update to current state of book

1 files changed, 132 insertions, 132 deletions

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