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+%
+% drehung.tex -- Drehung aus streckungen
+%
+% (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{Drehung aus Streckungen und Scherungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Drehung}
+\[
+\operatorname{SO}(2)
+=
+\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2)
+\]
+\end{block}
+\begin{block}{Zusammensetzung}
+Eine Drehung muss als Zusammensetzung geschrieben werden können:
+\[
+D_{\alpha} 
+=
+\begin{pmatrix}
+\cos\alpha & -\sin\alpha\\
+\sin\alpha &\phantom{-}\cos\alpha
+\end{pmatrix}
+=
+DST
+\]
+\end{block}
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix}
+}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{block}{Ansatz}
+\vspace{-12pt}
+\begin{align*}
+DST
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+1&-s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&0\\
+t&1
+\end{pmatrix}
+\\
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+-st&-s\\
+  t& 1
+\end{pmatrix}
+\\
+&=
+\begin{pmatrix}
+-stc^{-1}&{\color{darkgreen}sc^{-1}}\\
+{\color{blue}ct}&{\color{red}c}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\cos\alpha & {\color{darkgreen}- \sin\alpha} \\
+{\color{blue}\sin\alpha} & \phantom{-}  {\color{red}\cos\alpha}
+\end{pmatrix}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Koeffizientenvergleich}
+\vspace{-15pt}
+\begin{align*}
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }
+&&
+&
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+\\
+{\color{orange} -stc^{-t}}
+&=
+\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup