diff options
Diffstat (limited to '')
| -rw-r--r-- | buch/chapters/60-gruppen/images/Makefile | 50 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/karten.tex | 224 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/kartenkreis.tex | 378 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/phasenraum.tex | 186 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/scherungen.tex | 314 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/sl2.tex | 292 | ||||
| -rw-r--r-- | buch/chapters/60-gruppen/images/torus.pov | 378 |
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
+ }
+}
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