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Diffstat (limited to 'buch/chapters/60-gruppen/images/kartenkreis.tex')
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diff --git a/buch/chapters/60-gruppen/images/kartenkreis.tex b/buch/chapters/60-gruppen/images/kartenkreis.tex new file mode 100644 index 0000000..4f19937 --- /dev/null +++ b/buch/chapters/60-gruppen/images/kartenkreis.tex @@ -0,0 +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} + |