tangentialvektoren von o(n), sl_n

5 files changed, 157 insertions, 5 deletions

diff --git a/buch/chapters/60-gruppen/images/Makefile b/buch/chapters/60-gruppen/images/Makefile
index bc65a71..8824d75 100644
--- a/buch/chapters/60-gruppen/images/Makefile
+++ b/buch/chapters/60-gruppen/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	phasenraum.pdf kartenkreis.pdf karten.pdf
+all:	phasenraum.pdf kartenkreis.pdf karten.pdf sl2.pdf
 
 phasenraum.pdf:	phasenraum.tex
 	pdflatex phasenraum.tex
@@ -17,3 +17,6 @@ torus.png:	torus.pov
 karten.pdf:	karten.tex torus.png
 	pdflatex karten.tex
 
+sl2.pdf:	sl2.tex 
+	pdflatex sl2.tex
+
diff --git a/buch/chapters/60-gruppen/images/kartenkreis.pdf b/buch/chapters/60-gruppen/images/kartenkreis.pdf
index 3235779..4619b56 100644
--- a/buch/chapters/60-gruppen/images/kartenkreis.pdf
+++ b/buch/chapters/60-gruppen/images/kartenkreis.pdf
diff --git a/buch/chapters/60-gruppen/images/kartenkreis.tex b/buch/chapters/60-gruppen/images/kartenkreis.tex
index 8f2d9d5..4f19937 100644
--- a/buch/chapters/60-gruppen/images/kartenkreis.tex
+++ b/buch/chapters/60-gruppen/images/kartenkreis.tex
@@ -62,12 +62,13 @@
 \node at (1.6,-1.6) {$\varphi_{14}$};
 
 \node at (-1.24,1.32)
-	[left] {$\varphi_2\circ\varphi_4^{-1}(x)=\sqrt{1-x^2}$};
-\node at (-1.6,-1.6) {$\varphi_{42}$};
+	[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) {$\varphi_{23}$};
+\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]
@@ -90,7 +91,9 @@
 		({-\x},{-sqrt(1-\x*\x)}) -- ({-\x},-1.5);
 }
 
-\draw[color=gray!20,line width=3pt] (0,0) circle[radius=1];
+%\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}
 
diff --git a/buch/chapters/60-gruppen/images/sl2.pdf b/buch/chapters/60-gruppen/images/sl2.pdf
new file mode 100644
index 0000000..ffc0759
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/sl2.pdf
diff --git a/buch/chapters/60-gruppen/images/sl2.tex b/buch/chapters/60-gruppen/images/sl2.tex
new file mode 100644
index 0000000..0e44aa9
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/sl2.tex
@@ -0,0 +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}
+