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Diffstat (limited to 'vorlesungen/slides/7/drehung.tex')
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diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex new file mode 100644 index 0000000..2d7b317 --- /dev/null +++ b/vorlesungen/slides/7/drehung.tex @@ -0,0 +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 |