update to current state of book

7 files changed, 911 insertions, 911 deletions

diff --git a/buch/chapters/60-gruppen/images/Makefile b/buch/chapters/60-gruppen/images/Makefile
index 3ed39e5..8cd824f 100644
--- a/buch/chapters/60-gruppen/images/Makefile
+++ b/buch/chapters/60-gruppen/images/Makefile
@@ -1,25 +1,25 @@
-#
-# Makefile
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all:	phasenraum.pdf kartenkreis.pdf karten.pdf sl2.pdf scherungen.pdf
-
-phasenraum.pdf:	phasenraum.tex
-	pdflatex phasenraum.tex
-
-kartenkreis.pdf:	kartenkreis.tex
-	pdflatex kartenkreis.tex
-
-torus.png:	torus.pov
-	povray +A0.1 -W1920 -H1080 -Otorus.png torus.pov
-
-karten.pdf:	karten.tex torus.png
-	pdflatex karten.tex
-
-sl2.pdf:	sl2.tex 
-	pdflatex sl2.tex
-
-scherungen.pdf:	scherungen.tex 
-	pdflatex scherungen.tex
-
+#

+# Makefile

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+#

+all:	phasenraum.pdf kartenkreis.pdf karten.pdf sl2.pdf scherungen.pdf

+

+phasenraum.pdf:	phasenraum.tex

+	pdflatex phasenraum.tex

+

+kartenkreis.pdf:	kartenkreis.tex

+	pdflatex kartenkreis.tex

+

+torus.png:	torus.pov

+	povray +A0.1 -W1920 -H1080 -Otorus.png torus.pov

+

+karten.pdf:	karten.tex torus.png

+	pdflatex karten.tex

+

+sl2.pdf:	sl2.tex 

+	pdflatex sl2.tex

+

+scherungen.pdf:	scherungen.tex 

+	pdflatex scherungen.tex

+

diff --git a/buch/chapters/60-gruppen/images/karten.tex b/buch/chapters/60-gruppen/images/karten.tex
index c8eb4a3..67c8d70 100644
--- a/buch/chapters/60-gruppen/images/karten.tex
+++ b/buch/chapters/60-gruppen/images/karten.tex
@@ -1,112 +1,112 @@
-%
-% karten.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}
-\begin{tikzpicture}[>=latex,thick,scale=\skala]
-
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-
-\node at (0,0) {\includegraphics[width=10cm]{torus.png}};
-
-\def\s{3}
-
-\node at (-3.5,-0.4) {$U_\alpha$};
-\node at (2.0,-0.4) {$U_\beta$};
-
-\draw[->] (-2,-2.2) -- (-3,-4.3);
-\node at (-2.5,-3.25) [left] {$\varphi_\alpha$};
-
-\draw[->] (1.4,-1.7) -- (3,-4.3);
-\node at (2.5,-3.25) [right] {$\varphi_\beta$};
-
-\begin{scope}[xshift=-4.5cm,yshift=-8cm]
-	\begin{scope}
-		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
-		\begin{scope}[xshift=1.8cm,yshift=0.6cm,rotate=30]
-			\fill[color=gray!20]
-				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
-			\foreach \x in {0,0.2,...,1}{
-				\draw[color=darkgreen]
-					({\x*\s},{-0.2*\s})
-					--
-					({\x*\s},{1.2*\s});
-			}
-			\foreach \y in {-0.2,0,...,1.2}{
-				\draw[color=orange]
-					(0,{\y*\s})
-					--
-					({1*\s},{\y*\s});
-			}
-		\end{scope}
-	\end{scope}
-
-	\foreach \x in {0,0.2,...,1}{
-		\draw[color=blue,line width=1.4pt]
-			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});
-	}
-	\foreach \y in {-0.2,0,...,1.2}{
-		\draw[color=red,line width=1.4pt]
-			(0,{\y*\s}) -- ({1*\s},{\y*\s});
-	}
-
-	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];
-	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];
-
-	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};
-
-\end{scope}
-
-\begin{scope}[xshift=1.5cm,yshift=-8cm]
-	\begin{scope}
-		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
-		% x = - [ (sqrt(3)/2)*0.6+(1/2)*0.2 ] = -0.6196
-		% y = - [ (-1/2)*0.6 + (sqrt(3)/2)*0.2 ] = 
-		\begin{scope}[xshift=-1.8588cm,yshift=0.3804cm,rotate=-30]
-			\fill[color=gray!20]
-				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
-				\foreach \x in {0,0.2,...,1}{
-					\draw[color=blue]
-						({\x*\s},{-0.2*\s})
-						--
-						({\x*\s},{1.2*\s});
-				}
-				\foreach \y in {-0.2,0,...,1.2}{
-					\draw[color=red]
-						(0,{\y*\s})
-						--
-						({1*\s},{\y*\s});
-				}
-		\end{scope}
-	\end{scope}
-
-	\foreach \x in {0,0.2,...,1}{
-		\draw[color=darkgreen,line width=1.4pt]
-			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});
-	}
-	\foreach \y in {-0.2,0,...,1.2}{
-		\draw[color=orange,line width=1.4pt] (0,{\y*\s}) -- ({1*\s},{\y*\s});
-	}
-	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];
-	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];
-	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};
-\end{scope}
-
-\draw[<-,color=white,opacity=0.8,line width=5pt] (2.5,-6.5) arc (55:100:6.5);
-\draw[<-,shorten >= 0.1cm,shorten <= 0.3cm] (2.5,-6.5) arc (55:100:6.5);
-
-\node at (0,-5.9)
-	{$\varphi_{\beta\alpha}=\varphi_\beta\circ\varphi_\alpha^{-1}$};
-
-\end{tikzpicture}
-\end{document}
-
+%

+% karten.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}

+\begin{tikzpicture}[>=latex,thick,scale=\skala]

+

+\definecolor{darkgreen}{rgb}{0,0.6,0}

+

+\node at (0,0) {\includegraphics[width=10cm]{torus.png}};

+

+\def\s{3}

+

+\node at (-3.5,-0.4) {$U_\alpha$};

+\node at (2.0,-0.4) {$U_\beta$};

+

+\draw[->] (-2,-2.2) -- (-3,-4.3);

+\node at (-2.5,-3.25) [left] {$\varphi_\alpha$};

+

+\draw[->] (1.4,-1.7) -- (3,-4.3);

+\node at (2.5,-3.25) [right] {$\varphi_\beta$};

+

+\begin{scope}[xshift=-4.5cm,yshift=-8cm]

+	\begin{scope}

+		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});

+		\begin{scope}[xshift=1.8cm,yshift=0.6cm,rotate=30]

+			\fill[color=gray!20]

+				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});

+			\foreach \x in {0,0.2,...,1}{

+				\draw[color=darkgreen]

+					({\x*\s},{-0.2*\s})

+					--

+					({\x*\s},{1.2*\s});

+			}

+			\foreach \y in {-0.2,0,...,1.2}{

+				\draw[color=orange]

+					(0,{\y*\s})

+					--

+					({1*\s},{\y*\s});

+			}

+		\end{scope}

+	\end{scope}

+

+	\foreach \x in {0,0.2,...,1}{

+		\draw[color=blue,line width=1.4pt]

+			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});

+	}

+	\foreach \y in {-0.2,0,...,1.2}{

+		\draw[color=red,line width=1.4pt]

+			(0,{\y*\s}) -- ({1*\s},{\y*\s});

+	}

+

+	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];

+	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];

+

+	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};

+

+\end{scope}

+

+\begin{scope}[xshift=1.5cm,yshift=-8cm]

+	\begin{scope}

+		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});

+		% x = - [ (sqrt(3)/2)*0.6+(1/2)*0.2 ] = -0.6196

+		% y = - [ (-1/2)*0.6 + (sqrt(3)/2)*0.2 ] = 

+		\begin{scope}[xshift=-1.8588cm,yshift=0.3804cm,rotate=-30]

+			\fill[color=gray!20]

+				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});

+				\foreach \x in {0,0.2,...,1}{

+					\draw[color=blue]

+						({\x*\s},{-0.2*\s})

+						--

+						({\x*\s},{1.2*\s});

+				}

+				\foreach \y in {-0.2,0,...,1.2}{

+					\draw[color=red]

+						(0,{\y*\s})

+						--

+						({1*\s},{\y*\s});

+				}

+		\end{scope}

+	\end{scope}

+

+	\foreach \x in {0,0.2,...,1}{

+		\draw[color=darkgreen,line width=1.4pt]

+			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});

+	}

+	\foreach \y in {-0.2,0,...,1.2}{

+		\draw[color=orange,line width=1.4pt] (0,{\y*\s}) -- ({1*\s},{\y*\s});

+	}

+	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];

+	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];

+	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};

+\end{scope}

+

+\draw[<-,color=white,opacity=0.8,line width=5pt] (2.5,-6.5) arc (55:100:6.5);

+\draw[<-,shorten >= 0.1cm,shorten <= 0.3cm] (2.5,-6.5) arc (55:100:6.5);

+

+\node at (0,-5.9)

+	{$\varphi_{\beta\alpha}=\varphi_\beta\circ\varphi_\alpha^{-1}$};

+

+\end{tikzpicture}

+\end{document}

+

diff --git a/buch/chapters/60-gruppen/images/kartenkreis.tex b/buch/chapters/60-gruppen/images/kartenkreis.tex
index 4f19937..ff6331e 100644
--- a/buch/chapters/60-gruppen/images/kartenkreis.tex
+++ b/buch/chapters/60-gruppen/images/kartenkreis.tex
@@ -1,189 +1,189 @@
-%
-% kartenkreis.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{3}
-\begin{tikzpicture}[>=latex,thick,scale=\skala]
-
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-
-\fill[color=red!20] (0,-1) rectangle (1.5,1);
-\fill[color=blue!20] (-1.5,-1) rectangle (0,1);
-\fill[color=darkgreen!40,opacity=0.5] (-1,0) rectangle (1,1.5);
-\fill[color=orange!40,opacity=0.5] (-1,-1.5) rectangle (1,0);
-\fill[color=white] (0,0) circle[radius=1];
-
-\fill[color=gray!20]
-	(0,-1.5) -- (0.02,-1.6) -- (0.5,-1.8) -- (0.98,-1.6) -- (1,-1.5)
-	-- cycle;
-\fill[color=gray!20]
-	(0,1.5) -- (0.02,1.6) -- (0.5,1.8) -- (0.98,1.6) -- (1,1.5)
-	-- cycle;
-\fill[color=gray!20]
-	(0,-1.5) -- (-0.02,-1.6) -- (-0.5,-1.8) -- (-0.98,-1.6) -- (-1,-1.5)
-	-- cycle;
-\fill[color=gray!20]
-	(0,1.5) -- (-0.02,1.6) -- (-0.5,1.8) -- (-0.98,1.6) -- (-1,1.5)
-	-- cycle;
-
-\fill[color=gray!20]
-	(1.5,0) -- (1.6,0.02) -- (1.8,0.5) -- (1.6,0.98) -- (1.5,1)
-	-- cycle;
-\fill[color=gray!20]
-	(-1.5,0) -- (-1.6,0.02) -- (-1.8,0.5) -- (-1.6,0.98) -- (-1.5,1)
-	-- cycle;
-\fill[color=gray!20]
-	(1.5,0) -- (1.6,-0.02) -- (1.8,-0.5) -- (1.6,-0.98) -- (1.5,-1)
-	-- cycle;
-\fill[color=gray!20]
-	(-1.5,0) -- (-1.6,-0.02) -- (-1.8,-0.5) -- (-1.6,-0.98) -- (-1.5,-1)
-	-- cycle;
-
-\draw[->] (0.5,-1.8) arc (-180:-90:0.1) arc (-90:0:1.3) arc (0:90:0.1);
-\draw[->] (1.8,0.5) arc (-90:0:0.1) arc (0:90:1.3) arc (90:180:0.1);
-\draw[->] (-0.5,1.8) arc (0:90:0.1) arc (90:180:1.3) arc (180:270:0.1);
-\draw[->] (-1.8,-0.5) arc (90:180:0.1) arc (180:270:1.3) arc (270:360:0.1);
-
-\node at (1.01,1.32)
-	[right] {$\varphi_3\circ \varphi_1^{-1}(y)=\sqrt{1-y^2}$};
-\node at (1.6,1.6) {$\varphi_{31}$};
-
-\node at (1.01,-1.28)
-	[right] {$\varphi_1\circ \varphi_4^{-1}(x)=-\sqrt{1-x^2}$};
-\node at (1.6,-1.6) {$\varphi_{14}$};
-
-\node at (-1.24,1.32)
-	[left] {$\varphi_2\circ\varphi_3^{-1}(x)=\sqrt{1-x^2}$};
-\node at (-1.6,1.6) {$\varphi_{23}$};
-
-\node at (-1.18,-1.28)
-	[left] {$\varphi_4\circ\varphi_2^{-1}(y)=-\sqrt{1-y^2}$};
-\node at (-1.6,-1.6) {$\varphi_{42}$};
-
-
-\foreach \y in {0.1,0.3,...,0.9}{
-	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]
-		({sqrt(1-\y*\y)},{\y}) -- (1.5,\y);
-	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]
-		({sqrt(1-\y*\y)},{-\y}) -- (1.5,-\y);
-	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]
-		({-sqrt(1-\y*\y)},{\y}) -- (-1.5,\y);
-	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]
-		({-sqrt(1-\y*\y)},{-\y}) -- (-1.5,-\y);
-}
-\foreach \x in {0.1,0.3,...,0.9}{
-	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]
-		({\x},{sqrt(1-\x*\x)}) -- ({\x},1.5);
-	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]
-		({-\x},{sqrt(1-\x*\x)}) -- ({-\x},1.5);
-	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]
-		({\x},{-sqrt(1-\x*\x)}) -- ({\x},-1.5);
-	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]
-		({-\x},{-sqrt(1-\x*\x)}) -- ({-\x},-1.5);
-}
-
-%\draw[color=gray!50,line width=3pt] (0,0) circle[radius=1];
-\draw[color=yellow!30,line width=3pt] (0,0) circle[radius=1];
-\node[color=yellow] at ({1/sqrt(2)},{1/sqrt(2)}) [above right] {$S^1$};
-
-\def\r{1.02}
-
-\begin{scope}
-	\clip (0,-1.1) rectangle (1.1,1.1);
-	\draw[color=red,line width=1.4pt] (-89:\r) arc (-89:89:\r);
-	\draw[color=red,line width=1.4pt] (0,-\r) circle[radius=0.02];
-	\draw[color=red,line width=1.4pt] (0,\r) circle[radius=0.02];
-\end{scope}
-
-\begin{scope}
-	\clip (-1.1,-1.1) rectangle (0,1.1);
-	\draw[color=blue,line width=1.4pt] (91:\r) arc (91:269:\r);
-	\draw[color=blue,line width=1.4pt] (0,-\r) circle[radius=0.02];
-	\draw[color=blue,line width=1.4pt] (0,\r) circle[radius=0.02];
-\end{scope}
-
-\xdef\r{0.98}
-
-\begin{scope} 
-	\clip (-1.1,0) rectangle (1.1,1.1);
-	\draw[color=darkgreen,line width=1.4pt] (1:\r) arc (1:179:\r);
-	\draw[color=darkgreen,line width=1.4pt] (\r,0) circle[radius=0.02];
-	\draw[color=darkgreen,line width=1.4pt] (-\r,0) circle[radius=0.02];
-\end{scope}
-
-\begin{scope} 
-	\clip (-1.1,-1.1) rectangle (1.1,0);
-	\draw[color=orange,line width=1.4pt] (181:\r) arc (181:359:\r);
-	\draw[color=orange,line width=1.4pt] (\r,0) circle[radius=0.02];
-	\draw[color=orange,line width=1.4pt] (-\r,0) circle[radius=0.02];
-\end{scope}
-
-\begin{scope}[yshift=1.5cm]
-	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$\mathbb{R}$}];
-	\begin{scope}
-		\clip (-1,-0.1) rectangle (1,0.1);
-		\draw[color=darkgreen,line width=1.4pt] (-0.98,0) -- (0.98,0);
-		\draw[color=darkgreen,line width=1.4pt] (-1,0)
-			circle[radius=0.02];
-		\draw[color=darkgreen,line width=1.4pt] (1,0)
-			circle[radius=0.02];
-	\end{scope}
-\end{scope}
-
-\begin{scope}[yshift=-1.5cm]
-	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={below:$\mathbb{R}$}];
-	\begin{scope}
-		\clip (-1,-0.1) rectangle (1,0.1);
-		\draw[color=orange,line width=1.4pt] (-0.98,0) -- (0.98,0);
-		\draw[color=orange,line width=1.4pt] (-1,0) circle[radius=0.02];
-		\draw[color=orange,line width=1.4pt] (1,0) circle[radius=0.02];
-	\end{scope}
-\end{scope}
-
-\begin{scope}[xshift=1.5cm]
-	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$\mathbb{R}$}];
-	\begin{scope}
-		\clip (-0.1,-1) rectangle (0.1,1);
-		\draw[color=red,line width=1.4pt] (0,-0.98) -- (0,0.98);
-		\draw[color=red,line width=1.4pt] (0,-1) circle[radius=0.02];
-		\draw[color=red,line width=1.4pt] (0,1) circle[radius=0.02];
-	\end{scope}
-\end{scope}
-
-\begin{scope}[xshift=-1.5cm]
-	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={left:$\mathbb{R}$}];
-	\begin{scope}
-		\clip (-0.1,-1) rectangle (0.1,1);
-		\draw[color=blue,line width=1.4pt] (0,-0.98) -- (0,0.98);
-		\draw[color=blue,line width=1.4pt] (0,-1) circle[radius=0.02];
-		\draw[color=blue,line width=1.4pt] (0,1) circle[radius=0.02];
-	\end{scope}
-\end{scope}
-
-\node[color=red] at (23:1) [right] {$U_{x>0}$};
-\node[color=red] at (1.25,0) [right] {$\varphi_1$};
-
-\node[color=blue] at (157:1) [left] {$U_{x<0}$};
-\node[color=blue] at (-1.25,0) [left] {$\varphi_2$};
-
-\node[color=darkgreen] at (115:1) [below right] {$U_{y>0}$};
-\node[color=darkgreen] at (0,1.25) [above] {$\varphi_3$};
-
-\node[color=orange] at (-115:1) [above right] {$U_{y<0}$};
-\node[color=orange] at (0,-1.25) [below] {$\varphi_4$};
-
-\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$x$}];
-\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$y$}];
-
-\end{tikzpicture}
-\end{document}
-
+%

+% kartenkreis.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{3}

+\begin{tikzpicture}[>=latex,thick,scale=\skala]

+

+\definecolor{darkgreen}{rgb}{0,0.6,0}

+

+\fill[color=red!20] (0,-1) rectangle (1.5,1);

+\fill[color=blue!20] (-1.5,-1) rectangle (0,1);

+\fill[color=darkgreen!40,opacity=0.5] (-1,0) rectangle (1,1.5);

+\fill[color=orange!40,opacity=0.5] (-1,-1.5) rectangle (1,0);

+\fill[color=white] (0,0) circle[radius=1];

+

+\fill[color=gray!20]

+	(0,-1.5) -- (0.02,-1.6) -- (0.5,-1.8) -- (0.98,-1.6) -- (1,-1.5)

+	-- cycle;

+\fill[color=gray!20]

+	(0,1.5) -- (0.02,1.6) -- (0.5,1.8) -- (0.98,1.6) -- (1,1.5)

+	-- cycle;

+\fill[color=gray!20]

+	(0,-1.5) -- (-0.02,-1.6) -- (-0.5,-1.8) -- (-0.98,-1.6) -- (-1,-1.5)

+	-- cycle;

+\fill[color=gray!20]

+	(0,1.5) -- (-0.02,1.6) -- (-0.5,1.8) -- (-0.98,1.6) -- (-1,1.5)

+	-- cycle;

+

+\fill[color=gray!20]

+	(1.5,0) -- (1.6,0.02) -- (1.8,0.5) -- (1.6,0.98) -- (1.5,1)

+	-- cycle;

+\fill[color=gray!20]

+	(-1.5,0) -- (-1.6,0.02) -- (-1.8,0.5) -- (-1.6,0.98) -- (-1.5,1)

+	-- cycle;

+\fill[color=gray!20]

+	(1.5,0) -- (1.6,-0.02) -- (1.8,-0.5) -- (1.6,-0.98) -- (1.5,-1)

+	-- cycle;

+\fill[color=gray!20]

+	(-1.5,0) -- (-1.6,-0.02) -- (-1.8,-0.5) -- (-1.6,-0.98) -- (-1.5,-1)

+	-- cycle;

+

+\draw[->] (0.5,-1.8) arc (-180:-90:0.1) arc (-90:0:1.3) arc (0:90:0.1);

+\draw[->] (1.8,0.5) arc (-90:0:0.1) arc (0:90:1.3) arc (90:180:0.1);

+\draw[->] (-0.5,1.8) arc (0:90:0.1) arc (90:180:1.3) arc (180:270:0.1);

+\draw[->] (-1.8,-0.5) arc (90:180:0.1) arc (180:270:1.3) arc (270:360:0.1);

+

+\node at (1.01,1.32)

+	[right] {$\varphi_3\circ \varphi_1^{-1}(y)=\sqrt{1-y^2}$};

+\node at (1.6,1.6) {$\varphi_{31}$};

+

+\node at (1.01,-1.28)

+	[right] {$\varphi_1\circ \varphi_4^{-1}(x)=-\sqrt{1-x^2}$};

+\node at (1.6,-1.6) {$\varphi_{14}$};

+

+\node at (-1.24,1.32)

+	[left] {$\varphi_2\circ\varphi_3^{-1}(x)=\sqrt{1-x^2}$};

+\node at (-1.6,1.6) {$\varphi_{23}$};

+

+\node at (-1.18,-1.28)

+	[left] {$\varphi_4\circ\varphi_2^{-1}(y)=-\sqrt{1-y^2}$};

+\node at (-1.6,-1.6) {$\varphi_{42}$};

+

+

+\foreach \y in {0.1,0.3,...,0.9}{

+	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]

+		({sqrt(1-\y*\y)},{\y}) -- (1.5,\y);

+	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]

+		({sqrt(1-\y*\y)},{-\y}) -- (1.5,-\y);

+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]

+		({-sqrt(1-\y*\y)},{\y}) -- (-1.5,\y);

+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]

+		({-sqrt(1-\y*\y)},{-\y}) -- (-1.5,-\y);

+}

+\foreach \x in {0.1,0.3,...,0.9}{

+	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]

+		({\x},{sqrt(1-\x*\x)}) -- ({\x},1.5);

+	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]

+		({-\x},{sqrt(1-\x*\x)}) -- ({-\x},1.5);

+	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]

+		({\x},{-sqrt(1-\x*\x)}) -- ({\x},-1.5);

+	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]

+		({-\x},{-sqrt(1-\x*\x)}) -- ({-\x},-1.5);

+}

+

+%\draw[color=gray!50,line width=3pt] (0,0) circle[radius=1];

+\draw[color=yellow!30,line width=3pt] (0,0) circle[radius=1];

+\node[color=yellow] at ({1/sqrt(2)},{1/sqrt(2)}) [above right] {$S^1$};

+

+\def\r{1.02}

+

+\begin{scope}

+	\clip (0,-1.1) rectangle (1.1,1.1);

+	\draw[color=red,line width=1.4pt] (-89:\r) arc (-89:89:\r);

+	\draw[color=red,line width=1.4pt] (0,-\r) circle[radius=0.02];

+	\draw[color=red,line width=1.4pt] (0,\r) circle[radius=0.02];

+\end{scope}

+

+\begin{scope}

+	\clip (-1.1,-1.1) rectangle (0,1.1);

+	\draw[color=blue,line width=1.4pt] (91:\r) arc (91:269:\r);

+	\draw[color=blue,line width=1.4pt] (0,-\r) circle[radius=0.02];

+	\draw[color=blue,line width=1.4pt] (0,\r) circle[radius=0.02];

+\end{scope}

+

+\xdef\r{0.98}

+

+\begin{scope} 

+	\clip (-1.1,0) rectangle (1.1,1.1);

+	\draw[color=darkgreen,line width=1.4pt] (1:\r) arc (1:179:\r);

+	\draw[color=darkgreen,line width=1.4pt] (\r,0) circle[radius=0.02];

+	\draw[color=darkgreen,line width=1.4pt] (-\r,0) circle[radius=0.02];

+\end{scope}

+

+\begin{scope} 

+	\clip (-1.1,-1.1) rectangle (1.1,0);

+	\draw[color=orange,line width=1.4pt] (181:\r) arc (181:359:\r);

+	\draw[color=orange,line width=1.4pt] (\r,0) circle[radius=0.02];

+	\draw[color=orange,line width=1.4pt] (-\r,0) circle[radius=0.02];

+\end{scope}

+

+\begin{scope}[yshift=1.5cm]

+	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$\mathbb{R}$}];

+	\begin{scope}

+		\clip (-1,-0.1) rectangle (1,0.1);

+		\draw[color=darkgreen,line width=1.4pt] (-0.98,0) -- (0.98,0);

+		\draw[color=darkgreen,line width=1.4pt] (-1,0)

+			circle[radius=0.02];

+		\draw[color=darkgreen,line width=1.4pt] (1,0)

+			circle[radius=0.02];

+	\end{scope}

+\end{scope}

+

+\begin{scope}[yshift=-1.5cm]

+	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={below:$\mathbb{R}$}];

+	\begin{scope}

+		\clip (-1,-0.1) rectangle (1,0.1);

+		\draw[color=orange,line width=1.4pt] (-0.98,0) -- (0.98,0);

+		\draw[color=orange,line width=1.4pt] (-1,0) circle[radius=0.02];

+		\draw[color=orange,line width=1.4pt] (1,0) circle[radius=0.02];

+	\end{scope}

+\end{scope}

+

+\begin{scope}[xshift=1.5cm]

+	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$\mathbb{R}$}];

+	\begin{scope}

+		\clip (-0.1,-1) rectangle (0.1,1);

+		\draw[color=red,line width=1.4pt] (0,-0.98) -- (0,0.98);

+		\draw[color=red,line width=1.4pt] (0,-1) circle[radius=0.02];

+		\draw[color=red,line width=1.4pt] (0,1) circle[radius=0.02];

+	\end{scope}

+\end{scope}

+

+\begin{scope}[xshift=-1.5cm]

+	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={left:$\mathbb{R}$}];

+	\begin{scope}

+		\clip (-0.1,-1) rectangle (0.1,1);

+		\draw[color=blue,line width=1.4pt] (0,-0.98) -- (0,0.98);

+		\draw[color=blue,line width=1.4pt] (0,-1) circle[radius=0.02];

+		\draw[color=blue,line width=1.4pt] (0,1) circle[radius=0.02];

+	\end{scope}

+\end{scope}

+

+\node[color=red] at (23:1) [right] {$U_{x>0}$};

+\node[color=red] at (1.25,0) [right] {$\varphi_1$};

+

+\node[color=blue] at (157:1) [left] {$U_{x<0}$};

+\node[color=blue] at (-1.25,0) [left] {$\varphi_2$};

+

+\node[color=darkgreen] at (115:1) [below right] {$U_{y>0}$};

+\node[color=darkgreen] at (0,1.25) [above] {$\varphi_3$};

+

+\node[color=orange] at (-115:1) [above right] {$U_{y<0}$};

+\node[color=orange] at (0,-1.25) [below] {$\varphi_4$};

+

+\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$x$}];

+\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$y$}];

+

+\end{tikzpicture}

+\end{document}

+

diff --git a/buch/chapters/60-gruppen/images/phasenraum.tex b/buch/chapters/60-gruppen/images/phasenraum.tex
index 2bccc27..2305b26 100644
--- a/buch/chapters/60-gruppen/images/phasenraum.tex
+++ b/buch/chapters/60-gruppen/images/phasenraum.tex
@@ -1,93 +1,93 @@
-%
-% phasenraum.tex -- 
-%
-% (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]
-
-\def\m{1}
-\def\K{0.444}
-
-\pgfmathparse{sqrt(\K/\m)}
-\xdef\o{\pgfmathresult}
-
-\def\punkt#1#2{ ({#2*cos(#1)},{\o*#2*sin(#1)}) }
-
-\foreach \r in {0.5,1,...,6}{
-	\draw plot[domain=0:359,samples=360]
-			({\r*cos(\x)},{\o*\r*sin(\x)}) -- cycle;
-}
-
-\def\tangente#1#2{
-		\pgfmathparse{#2/\m}
-		\xdef\u{\pgfmathresult}
-
-		\pgfmathparse{-#1*\K}
-		\xdef\v{\pgfmathresult}
-
-		\pgfmathparse{sqrt(\u*\u+\v*\v)}
-		\xdef\l{\pgfmathresult}
-
-		\fill[color=blue] (#1,#2) circle[radius=0.03];
-		\draw[color=blue,line width=0.5pt]
-			({#1-0.2*\u/\l},{#2-0.2*\v/\l})
-			--
-			({#1+0.2*\u/\l},{#2+0.2*\v/\l});
-}
-
-\foreach \x in {-6.25,-5.75,...,6.3}{
-	\foreach \y in {-4.25,-3.75,...,4.3}{
-		\tangente{\x}{\y}
-	}
-}
-
-%\foreach \x in {0.5,1,...,5.5,6}{
-%	\tangente{\x}{0}
-%	\tangente{-\x}{0}
-%	\foreach \y in {0.5,1,...,4}{
-%		\tangente{\x}{\y}
-%		\tangente{-\x}{\y}
-%		\tangente{\x}{-\y}
-%		\tangente{-\x}{-\y}
-%	}
-%}
-%\foreach \y in {0.5,1,...,4}{
-%	\tangente{0}{\y}
-%	\tangente{0}{-\y}
-%}
-
-\fill[color=white,opacity=0.7] \punkt{60}{4} rectangle \punkt{59}{5.8};
-\fill[color=white,opacity=0.7] \punkt{0}{4} rectangle \punkt{18}{4.9};
-
-\draw[->,color=red,line width=1.4pt]
-	plot[domain=0:60,samples=360]
-		({4*cos(\x)},{\o*4*sin(\x)});
-
-\draw[->] (-6.5,0) -- (6.7,0) coordinate[label={$x$}];
-\draw[->] (0,-4.5) -- (0,4.7) coordinate[label={right:$p$}];
-
-\fill[color=red] \punkt{60}{4} circle[radius=0.08];
-\node[color=red] at \punkt{60}{4} [above right]
-	{$\begin{pmatrix}x(t)\\p(t)\end{pmatrix}$};
-
-\fill[color=red] \punkt{0}{4} circle[radius=0.08];
-\node[color=red] at \punkt{0}{4} [above right]
-	{$\begin{pmatrix}x_0\\0\end{pmatrix}$};
-
-\fill[color=white] (4,0) circle[radius=0.05];
-\node at (3.9,0) [below right] {$x_0$};
-\fill (0,{\o*4}) circle[radius=0.05];
-\node at (0.1,{\o*4+0.05}) [below left] {$\omega x_0$};
-
-\end{tikzpicture}
-\end{document}
-
+%

+% phasenraum.tex -- 

+%

+% (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]

+

+\def\m{1}

+\def\K{0.444}

+

+\pgfmathparse{sqrt(\K/\m)}

+\xdef\o{\pgfmathresult}

+

+\def\punkt#1#2{ ({#2*cos(#1)},{\o*#2*sin(#1)}) }

+

+\foreach \r in {0.5,1,...,6}{

+	\draw plot[domain=0:359,samples=360]

+			({\r*cos(\x)},{\o*\r*sin(\x)}) -- cycle;

+}

+

+\def\tangente#1#2{

+		\pgfmathparse{#2/\m}

+		\xdef\u{\pgfmathresult}

+

+		\pgfmathparse{-#1*\K}

+		\xdef\v{\pgfmathresult}

+

+		\pgfmathparse{sqrt(\u*\u+\v*\v)}

+		\xdef\l{\pgfmathresult}

+

+		\fill[color=blue] (#1,#2) circle[radius=0.03];

+		\draw[color=blue,line width=0.5pt]

+			({#1-0.2*\u/\l},{#2-0.2*\v/\l})

+			--

+			({#1+0.2*\u/\l},{#2+0.2*\v/\l});

+}

+

+\foreach \x in {-6.25,-5.75,...,6.3}{

+	\foreach \y in {-4.25,-3.75,...,4.3}{

+		\tangente{\x}{\y}

+	}

+}

+

+%\foreach \x in {0.5,1,...,5.5,6}{

+%	\tangente{\x}{0}

+%	\tangente{-\x}{0}

+%	\foreach \y in {0.5,1,...,4}{

+%		\tangente{\x}{\y}

+%		\tangente{-\x}{\y}

+%		\tangente{\x}{-\y}

+%		\tangente{-\x}{-\y}

+%	}

+%}

+%\foreach \y in {0.5,1,...,4}{

+%	\tangente{0}{\y}

+%	\tangente{0}{-\y}

+%}

+

+\fill[color=white,opacity=0.7] \punkt{60}{4} rectangle \punkt{59}{5.8};

+\fill[color=white,opacity=0.7] \punkt{0}{4} rectangle \punkt{18}{4.9};

+

+\draw[->,color=red,line width=1.4pt]

+	plot[domain=0:60,samples=360]

+		({4*cos(\x)},{\o*4*sin(\x)});

+

+\draw[->] (-6.5,0) -- (6.7,0) coordinate[label={$x$}];

+\draw[->] (0,-4.5) -- (0,4.7) coordinate[label={right:$p$}];

+

+\fill[color=red] \punkt{60}{4} circle[radius=0.08];

+\node[color=red] at \punkt{60}{4} [above right]

+	{$\begin{pmatrix}x(t)\\p(t)\end{pmatrix}$};

+

+\fill[color=red] \punkt{0}{4} circle[radius=0.08];

+\node[color=red] at \punkt{0}{4} [above right]

+	{$\begin{pmatrix}x_0\\0\end{pmatrix}$};

+

+\fill[color=white] (4,0) circle[radius=0.05];

+\node at (3.9,0) [below right] {$x_0$};

+\fill (0,{\o*4}) circle[radius=0.05];

+\node at (0.1,{\o*4+0.05}) [below left] {$\omega x_0$};

+

+\end{tikzpicture}

+\end{document}

+

diff --git a/buch/chapters/60-gruppen/images/scherungen.tex b/buch/chapters/60-gruppen/images/scherungen.tex
index 893bd12..f6df172 100644
--- a/buch/chapters/60-gruppen/images/scherungen.tex
+++ b/buch/chapters/60-gruppen/images/scherungen.tex
@@ -1,157 +1,157 @@
-%
-% scherungen.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}
-\begin{tikzpicture}[>=latex,thick,scale=\skala]
-
-\definecolor{blau}{rgb}{0,0.8,1}
-\definecolor{blau}{rgb}{0,0.6,0}
-\def\s{1.1}
-
-\begin{scope}[xshift=-4.6cm]
-
-	\fill[color=blue!20] (0,0) rectangle (2,2);
-	\fill[color=red!40,opacity=0.5] (0,0) -- (2,\s) -- (2,{2+\s}) -- (0,2)
-		-- cycle;
-
-	\foreach \x in {-1,...,3}{
-		\draw[color=blau]  (\x,-1) -- (\x,3);
-		\draw[color=blau]  (-1,\x) -- (3,\x);
-	}
-
-	\begin{scope}
-		\clip (-1,-1) rectangle (3,3);
-		\foreach \x in {-1,...,3}{
-			\draw[color=orange] (\x,-1) -- (\x,3);
-			\draw[color=orange] (-1,{\x-0.5*\s}) -- (3,{\x+1.5*\s});
-		}
-	\end{scope}
-
-	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
-
-	\node[color=blue] at (0,2) [above left] {$1$};
-	\node[color=blue] at (2,0) [below right] {$1$};
-	\draw[->,color=blue] (0,0) -- (2,0);
-	\draw[->,color=blue] (0,0) -- (0,2);
-
-	\draw[->,color=red] (0,0) -- (2,\s);
-	\draw[->,color=red] (0,0) -- (0,2);
-
-	\node[color=red] at (2,\s) [below right] {$(1,t)$};
-
-	\node at (0,0) [below right] {$O$};
-	\node at (1,-1.1) [below] {$\displaystyle
-		\begin{aligned}
-		M &= \begin{pmatrix}0&0\\1&0 \end{pmatrix}
-		\\
-		e^{Mt}
-		&=
-		\begin{pmatrix}1&0\\t&1 \end{pmatrix}
-		\end{aligned}
-	$};
-\end{scope}
-
-\begin{scope}
-	\fill[color=blue!20] (0,0) rectangle (2,2);
-	\fill[color=red!40,opacity=0.5] (0,0) -- (2,0) -- ({2+\s},2) -- (\s,2)
-		-- cycle;
-
-	\foreach \x in {-1,...,3}{
-		\draw[color=blau]  (\x,-1) -- (\x,3);
-		\draw[color=blau]  (-1,\x) -- (3,\x);
-	}
-
-	\begin{scope}
-		\clip (-1,-1) rectangle (3,3);
-		\foreach \x in {-1,...,3}{
-			\draw[color=orange] (-1,\x) -- (3,\x);
-			\draw[color=orange] ({\x-0.5*\s},-1) -- ({\x+1.5*\s},3);
-		}
-	\end{scope}
-
-	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
-
-	\node[color=blue] at (0,2) [above left] {$1$};
-	\node[color=blue] at (2,0) [below right] {$1$};
-	\draw[->,color=blue] (0,0) -- (2,0);
-	\draw[->,color=blue] (0,0) -- (0,2);
-
-	\draw[->,color=red] (0,0) -- (2,0);
-	\draw[->,color=red] (0,0) -- (\s,2);
-
-	\node[color=red] at (\s,2) [above left] {$(t,1)$};
-
-	\node at (0,0) [below right] {$O$};
-	
-	\node at (1,-1.1) [below] {$\displaystyle
-		\begin{aligned} N &= \begin{pmatrix}0&1\\0&0 \end{pmatrix}
-		\\
-		e^{Nt}
-		&=
-		\begin{pmatrix}1&t\\0&1 \end{pmatrix}
-		\end{aligned}
-	$};
-\end{scope}
-
-\begin{scope}[xshift=3.6cm,yshift=0cm]
-	\def\punkt#1#2{({1.6005*(#1)+0.4114*(#2)},{-0.2057*(#1)+0.5719*(#2)})}
-	\fill[color=blue!20] (0,0) rectangle (2,2);
-	\fill[color=red!40,opacity=0.5]
-		(0,0) -- \punkt{2}{0} -- \punkt{2}{2} -- \punkt{0}{2} -- cycle;
-
-	\foreach \x in {0,...,4}{
-		\draw[color=blau]  (\x,-1) -- (\x,3);
-	}
-	\foreach \y in {-1,...,3}{
-		\draw[color=blau]  (0,\y) -- (4,\y);
-	}
-
-	\begin{scope}
-		\clip (-0,-1) rectangle (4,3);
-		\foreach \x in {-1,...,6}{
-			\draw[color=orange] \punkt{\x}{-3} -- \punkt{\x}{6};
-			\draw[color=orange] \punkt{-3}{\x} -- \punkt{6}{\x};
-		}
-	\end{scope}
-
-	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
-
-	\node[color=blue] at (0,2) [above left] {$1$};
-	\node[color=blue] at (2,0) [below right] {$1$};
-	\draw[->,color=blue] (0,0) -- (2,0);
-	\draw[->,color=blue] (0,0) -- (0,2);
-
-	\draw[->,color=red] (0,0) -- \punkt{2}{0};
-	\draw[->,color=red] (0,0) -- \punkt{0}{2};
-
-	\node at (0,0) [below right] {$O$};
-	
-	\node at (2,-1.1) [below] {$\displaystyle
-		\begin{aligned} D &= \begin{pmatrix}0.5&0.4\\-0.2&-0.5 \end{pmatrix}
-		\\
-		e^{D\cdot 1}
-		&=
-		\begin{pmatrix}
-			   1.6005 & 0.4114\\
-			  -0.2057 & 0.5719
-		\end{pmatrix}
-		\end{aligned}
-	$};
-\end{scope}
-
-\end{tikzpicture}
-\end{document}
-
+%

+% scherungen.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}

+\begin{tikzpicture}[>=latex,thick,scale=\skala]

+

+\definecolor{blau}{rgb}{0,0.8,1}

+\definecolor{blau}{rgb}{0,0.6,0}

+\def\s{1.1}

+

+\begin{scope}[xshift=-4.6cm]

+

+	\fill[color=blue!20] (0,0) rectangle (2,2);

+	\fill[color=red!40,opacity=0.5] (0,0) -- (2,\s) -- (2,{2+\s}) -- (0,2)

+		-- cycle;

+

+	\foreach \x in {-1,...,3}{

+		\draw[color=blau]  (\x,-1) -- (\x,3);

+		\draw[color=blau]  (-1,\x) -- (3,\x);

+	}

+

+	\begin{scope}

+		\clip (-1,-1) rectangle (3,3);

+		\foreach \x in {-1,...,3}{

+			\draw[color=orange] (\x,-1) -- (\x,3);

+			\draw[color=orange] (-1,{\x-0.5*\s}) -- (3,{\x+1.5*\s});

+		}

+	\end{scope}

+

+	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];

+

+	\node[color=blue] at (0,2) [above left] {$1$};

+	\node[color=blue] at (2,0) [below right] {$1$};

+	\draw[->,color=blue] (0,0) -- (2,0);

+	\draw[->,color=blue] (0,0) -- (0,2);

+

+	\draw[->,color=red] (0,0) -- (2,\s);

+	\draw[->,color=red] (0,0) -- (0,2);

+

+	\node[color=red] at (2,\s) [below right] {$(1,t)$};

+

+	\node at (0,0) [below right] {$O$};

+	\node at (1,-1.1) [below] {$\displaystyle

+		\begin{aligned}

+		M &= \begin{pmatrix}0&0\\1&0 \end{pmatrix}

+		\\

+		e^{Mt}

+		&=

+		\begin{pmatrix}1&0\\t&1 \end{pmatrix}

+		\end{aligned}

+	$};

+\end{scope}

+

+\begin{scope}

+	\fill[color=blue!20] (0,0) rectangle (2,2);

+	\fill[color=red!40,opacity=0.5] (0,0) -- (2,0) -- ({2+\s},2) -- (\s,2)

+		-- cycle;

+

+	\foreach \x in {-1,...,3}{

+		\draw[color=blau]  (\x,-1) -- (\x,3);

+		\draw[color=blau]  (-1,\x) -- (3,\x);

+	}

+

+	\begin{scope}

+		\clip (-1,-1) rectangle (3,3);

+		\foreach \x in {-1,...,3}{

+			\draw[color=orange] (-1,\x) -- (3,\x);

+			\draw[color=orange] ({\x-0.5*\s},-1) -- ({\x+1.5*\s},3);

+		}

+	\end{scope}

+

+	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];

+

+	\node[color=blue] at (0,2) [above left] {$1$};

+	\node[color=blue] at (2,0) [below right] {$1$};

+	\draw[->,color=blue] (0,0) -- (2,0);

+	\draw[->,color=blue] (0,0) -- (0,2);

+

+	\draw[->,color=red] (0,0) -- (2,0);

+	\draw[->,color=red] (0,0) -- (\s,2);

+

+	\node[color=red] at (\s,2) [above left] {$(t,1)$};

+

+	\node at (0,0) [below right] {$O$};

+	

+	\node at (1,-1.1) [below] {$\displaystyle

+		\begin{aligned} N &= \begin{pmatrix}0&1\\0&0 \end{pmatrix}

+		\\

+		e^{Nt}

+		&=

+		\begin{pmatrix}1&t\\0&1 \end{pmatrix}

+		\end{aligned}

+	$};

+\end{scope}

+

+\begin{scope}[xshift=3.6cm,yshift=0cm]

+	\def\punkt#1#2{({1.6005*(#1)+0.4114*(#2)},{-0.2057*(#1)+0.5719*(#2)})}

+	\fill[color=blue!20] (0,0) rectangle (2,2);

+	\fill[color=red!40,opacity=0.5]

+		(0,0) -- \punkt{2}{0} -- \punkt{2}{2} -- \punkt{0}{2} -- cycle;

+

+	\foreach \x in {0,...,4}{

+		\draw[color=blau]  (\x,-1) -- (\x,3);

+	}

+	\foreach \y in {-1,...,3}{

+		\draw[color=blau]  (0,\y) -- (4,\y);

+	}

+

+	\begin{scope}

+		\clip (-0,-1) rectangle (4,3);

+		\foreach \x in {-1,...,6}{

+			\draw[color=orange] \punkt{\x}{-3} -- \punkt{\x}{6};

+			\draw[color=orange] \punkt{-3}{\x} -- \punkt{6}{\x};

+		}

+	\end{scope}

+

+	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];

+

+	\node[color=blue] at (0,2) [above left] {$1$};

+	\node[color=blue] at (2,0) [below right] {$1$};

+	\draw[->,color=blue] (0,0) -- (2,0);

+	\draw[->,color=blue] (0,0) -- (0,2);

+

+	\draw[->,color=red] (0,0) -- \punkt{2}{0};

+	\draw[->,color=red] (0,0) -- \punkt{0}{2};

+

+	\node at (0,0) [below right] {$O$};

+	

+	\node at (2,-1.1) [below] {$\displaystyle

+		\begin{aligned} D &= \begin{pmatrix}0.5&0.4\\-0.2&-0.5 \end{pmatrix}

+		\\

+		e^{D\cdot 1}

+		&=

+		\begin{pmatrix}

+			   1.6005 & 0.4114\\

+			  -0.2057 & 0.5719

+		\end{pmatrix}

+		\end{aligned}

+	$};

+\end{scope}

+

+\end{tikzpicture}

+\end{document}

+

diff --git a/buch/chapters/60-gruppen/images/sl2.tex b/buch/chapters/60-gruppen/images/sl2.tex
index 0e44aa9..c41308c 100644
--- a/buch/chapters/60-gruppen/images/sl2.tex
+++ b/buch/chapters/60-gruppen/images/sl2.tex
@@ -1,146 +1,146 @@
-%
-% sl2.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}
-\begin{tikzpicture}[>=latex,thick,scale=\skala]
-
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-
-\begin{scope}[xshift=-4.5cm]
-	\fill[color=blue!20]
-		(1.4,0) -- (0,1.4) -- (-1.4,0) -- (0,-1.4) -- cycle;
-	\fill[color=red!40,opacity=0.5]
-		(1.96,0) -- (0,1) -- (-1.96,0) -- (0,-1) -- cycle;
-
-	\begin{scope}
-		\clip (-2.1,-2.1) rectangle (2.3,2.3);
-		\draw[color=darkgreen]
-			plot[domain=-1:1,samples=100]
-				({(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});
-		\draw[color=darkgreen]
-			plot[domain=-1:1,samples=100]
-				({(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});
-		\draw[color=darkgreen]
-			plot[domain=-1:1,samples=100]
-				({-(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});
-		\draw[color=darkgreen]
-			plot[domain=-1:1,samples=100]
-				({-(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});
-	\end{scope}
-
-	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-
-	\draw[->,color=blue] (0,0) -- (1.4,0);
-	\draw[->,color=blue] (0,0) -- (0,1.4);
-
-	\draw[->,color=red] (0,0) -- (1.96,0);
-	\draw[->,color=red] (0,0) -- (0,1);
-	\node at (0,-3.2) 
-		{$\displaystyle
-		\begin{aligned}
-		A&=\begin{pmatrix}1&0\\0&-1\end{pmatrix}
-		\\
-		e^{At}
-		&=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}
-		\end{aligned}
-		$};
-	
-\end{scope}
-
-
-\begin{scope}
-	\fill[color=blue!20]
-		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;
-	\fill[color=red!40,opacity=0.5]
-		(33:1.4) -- (123:1.4) -- (213:1.4) -- (303:1.4) -- cycle;
-
-	\draw[color=darkgreen] (0,0) circle[radius=1.4];
-
-	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-
-	\draw[->,color=blue] (0,0) -- (1.4,0);
-	\draw[->,color=blue] (0,0) -- (0,1.4);
-
-	\draw[->,color=red] (0,0) -- (33:1.4);
-	\draw[->,color=red] (0,0) -- (123:1.4);
-
-	\node at (0,-3.2)
-		{$\displaystyle
-		\begin{aligned}
-		B
-		&=\begin{pmatrix}0&-1\\1&0 \end{pmatrix}
-		\\
-		e^{Bt}
-		&=
-		\begin{pmatrix}
-		\cos t&-\sin t\\
-		\sin t& \cos t
-		\end{pmatrix}
-		\end{aligned}$};
-\end{scope}
-
-
-\begin{scope}[xshift=4.5cm]
-	\fill[color=blue!20]
-		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;
-	\def\x{0.5}
-	\fill[color=red!40,opacity=0.5]
-		({1.4*cosh(\x)},{1.4*sinh(\x})
-		--
-		({1.4*sinh(\x},{1.4*cosh(\x)})
-		--
-		({-1.4*cosh(\x)},{-1.4*sinh(\x})
-		--
-		({-1.4*sinh(\x},{-1.4*cosh(\x)})
-		-- cycle;
-
-	\begin{scope}
-	\clip (-2.1,-2.1) rectangle (2.2,2.2);
-	\draw[color=darkgreen]
-		plot[domain=-1:1,samples=100] ({1.4*cosh(\x)},{1.4*sinh(\x)});
-	\draw[color=darkgreen]
-		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{1.4*cosh(\x)});
-	\draw[color=darkgreen]
-		plot[domain=-1:1,samples=100] ({-1.4*cosh(\x)},{1.4*sinh(\x)});
-	\draw[color=darkgreen]
-		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{-1.4*cosh(\x)});
-	\end{scope}
-
-	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-
-	\draw[->,color=blue] (0,0) -- (1.4,0);
-	\draw[->,color=blue] (0,0) -- (0,1.4);
-
-	\draw[->,color=red] (0,0) -- ({1.4*cosh(\x)},{1.4*sinh(\x)});
-	\draw[->,color=red] (0,0) -- ({1.4*sinh(\x)},{1.4*cosh(\x)});
-
-	\node at (0,-3.2) {$\displaystyle
-		\begin{aligned}
-		C&=\begin{pmatrix}0&1\\1&0\end{pmatrix}
-		\\
-		e^{Ct}
-		&=
-		\begin{pmatrix}
-		\cosh t&\sinh t\\
-		\sinh t&\cosh t
-		\end{pmatrix}
-		\end{aligned}
-		$};
-\end{scope}
-
-\end{tikzpicture}
-\end{document}
-
+%

+% sl2.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}

+\begin{tikzpicture}[>=latex,thick,scale=\skala]

+

+\definecolor{darkgreen}{rgb}{0,0.6,0}

+

+\begin{scope}[xshift=-4.5cm]

+	\fill[color=blue!20]

+		(1.4,0) -- (0,1.4) -- (-1.4,0) -- (0,-1.4) -- cycle;

+	\fill[color=red!40,opacity=0.5]

+		(1.96,0) -- (0,1) -- (-1.96,0) -- (0,-1) -- cycle;

+

+	\begin{scope}

+		\clip (-2.1,-2.1) rectangle (2.3,2.3);

+		\draw[color=darkgreen]

+			plot[domain=-1:1,samples=100]

+				({(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});

+		\draw[color=darkgreen]

+			plot[domain=-1:1,samples=100]

+				({(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});

+		\draw[color=darkgreen]

+			plot[domain=-1:1,samples=100]

+				({-(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});

+		\draw[color=darkgreen]

+			plot[domain=-1:1,samples=100]

+				({-(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});

+	\end{scope}

+

+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];

+

+	\draw[->,color=blue] (0,0) -- (1.4,0);

+	\draw[->,color=blue] (0,0) -- (0,1.4);

+

+	\draw[->,color=red] (0,0) -- (1.96,0);

+	\draw[->,color=red] (0,0) -- (0,1);

+	\node at (0,-3.2) 

+		{$\displaystyle

+		\begin{aligned}

+		A&=\begin{pmatrix}1&0\\0&-1\end{pmatrix}

+		\\

+		e^{At}

+		&=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}

+		\end{aligned}

+		$};

+	

+\end{scope}

+

+

+\begin{scope}

+	\fill[color=blue!20]

+		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;

+	\fill[color=red!40,opacity=0.5]

+		(33:1.4) -- (123:1.4) -- (213:1.4) -- (303:1.4) -- cycle;

+

+	\draw[color=darkgreen] (0,0) circle[radius=1.4];

+

+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];

+

+	\draw[->,color=blue] (0,0) -- (1.4,0);

+	\draw[->,color=blue] (0,0) -- (0,1.4);

+

+	\draw[->,color=red] (0,0) -- (33:1.4);

+	\draw[->,color=red] (0,0) -- (123:1.4);

+

+	\node at (0,-3.2)

+		{$\displaystyle

+		\begin{aligned}

+		B

+		&=\begin{pmatrix}0&-1\\1&0 \end{pmatrix}

+		\\

+		e^{Bt}

+		&=

+		\begin{pmatrix}

+		\cos t&-\sin t\\

+		\sin t& \cos t

+		\end{pmatrix}

+		\end{aligned}$};

+\end{scope}

+

+

+\begin{scope}[xshift=4.5cm]

+	\fill[color=blue!20]

+		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;

+	\def\x{0.5}

+	\fill[color=red!40,opacity=0.5]

+		({1.4*cosh(\x)},{1.4*sinh(\x})

+		--

+		({1.4*sinh(\x},{1.4*cosh(\x)})

+		--

+		({-1.4*cosh(\x)},{-1.4*sinh(\x})

+		--

+		({-1.4*sinh(\x},{-1.4*cosh(\x)})

+		-- cycle;

+

+	\begin{scope}

+	\clip (-2.1,-2.1) rectangle (2.2,2.2);

+	\draw[color=darkgreen]

+		plot[domain=-1:1,samples=100] ({1.4*cosh(\x)},{1.4*sinh(\x)});

+	\draw[color=darkgreen]

+		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{1.4*cosh(\x)});

+	\draw[color=darkgreen]

+		plot[domain=-1:1,samples=100] ({-1.4*cosh(\x)},{1.4*sinh(\x)});

+	\draw[color=darkgreen]

+		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{-1.4*cosh(\x)});

+	\end{scope}

+

+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];

+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];

+

+	\draw[->,color=blue] (0,0) -- (1.4,0);

+	\draw[->,color=blue] (0,0) -- (0,1.4);

+

+	\draw[->,color=red] (0,0) -- ({1.4*cosh(\x)},{1.4*sinh(\x)});

+	\draw[->,color=red] (0,0) -- ({1.4*sinh(\x)},{1.4*cosh(\x)});

+

+	\node at (0,-3.2) {$\displaystyle

+		\begin{aligned}

+		C&=\begin{pmatrix}0&1\\1&0\end{pmatrix}

+		\\

+		e^{Ct}

+		&=

+		\begin{pmatrix}

+		\cosh t&\sinh t\\

+		\sinh t&\cosh t

+		\end{pmatrix}

+		\end{aligned}

+		$};

+\end{scope}

+

+\end{tikzpicture}

+\end{document}

+

diff --git a/buch/chapters/60-gruppen/images/torus.pov b/buch/chapters/60-gruppen/images/torus.pov
index 3a8e327..ee09c36 100644
--- a/buch/chapters/60-gruppen/images/torus.pov
+++ b/buch/chapters/60-gruppen/images/torus.pov
@@ -1,189 +1,189 @@
-//
-// diffusion.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
-	assumed_gamma 1
-}
-
-#declare imagescale = 0.034;
-#declare N = 100;
-#declare r = 0.43;
-#declare R = 1;
-
-camera {
-        location <43, 25, -20>
-        look_at <0, -0.01, 0>
-        right 16/9 * x * imagescale
-        up y * imagescale
-}
-
-light_source {
-        <10, 20, -40> color White
-        area_light <1,0,0> <0,0,1>, 10, 10
-        adaptive 1
-        jitter
-}
-
-sky_sphere {
-        pigment {
-                color rgb<1,1,1>
-        }
-}
-
-#macro rotiere(phi, vv)
-	< cos(phi) * vv.x - sin(phi) * vv.z, vv.y, sin(phi) * vv.x + cos(phi) * vv.z >
-#end
-
-#macro punkt(phi,theta)
-	rotiere(phi, < R + r * cos(theta), r * sin(theta), 0 >)
-#end
-
-mesh {
-	#declare phistep = 2 * pi / N;
-	#declare thetastep = 2 * 2 * pi / N;
-	#declare phi = 0;
-	#while (phi < 2 * pi - phistep/2)
-		#declare theta = 0;
-		#while (theta < 2 * pi - thetastep/2)
-			triangle {
-				punkt(phi          , theta            ), 
-				punkt(phi + phistep, theta            ), 
-				punkt(phi + phistep, theta + thetastep)
-			}
-			triangle {
-				punkt(phi          , theta            ), 
-				punkt(phi + phistep, theta + thetastep), 
-				punkt(phi          , theta + thetastep)
-			}
-			#declare theta = theta + thetastep;
-		#end
-		#declare phi = phi + phistep;
-	#end
-	pigment {
-		color Gray
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-#declare thetastart = -0.2;
-#declare thetaend = 1.2;
-#declare phistart = 5;
-#declare phiend = 6;
-
-union {
-	#declare thetastep = 0.2;
-	#declare theta = thetastart;
-	#while (theta < thetaend + thetastep/2)
-		#declare phistep = (phiend-phistart)/N;
-		#declare phi = phistart;
-		#while (phi < phiend - phistep/2)
-			sphere { punkt(phi,theta), 0.01 }
-			cylinder {
-				punkt(phi,theta),
-				punkt(phi+phistep,theta),
-				0.01
-			}
-			#declare phi = phi + phistep;
-		#end
-		sphere { punkt(phi,theta), 0.01 }
-		#declare theta = theta + thetastep;
-	#end
-
-	pigment {
-		color Red
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-union {
-	#declare phistep = 0.2;
-	#declare phi = phistart;
-	#while (phi < phiend + phistep/2)
-		#declare thetastep = (thetaend-thetastart)/N;
-		#declare theta = thetastart;
-		#while (theta < thetaend - thetastep/2)
-			sphere { punkt(phi,theta), 0.01 }
-			cylinder {
-				punkt(phi,theta),
-				punkt(phi,theta+thetastep),
-				0.01
-			}
-			#declare theta = theta + thetastep;
-		#end
-		sphere { punkt(phi,theta), 0.01 }
-		#declare phi = phi + phistep;
-	#end
-	pigment {
-		color Blue
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-#macro punkt2(a,b)
-	punkt(5.6+a*sqrt(3)/2-b/2,0.2+a/2 + b*sqrt(3)/2)
-#end
-
-#declare darkgreen = rgb<0,0.6,0>;
-
-#declare astart = 0;
-#declare aend = 1;
-#declare bstart = -0.2;
-#declare bend = 1.2;
-union {
-	#declare a = astart;
-	#declare astep = 0.2;
-	#while (a < aend + astep/2)
-		#declare b = bstart;
-		#declare bstep = (bend - bstart)/N;
-		#while (b < bend - bstep/2)
-			sphere { punkt2(a,b), 0.01 }
-			cylinder { punkt2(a,b), punkt2(a,b+bstep), 0.01 }
-			#declare b = b + bstep;
-		#end
-		sphere { punkt2(a,b), 0.01 }
-		#declare a = a + astep;
-	#end
-	pigment {
-		color darkgreen
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-union {
-	#declare b = bstart;
-	#declare bstep = 0.2;
-	#while (b < bend + bstep/2)
-		#declare a = astart;
-		#declare astep = (aend - astart)/N;
-		#while (a < aend - astep/2)
-			sphere { punkt2(a,b), 0.01 }
-			cylinder { punkt2(a,b), punkt2(a+astep,b), 0.01 }
-			#declare a = a + astep;
-		#end
-		sphere { punkt2(a,b), 0.01 }
-		#declare b = b + bstep;
-	#end
-	pigment {
-		color Orange
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
+//

+// diffusion.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule

+//

+#version 3.7;

+#include "colors.inc"

+

+global_settings {

+	assumed_gamma 1

+}

+

+#declare imagescale = 0.034;

+#declare N = 100;

+#declare r = 0.43;

+#declare R = 1;

+

+camera {

+        location <43, 25, -20>

+        look_at <0, -0.01, 0>

+        right 16/9 * x * imagescale

+        up y * imagescale

+}

+

+light_source {

+        <10, 20, -40> color White

+        area_light <1,0,0> <0,0,1>, 10, 10

+        adaptive 1

+        jitter

+}

+

+sky_sphere {

+        pigment {

+                color rgb<1,1,1>

+        }

+}

+

+#macro rotiere(phi, vv)

+	< cos(phi) * vv.x - sin(phi) * vv.z, vv.y, sin(phi) * vv.x + cos(phi) * vv.z >

+#end

+

+#macro punkt(phi,theta)

+	rotiere(phi, < R + r * cos(theta), r * sin(theta), 0 >)

+#end

+

+mesh {

+	#declare phistep = 2 * pi / N;

+	#declare thetastep = 2 * 2 * pi / N;

+	#declare phi = 0;

+	#while (phi < 2 * pi - phistep/2)

+		#declare theta = 0;

+		#while (theta < 2 * pi - thetastep/2)

+			triangle {

+				punkt(phi          , theta            ), 

+				punkt(phi + phistep, theta            ), 

+				punkt(phi + phistep, theta + thetastep)

+			}

+			triangle {

+				punkt(phi          , theta            ), 

+				punkt(phi + phistep, theta + thetastep), 

+				punkt(phi          , theta + thetastep)

+			}

+			#declare theta = theta + thetastep;

+		#end

+		#declare phi = phi + phistep;

+	#end

+	pigment {

+		color Gray

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+#declare thetastart = -0.2;

+#declare thetaend = 1.2;

+#declare phistart = 5;

+#declare phiend = 6;

+

+union {

+	#declare thetastep = 0.2;

+	#declare theta = thetastart;

+	#while (theta < thetaend + thetastep/2)

+		#declare phistep = (phiend-phistart)/N;

+		#declare phi = phistart;

+		#while (phi < phiend - phistep/2)

+			sphere { punkt(phi,theta), 0.01 }

+			cylinder {

+				punkt(phi,theta),

+				punkt(phi+phistep,theta),

+				0.01

+			}

+			#declare phi = phi + phistep;

+		#end

+		sphere { punkt(phi,theta), 0.01 }

+		#declare theta = theta + thetastep;

+	#end

+

+	pigment {

+		color Red

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+union {

+	#declare phistep = 0.2;

+	#declare phi = phistart;

+	#while (phi < phiend + phistep/2)

+		#declare thetastep = (thetaend-thetastart)/N;

+		#declare theta = thetastart;

+		#while (theta < thetaend - thetastep/2)

+			sphere { punkt(phi,theta), 0.01 }

+			cylinder {

+				punkt(phi,theta),

+				punkt(phi,theta+thetastep),

+				0.01

+			}

+			#declare theta = theta + thetastep;

+		#end

+		sphere { punkt(phi,theta), 0.01 }

+		#declare phi = phi + phistep;

+	#end

+	pigment {

+		color Blue

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+#macro punkt2(a,b)

+	punkt(5.6+a*sqrt(3)/2-b/2,0.2+a/2 + b*sqrt(3)/2)

+#end

+

+#declare darkgreen = rgb<0,0.6,0>;

+

+#declare astart = 0;

+#declare aend = 1;

+#declare bstart = -0.2;

+#declare bend = 1.2;

+union {

+	#declare a = astart;

+	#declare astep = 0.2;

+	#while (a < aend + astep/2)

+		#declare b = bstart;

+		#declare bstep = (bend - bstart)/N;

+		#while (b < bend - bstep/2)

+			sphere { punkt2(a,b), 0.01 }

+			cylinder { punkt2(a,b), punkt2(a,b+bstep), 0.01 }

+			#declare b = b + bstep;

+		#end

+		sphere { punkt2(a,b), 0.01 }

+		#declare a = a + astep;

+	#end

+	pigment {

+		color darkgreen

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+union {

+	#declare b = bstart;

+	#declare bstep = 0.2;

+	#while (b < bend + bstep/2)

+		#declare a = astart;

+		#declare astep = (aend - astart)/N;

+		#while (a < aend - astep/2)

+			sphere { punkt2(a,b), 0.01 }

+			cylinder { punkt2(a,b), punkt2(a+astep,b), 0.01 }

+			#declare a = a + astep;

+		#end

+		sphere { punkt2(a,b), 0.01 }

+		#declare b = b + bstep;

+	#end

+	pigment {

+		color Orange

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}