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%
% parameter.tex -- Parametrisierung der Matrizen
%
% (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{Drehungen Parametrisieren}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.4\textwidth}
\begin{block}{Drehung um Achsen}
\begin{align*}
D_{x,\alpha}
&=
\begin{pmatrix}
1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
\end{pmatrix}
\\
D_{y,\beta}
&=
\begin{pmatrix}
\cos\beta&0&-\sin\beta\\0&1&0\\\sin\beta&0&\cos\beta
\end{pmatrix}
\\
D_{z,\gamma}
&=
\begin{pmatrix}
\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
\end{pmatrix}
\end{align*}
\end{block}
\end{column}
\begin{column}{0.56\textwidth}
\begin{block}{Drehung um $\vec{\omega}$}
$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
\\
$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
\[
\vec{x}
\mapsto
(\vec{x} -(\vec{k}\cdot\vec{x})\vec{k})
\cos\omega
+
(\vec{k}\times\vec{x})\sin\omega
+
\vec{k}(\vec{k}\cdot\vec{x})
\]
\vspace{-40pt}
\begin{center}
\begin{tikzpicture}[>=latex,thick]
\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) {$\vec{k}$};
\node[color=darkgreen] at (-3,1.1) {$\vec{k}\times\vec{x}$};
\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}
{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
dreidimensionale Gruppe
\end{frame}
\egroup
|
