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+%
+% symmetrien.tex -- Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Symmetrien}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diskrete Symmetrien}
+\begin{itemize}
+\item
+Ebenen-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+{\tiny
+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*},
+}
+\;
+\vec{x}
+\mapsto
+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
+\]
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{10}
+\def\b{50}
+\def\r{2}
+\coordinate (O) at (0,0);
+\coordinate (A) at (\b:\r);
+\coordinate (B) at ({180+2*\a-\b}:\r);
+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
+\coordinate (N) at (\a:2);
+\coordinate (D) at (\a:{\r*cos(\b-\a)});
+\clip (-2.5,-0.45) rectangle (2.5,1.95);
+
+\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) -- cycle;
+\draw[->,color=darkgreen] (O) -- (N);
+\node[color=darkgreen] at (N) [above] {$\vec{n}$};
+
+
+\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) -- cycle;
+\fill[color=red] (O) circle[radius=0.06];
+\draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
+\fill[color=blue] (C) circle[radius=0.06];
+\draw[color=blue,line width=0.1pt] (A) -- (D);
+\node[color=darkgreen] at (D) [below,rotate=\a] {$(\vec{n}\cdot\vec{x})\vec{n}$};
+\draw[color=blue,line width=0.5pt] (A)--(B);
+
+\node[color=blue] at (A) [above right] {$\vec{x}$};
+\node[color=blue] at (B) [above left] {$\vec{x}'$};
+
+\node[color=red] at (O) [below left] {$O$};
+
+\draw[->,color=blue,shorten <= 0.06cm] (O) -- (A);
+\draw[->,color=blue,shorten <= 0.06cm] (O) -- (B);
+
+\end{tikzpicture}
+\end{center}
+\vspace{-5pt}
+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
+\item
+Punkt-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+-
+{\tiny
+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Kontinuierliche Symmetrien}
+\begin{itemize}
+\item Translation:
+\(
+\vec{x} \mapsto \vec{x} + \vec{t}
+\)
+\item Drehung:
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{25}
+\def\r{1.3}
+\coordinate (O) at (0,0);
+\begin{scope}
+\clip (-1.1,-0.1) rectangle (2.3,2.3);
+\draw[color=red] (O) circle[radius=2];
+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
+\node at ({0.5*\a}:1) {$\alpha$};
+\node at ({90+0.5*\a}:1) {$\alpha$};
+\draw[->,color=blue] (O) -- (\a:2);
+\draw[->,color=darkgreen] (O) -- ({90+\a}:2);
+\end{scope}
+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
+\end{tikzpicture}
+\end{center}
+\[
+\begin{pmatrix}x\\y\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
+\end{pmatrix}
+\begin{pmatrix}x\\y\end{pmatrix}
+\]
+\end{itemize}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Definition}
+Längen/Winkel bleiben erhalten
+\\
+$\Rightarrow$ $\exists$ Erhaltungsgrösse
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup