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-rw-r--r-- | vorlesungen/slides/7/parameter.tex | 107 |
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diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex new file mode 100644 index 0000000..52c8e4a --- /dev/null +++ b/vorlesungen/slides/7/parameter.tex @@ -0,0 +1,107 @@ +% +% parameter.tex -- Parametrisierung der Matrizen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkyellow}{rgb}{1,0.8,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen Parametrisieren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.4\textwidth} +\begin{block}{Drehung um Achsen} +\vspace{-12pt} +\begin{align*} +\uncover<2->{ +D_{x,\alpha} +&= +\begin{pmatrix} +1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha +\end{pmatrix} +} +\\ +\uncover<3->{ +D_{y,\beta} +&= +\begin{pmatrix} +\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta +\end{pmatrix} +} +\\ +\uncover<4->{ +D_{z,\gamma} +&= +\begin{pmatrix} +\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 +\end{pmatrix} +} +\intertext{\uncover<5->{beliebige Drehung:}} +\uncover<5->{ +D +&= +D_{x,\alpha} +D_{y,\beta} +D_{z,\gamma} +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<6->{% +\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} +\uncover<7->{% +$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel +} +\\ +\uncover<8->{% +$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse +} +\[ +\uncover<9->{ +{\color{red}\vec{x}} +\mapsto +} +\uncover<10->{ +({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) +\cos\omega ++ +} +\uncover<11->{ +({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega ++ +} +\uncover<9->{ +{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) +} +\] +\vspace{-40pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \node at (0,0) + {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; + \node[color=red] at (1.6,-0.9) {$\vec{x}$}; + \node[color=blue] at (0.5,2) {$\vec{k}$}; +} +\uncover<11->{ + \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; +} +\uncover<10->{ + \node[color=yellow] at (2.2,-0.2) + {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\uncover<5->{% +{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine +dreidimensionale Gruppe} +\end{frame} +\egroup |