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%
% template.tex -- slide template
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
\makeatletter
\hoffset=-2cm
\advance\textwidth2cm
\hsize\textwidth
\columnwidth\textwidth
\makeatother
\begin{frame}[t,plain]
\vspace{-5pt}
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\begin{center}
\begin{tikzpicture}[>=latex,thick]
\fill[color=white] (-4,-4) rectangle (9,4.5);
\def\a{60}
\pgfmathparse{tan(\a)}
\xdef\T{\pgfmathresult}
\pgfmathparse{-sin(\a)*cos(\a)}
\xdef\S{\pgfmathresult}
\pgfmathparse{1/cos(\a)}
\xdef\E{\pgfmathresult}
\def\N{20}
\pgfmathparse{2*\N}
\xdef\Nzwei{\pgfmathresult}
\pgfmathparse{3*\N}
\xdef\Ndrei{\pgfmathresult}
\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
\begin{itemize}
\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
as shear mappings and axis rescalings
\[
A=
\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
\begin{pmatrix}1&s\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\t&1\end{pmatrix}
\]
\item Most rotations can be decomposed into a product of
shear mappings and axis rescalings
\end{itemize}
\end{block}
\end{minipage}};
\foreach \d in {1,2,...,\Ndrei}{
% Scherung in Y-Richtung
\ifnum \d>\N
\pgfmathparse{\T}
\else
\pgfmathparse{\T*(\d-1)/(\N-1)}
\fi
\xdef\t{\pgfmathresult}
% Scherung in X-Richtung
\ifnum \d>\Nzwei
\xdef\s{\S}
\else
\ifnum \d<\N
\xdef\s{0}
\else
\ifnum \d=\N
\xdef\s{0}
\else
\pgfmathparse{\S*(\d-\N-1)/(\N-1)}
\xdef\s{\pgfmathresult}
\fi
\fi
\fi
% Reskalierung der Achsen
\ifnum \d>\Nzwei
\pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
\else
\pgfmathparse{1}
\fi
\xdef\e{\pgfmathresult}
% Matrixelemente
\pgfmathparse{(\e)*((\s)*(\t)+1)}
\xdef\A{\pgfmathresult}
\pgfmathparse{(\e)*(\s)}
\xdef\B{\pgfmathresult}
\pgfmathparse{(\t)/(\e)}
\xdef\C{\pgfmathresult}
\pgfmathparse{1/(\e)}
\xdef\D{\pgfmathresult}
\only<\d>{
\node at (5.0,-0.9) [below right] {$
\begin{aligned}
t &= \t \\
s &= \s \\
d &= \e \\
D &= \begin{pmatrix}
\A&\B\\
\C&\D
\end{pmatrix}
\qquad
\only<60>{\checkmark}
\end{aligned}
$};
}
\begin{scope}
\clip (-4.05,-4.05) rectangle (4.05,4.05);
\only<\d>{
\foreach \x in {-6,...,6}{
\draw[color=blue,line width=0.5pt]
\punkt{\x}{-12} -- \punkt{\x}{12};
}
\foreach \y in {-12,...,12}{
\draw[color=darkgreen,line width=0.5pt]
\punkt{-6}{\y} -- \punkt{6}{\y};
}
\foreach \r in {1,2,3,4}{
\draw[color=red] plot[domain=0:359,samples=360]
({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
--
cycle;
}
}
\end{scope}
% \uncover<\d>{
% \node at (5,4) {\d};
% }
}
\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
\end{tikzpicture}
\end{center}
\end{frame}
\egroup
|
