From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001 From: Joshua Baer 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