From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001
From: Joshua Baer <the.baer.joshua@gmail.ch>
Date: Mon, 12 Apr 2021 21:51:55 +0200
Subject: update to current state of book

---
 vorlesungen/slides/7/drehung.tex | 264 +++++++++++++++++++--------------------
 1 file changed, 132 insertions(+), 132 deletions(-)

(limited to 'vorlesungen/slides/7/drehung.tex')

diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
index 2d7b317..e7b4a92 100644
--- a/vorlesungen/slides/7/drehung.tex
+++ b/vorlesungen/slides/7/drehung.tex
@@ -1,132 +1,132 @@
-%
-% 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}
-{\color{blue}Längen}, {\color<2->{blue}Winkel},
-{\color<2->{darkgreen}Orientierung}
-erhalten
-\uncover<2->{
-\[
-\operatorname{SO}(2)
-=
-{\color{blue}\operatorname{O}(2)}
-\cap
-{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
-\]}
-\vspace{-20pt}
-\end{block}
-\uncover<3->{%
-\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}}
-\vspace{-10pt}
-\uncover<12->{%
-\begin{block}{Beispiel}
-\vspace{-12pt}
-\[
-D_{60^\circ}
-=
-{\tiny
-\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
-\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
-}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.58\textwidth}
-\uncover<4->{%
-\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}
-\\
-&\uncover<5->{=
-\begin{pmatrix}
-c^{-1}&0\\
-  0   &c
-\end{pmatrix}
-\begin{pmatrix}
-1-st&-s\\
-   t& 1
-\end{pmatrix}
-}
-\\
-&\uncover<6->{=
-\begin{pmatrix}
-{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
-{\color<9->{blue}ct}&{\color<8->{red}c}
-\end{pmatrix}}
-\uncover<7->{=
-\begin{pmatrix}
-{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
-{\color<9->{blue}\sin\alpha} & \phantom{-}  {\color<8->{red}\cos\alpha}
-\end{pmatrix}}
-\end{align*}
-\end{block}}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Koeffizientenvergleich}
-\vspace{-15pt}
-\begin{align*}
-\uncover<8->{
-{\color{red} c}
-&=
-{\color{red}\cos\alpha }}
-&&
-&
-\uncover<9->{
-{\color{blue}
-t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
-\uncover<10->{
-{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
-&
-&\Rightarrow&
-{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
-}
-\\
-\uncover<11->{
-{\color{orange} (1-st)c^{-t}}
-&=
-\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% 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}
+{\color{blue}Längen}, {\color<2->{blue}Winkel},
+{\color<2->{darkgreen}Orientierung}
+erhalten
+\uncover<2->{
+\[
+\operatorname{SO}(2)
+=
+{\color{blue}\operatorname{O}(2)}
+\cap
+{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
+\]}
+\vspace{-20pt}
+\end{block}
+\uncover<3->{%
+\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}}
+\vspace{-10pt}
+\uncover<12->{%
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
+}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\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}
+\\
+&\uncover<5->{=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+1-st&-s\\
+   t& 1
+\end{pmatrix}
+}
+\\
+&\uncover<6->{=
+\begin{pmatrix}
+{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
+{\color<9->{blue}ct}&{\color<8->{red}c}
+\end{pmatrix}}
+\uncover<7->{=
+\begin{pmatrix}
+{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
+{\color<9->{blue}\sin\alpha} & \phantom{-}  {\color<8->{red}\cos\alpha}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Koeffizientenvergleich}
+\vspace{-15pt}
+\begin{align*}
+\uncover<8->{
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }}
+&&
+&
+\uncover<9->{
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
+\uncover<10->{
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+}
+\\
+\uncover<11->{
+{\color{orange} (1-st)c^{-t}}
+&=
+\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
-- 
cgit v1.2.1

