diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/7/drehanim.tex | 310 |
1 files changed, 155 insertions, 155 deletions
diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex index ac136f1..776617f 100644 --- a/vorlesungen/slides/7/drehanim.tex +++ b/vorlesungen/slides/7/drehanim.tex @@ -1,155 +1,155 @@ -% -% 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$}]; - -\end{tikzpicture} -\end{center} -\end{frame} -\egroup +%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
+
+\makeatletter
+\hoffset=-2cm
+\advance\textwidth2cm
+\hsize\textwidth
+\columnwidth\textwidth
+\makeatother
+
+\begin{frame}[t,plain]
+\vspace{-5pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=white] (-4,-4) rectangle (9,4.5);
+
+\def\a{60}
+
+\pgfmathparse{tan(\a)}
+\xdef\T{\pgfmathresult}
+
+\pgfmathparse{-sin(\a)*cos(\a)}
+\xdef\S{\pgfmathresult}
+
+\pgfmathparse{1/cos(\a)}
+\xdef\E{\pgfmathresult}
+
+\def\N{20}
+\pgfmathparse{2*\N}
+\xdef\Nzwei{\pgfmathresult}
+\pgfmathparse{3*\N}
+\xdef\Ndrei{\pgfmathresult}
+
+\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
+\begin{block}{$\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
|