diff options
| -rw-r--r-- | vorlesungen/07_lie/slides.tex | 3 | ||||
| -rw-r--r-- | vorlesungen/common/slide-template.tex (renamed from vorlesungen/common/template.tex) | 0 | ||||
| -rw-r--r-- | vorlesungen/slides/7/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/drehanim.tex | 155 | ||||
| -rw-r--r-- | vorlesungen/slides/7/drehung.tex | 12 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 5 |
7 files changed, 169 insertions, 8 deletions
diff --git a/vorlesungen/07_lie/slides.tex b/vorlesungen/07_lie/slides.tex index db4a3fc..19131d8 100644 --- a/vorlesungen/07_lie/slides.tex +++ b/vorlesungen/07_lie/slides.tex @@ -10,6 +10,9 @@ \folie{7/mannigfaltigkeit.tex} \folie{7/sl2.tex} \folie{7/drehung.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{7/drehanim.tex} +}{} \folie{7/semi.tex} \section{Ableitungen} diff --git a/vorlesungen/common/template.tex b/vorlesungen/common/slide-template.tex index a1343f8..a1343f8 100644 --- a/vorlesungen/common/template.tex +++ b/vorlesungen/common/slide-template.tex diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 540b080..7afeea1 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -10,6 +10,7 @@ chapter5 = \ ../slides/7/mannigfaltigkeit.tex \ ../slides/7/sl2.tex \ ../slides/7/drehung.tex \ + ../slides/7/drehanim.tex \ ../slides/7/semi.tex \ ../slides/7/kurven.tex \ ../slides/7/einparameter.tex \ diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 5878164..079cf16 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -9,6 +9,7 @@ \folie{7/mannigfaltigkeit.tex} \folie{7/sl2.tex} \folie{7/drehung.tex} +\folie{7/drehanim.tex} \folie{7/semi.tex} \folie{7/kurven.tex} \folie{7/einparameter.tex} diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex new file mode 100644 index 0000000..ac136f1 --- /dev/null +++ b/vorlesungen/slides/7/drehanim.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 7744e99..2d7b317 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -50,7 +50,7 @@ D_{60^\circ} {\tiny \begin{pmatrix}2&0\\0&\frac12\end{pmatrix} \begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} -\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} +\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} } \] \end{block}} @@ -81,14 +81,14 @@ c^{-1}&0\\ 0 &c \end{pmatrix} \begin{pmatrix} --st&-s\\ - t& 1 +1-st&-s\\ + t& 1 \end{pmatrix} } \\ &\uncover<6->{= \begin{pmatrix} -{\color<11->{orange}-stc^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ {\color<9->{blue}ct}&{\color<8->{red}c} \end{pmatrix}} \uncover<7->{= @@ -120,9 +120,9 @@ t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ } \\ \uncover<11->{ -{\color{orange} -stc^{-t}} +{\color{orange} (1-st)c^{-t}} &= -\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} } \end{align*} \end{block}} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 0cab79f..4673f76 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -16,7 +16,8 @@ % Weitere Beispiele % SL_2(R) %\folie{7/sl2.tex} -%\folie{7/drehung.tex} +\folie{7/drehung.tex} +%\folie{7/drehanim.tex} % Semidirekte Produkte SO(2) x R^2, R^+ x R %\folie{7/semi.tex} @@ -34,5 +35,5 @@ \section{Exponentialabbildung} % Differentialgleichung für die Exponentialabbildung -\folie{7/dg.tex} +%\folie{7/dg.tex} |