch/papers/fresnel/images/schale.pdf
new file mode 100644
index 0000000..9c21951
--- /dev/null
+++ b/buch/papers/fresnel/images/schale.pdf
diff --git a/buch/papers/fresnel/images/schale.pov b/buch/papers/fresnel/images/schale.pov
new file mode 100644
index 0000000..085a6a4
--- /dev/null
+++ b/buch/papers/fresnel/images/schale.pov
@@ -0,0 +1,191 @@
+//
+// schale.pov -- 
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+#declare O = <0,0,0>;
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.036;
+
+camera {
+        location <40, 20, -20>
+        look_at <0, 0.5, 0>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <10, 10, -40> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+sphere {
+	<0, 0, 0>, 1
+	pigment {
+		color rgb<0.8,0.8,0.8>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+#declare stripcolor = rgb<0.2,0.2,0.8>;
+
+#declare R = 1.002;
+
+#macro punkt(phi,theta)
+R * < cos(phi) * cos(theta), sin(theta), sin(phi) * cos(theta) > 
+#end
+
+#declare N = 24;
+#declare thetaphi = 0.01;
+#declare thetawidth = pi * 0.008;
+#declare theta = function(phi) { phi * thetaphi }
+
+#declare axisdiameter = 0.007;
+
+cylinder {
+	< 0, -2, 0>, < 0, 2, 0>, axisdiameter
+	pigment {
+		color White
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+#declare curvaturecircle = 0.008;
+#declare curvaturecirclecolor = rgb<0.4,0.8,0.4>;
+
+#declare phit = 12.8 * 2 * pi;
+#declare P = punkt(phit, theta(phit));
+#declare Q = <0, R / sin(theta(phit)), 0>;
+
+#declare e1 = vnormalize(P - Q) / tan(theta(phit));
+#declare e2 = vnormalize(vcross(e1, <0,1,0>)) / tan(theta(phit));
+#declare psimin = -0.1 * pi;
+#declare psimax = 0.1 * pi;
+#declare psistep = (psimax - psimin) / 30;
+
+union {
+	#declare psi = psimin;
+	#declare K = Q + cos(psi) * e1 + sin(psi) * e2;
+	#while (psi < psimax - psistep/2)
+		sphere { K, curvaturecircle }
+		#declare psi = psi + psistep;
+		#declare K2 = Q + cos(psi) * e1 + sin(psi) * e2;
+		cylinder { K, K2, curvaturecircle }
+		#declare K = K2;
+	#end
+	sphere { K, curvaturecircle }
+	pigment {
+		color curvaturecirclecolor
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	mesh {
+		#declare psi = psimin;
+		#declare K = Q + cos(psi) * e1 + sin(psi) * e2;
+		#while (psi < psimax - psistep/2)
+			#declare psi = psi + psistep;
+			#declare K2 = Q + cos(psi) * e1 + sin(psi) * e2;
+			triangle { K, K2, Q }
+			#declare K = K2;
+		#end
+	}
+	pigment {
+		color rgbt<0.4,0.8,0.4,0.5>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	sphere { P, 0.02 }
+	sphere { Q, 0.02 }
+	cylinder { P, Q, 0.01 }
+	pigment {
+		color Red
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+#declare phisteps = 300;
+#declare phistep = 2 * pi / phisteps;
+#declare phimin = 0;
+#declare phimax = N * 2 * pi;
+
+object {
+	mesh {
+		#declare phi = phimin;
+		#declare Poben = punkt(phi, theta(phi) + thetawidth);
+		#declare Punten = punkt(phi, theta(phi) - thetawidth);
+		triangle { O, Punten, Poben }
+		#while (phi < phimax - phistep/2)
+			#declare phi = phi + phistep;
+			#declare Poben2 = punkt(phi, theta(phi) + thetawidth);
+			#declare Punten2 = punkt(phi, theta(phi) - thetawidth);
+			triangle { O, Punten, Punten2 }
+			triangle { O, Poben, Poben2 }
+			triangle { Punten, Punten2, Poben }
+			triangle { Punten2, Poben2, Poben }
+			#declare Poben = Poben2;
+			#declare Punten = Punten2;
+		#end
+		triangle { O, Punten, Poben }
+	}
+	pigment {
+		color stripcolor
+	}
+	finish {
+		specular 0.8
+		metallic
+	}
+}
+
+union {
+	#declare phi = phimin;
+	#declare P = punkt(phi, theta(phi));
+	#while (phi < phimax - phistep/2)
+		sphere { P, 0.003 }
+		#declare phi = phi + phistep;
+		#declare P2 = punkt(phi, theta(phi));
+		cylinder { P, P2, 0.003 }
+		#declare P = P2;
+	#end
+	sphere { P, 0.003 }
+	pigment {
+		color stripcolor
+	}
+	finish {
+		specular 0.8
+		metallic
+	}
+}
diff --git a/buch/papers/fresnel/images/schale.tex b/buch/papers/fresnel/images/schale.tex
new file mode 100644
index 0000000..577ede4
--- /dev/null
+++ b/buch/papers/fresnel/images/schale.tex
@@ -0,0 +1,77 @@
+%
+% schlange.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math,calc}
+\usepackage{ifthen}
+\begin{document}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+\def\a{47}
+\def\r{3.3}
+\def\skala{0.95}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}[xshift=-7.4cm,yshift=-1.2cm]
+	\clip (-3.6,-2.2) rectangle (3.6,5.1);
+
+	\fill[color=blue!20] (0,0)
+		-- ({180-\a}:{0.4*\r}) arc ({180-\a}:180:{0.4*\r})
+		-- cycle;
+	\node[color=blue] at ({180-\a/2}:{0.3*\r}) {$\vartheta$};
+
+	\fill[color=blue!20] (0,{\r/sin(\a)})
+		-- ($(0,{\r/sin(\a)})+({270-\a}:{0.3*\r})$)
+			arc ({270-\a}:270:{0.3*\r})
+		-- cycle;
+	\node[color=blue] at ($(0,{\r/sin(\a)})+({270-\a/2}:{0.2*\r})$)
+		{$\vartheta$};
+
+
+	\draw (0,0) circle[radius=\r];
+	\draw[->] (0,-3.0) -- (0,5);
+	\draw ({-\r-0.2},0) -- ({\r+0.2},0);
+	\fill (0,0) circle[radius=0.06];
+
+	\draw (0,0) -- ({180-\a}:\r);
+	\node at ({180-\a+3}:{0.65*\r}) [above right] {$1$};
+
+	\draw[color=red,line width=1.4pt]
+		({180-\a}:\r) -- (0,{\r/cos(90-\a)});
+	\fill[color=red] ({180-\a}:\r) circle[radius=0.08];
+	\fill[color=red] (0,{\r/cos(90-\a)}) circle[radius=0.08];
+	\node[color=red] at (-1.0,3.7) [left] {$r=\cot\vartheta$};
+	\node[color=red] at ({180-\a}:\r) [above left] {$P$};
+	\node[color=red] at (0,{\r/sin(\a)}) [right] {$Q$};
+\end{scope}
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=7.6cm]{schale.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\node[color=red] at (-1.4,1.4) {$r$};
+\node[color=red] at (-2.2,-0.2) {$P$};
+\node[color=red] at (0,3.3) [right] {$Q$};
+
+\end{tikzpicture}
+
+\end{document}
+