From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001 From: Joshua Baer Date: Mon, 12 Apr 2021 21:51:55 +0200 Subject: update to current state of book --- vorlesungen/slides/7/drehanim.tex | 310 +++++++++++++++++++------------------- 1 file changed, 155 insertions(+), 155 deletions(-) (limited to 'vorlesungen/slides/7/drehanim.tex') 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 -- cgit v1.2.1