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-rw-r--r--vorlesungen/slides/7/sl2.tex484
1 files changed, 242 insertions, 242 deletions
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
index a65b4f6..58e87a1 100644
--- a/vorlesungen/slides/7/sl2.tex
+++ b/vorlesungen/slides/7/sl2.tex
@@ -1,242 +1,242 @@
-%
-% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t,fragile]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{block}{Determinante}
-\[
-A=\begin{pmatrix}
-a&b\\
-c&d
-\end{pmatrix}
-\;\Rightarrow\;
-\det A = ad-bc
-\]
-\end{block}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Dimension}
-\[
-4\; \text{Variablen}
--
-1\; \text{Bedingung}
-=
-3\; \text{Dimensionen}
-\]
-\end{block}
-\end{column}
-\end{columns}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{columns}[t,onlytextwidth]
-\def\s{0.94}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \fill[color=blue!20] (-1,-1) rectangle (1,1);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {4,...,10}{
- \only<\d>{
- \pgfmathparse{1+(\d-4)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
- }
- }{}
- \uncover<11->{
- \xdef\t{1.6}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<3->{%
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- D
- =
- \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {11,...,17}{
- \only<\d>{
- \pgfmathparse{(\d-11)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
- }
- }{}
- \uncover<18->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<11->{
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- S
- =
- \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {18,...,24}{
- \only<\d>{
- \pgfmathparse{(\d-18)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
- }
- }{}
- \uncover<25->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<18->{%
-\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- T
- =
- \begin{pmatrix} 1&0\\t&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\end{columns}}
-\end{frame}
-\egroup
+%
+% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t,fragile]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Determinante}
+\[
+A=\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\;\Rightarrow\;
+\det A = ad-bc
+\]
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Dimension}
+\[
+4\; \text{Variablen}
+-
+1\; \text{Bedingung}
+=
+3\; \text{Dimensionen}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{columns}[t,onlytextwidth]
+\def\s{0.94}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \fill[color=blue!20] (-1,-1) rectangle (1,1);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {4,...,10}{
+ \only<\d>{
+ \pgfmathparse{1+(\d-4)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+ }
+ }{}
+ \uncover<11->{
+ \xdef\t{1.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<3->{%
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ D
+ =
+ \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {11,...,17}{
+ \only<\d>{
+ \pgfmathparse{(\d-11)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+ }
+ }{}
+ \uncover<18->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<11->{
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ S
+ =
+ \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {18,...,24}{
+ \only<\d>{
+ \pgfmathparse{(\d-18)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+ }
+ }{}
+ \uncover<25->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<18->{%
+\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ T
+ =
+ \begin{pmatrix} 1&0\\t&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}}
+\end{frame}
+\egroup
generated by cgit v1.2.1 (git 2.18.0) at 2024-11-02 22:17:42 +0100