diff options
Diffstat (limited to 'vorlesungen/slides/7')
22 files changed, 2093 insertions, 2093 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 7afeea1..2391099 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -1,22 +1,22 @@ -# -# Makefile.inc -- additional depencencies -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -chapter5 = \ - ../slides/7/symmetrien.tex \ - ../slides/7/algebraisch.tex \ - ../slides/7/parameter.tex \ - ../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 \ - ../slides/7/ableitung.tex \ - ../slides/7/liealgebra.tex \ - ../slides/7/kommutator.tex \ - ../slides/7/dg.tex \ - ../slides/7/chapter.tex - +#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 = \
+ ../slides/7/symmetrien.tex \
+ ../slides/7/algebraisch.tex \
+ ../slides/7/parameter.tex \
+ ../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 \
+ ../slides/7/ableitung.tex \
+ ../slides/7/liealgebra.tex \
+ ../slides/7/kommutator.tex \
+ ../slides/7/dg.tex \
+ ../slides/7/chapter.tex
+
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index 12f9084..5a4b94e 100644 --- a/vorlesungen/slides/7/ableitung.tex +++ b/vorlesungen/slides/7/ableitung.tex @@ -1,68 +1,68 @@ -% -% ableitung.tex -- Ableitung in der Lie-Gruppe -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Ableitung in der Matrix-Gruppe} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Ableitung in $\operatorname{O}(n)$} -\uncover<2->{% -$s \mapsto A(s)\in\operatorname{O}(n)$ -} -\begin{align*} -\uncover<3->{I -&= -A(s)^tA(s)} -\\ -\uncover<4->{0 -= -\frac{d}{ds} I -&= -\frac{d}{ds} (A(s)^t A(s))} -\\ -&\uncover<5->{= -\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} -\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} -\uncover<7->{0 -&= -\dot{A}(0)^t -+ -\dot{A}(0)} -\\ -\uncover<8->{\Leftrightarrow -\qquad -\dot{A}(0)^t &= -\dot{A}(0)} -\end{align*} -\uncover<9->{% -``Tangentialvektoren'' sind antisymmetrische Matrizen -} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} -\uncover<2->{% -$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ -} -\begin{align*} -\uncover<3->{1 &= \det A(t)} -\\ -\uncover<10->{0 -= -\frac{d}{dt}1 -&= -\frac{d}{dt} \det A(t)} -\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} -\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} -\end{align*} -\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} -\end{block} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% ableitung.tex -- Ableitung in der Lie-Gruppe
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ableitung in der Matrix-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{O}(n)$}
+\uncover<2->{%
+$s \mapsto A(s)\in\operatorname{O}(n)$
+}
+\begin{align*}
+\uncover<3->{I
+&=
+A(s)^tA(s)}
+\\
+\uncover<4->{0
+=
+\frac{d}{ds} I
+&=
+\frac{d}{ds} (A(s)^t A(s))}
+\\
+&\uncover<5->{=
+\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)}
+\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}}
+\uncover<7->{0
+&=
+\dot{A}(0)^t
++
+\dot{A}(0)}
+\\
+\uncover<8->{\Leftrightarrow
+\qquad
+\dot{A}(0)^t &= -\dot{A}(0)}
+\end{align*}
+\uncover<9->{%
+``Tangentialvektoren'' sind antisymmetrische Matrizen
+}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
+\uncover<2->{%
+$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
+}
+\begin{align*}
+\uncover<3->{1 &= \det A(t)}
+\\
+\uncover<10->{0
+=
+\frac{d}{dt}1
+&=
+\frac{d}{dt} \det A(t)}
+\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}}
+\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)}
+\end{align*}
+\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex index 31d209a..fba42cf 100644 --- a/vorlesungen/slides/7/algebraisch.tex +++ b/vorlesungen/slides/7/algebraisch.tex @@ -1,115 +1,115 @@ -% -% algebraisch.tex -- algebraische Definition der Symmetrien -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Erhaltungsgrössen und Algebra} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Längen und Winkel} -Längenmessung mit Skalarprodukt -\begin{align*} -\|\vec{v}\|^2 -&= -\langle \vec{v},\vec{v}\rangle -= -\vec{v}\cdot \vec{v} -\uncover<2->{= -\vec{v}^t\vec{v}} -\end{align*} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<3->{% -\begin{block}{Flächeninhalt/Volumen} -$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ -\\ -Volumen des Parallelepipeds: $\det V$ -\end{block}} -\end{column} -\end{columns} -% -\vspace{-7pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Längenerhaltende Transformationen} -$A\in\operatorname{GL}_n(\mathbb{R})$ -\begin{align*} -\vec{x}^t\vec{y} -&= -(A\vec{x}) -\cdot -(A\vec{y}) -\uncover<5->{= -(A\vec{x})^t -(A\vec{y})} -\\ -\uncover<6->{ -\vec{x}^tI\vec{y} -&= -\vec{x}^tA^tA\vec{y}} -\uncover<7->{ -\Rightarrow I=A^tA} -\end{align*} -\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<9->{% -\begin{block}{Volumenerhaltende Transformationen} -$A\in\operatorname{GL}_n(\mathbb{R})$ -\begin{align*} -\det(V) -&= -\det(AV) -\uncover<10->{= -\det(A)\det(V)} -\\ -\uncover<11->{ -1&=\det(A)} -\end{align*} -\uncover<10->{ -(Produktsatz für Determinante) -} -\end{block}} -\end{column} -\end{columns} -% -\vspace{-3pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<12->{% -\begin{block}{Orthogonale Matrizen} -Längentreue Abbildungen = orthogonale Matrizen: -\[ -O(n) -= -\{ -A \in \operatorname{GL}_n(\mathbb{R}) -\;|\; -A^tA=I -\} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<13->{% -\begin{block}{``Spezielle'' Matrizen} -Volumen-/Orientierungserhaltende Transformationen: -\[ -\operatorname{SL}_n(\mathbb R) -= -\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} -\] -\end{block}} -\end{column} -\end{columns} - -\end{frame} -\egroup +%
+% algebraisch.tex -- algebraische Definition der Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erhaltungsgrössen und Algebra}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Längen und Winkel}
+Längenmessung mit Skalarprodukt
+\begin{align*}
+\|\vec{v}\|^2
+&=
+\langle \vec{v},\vec{v}\rangle
+=
+\vec{v}\cdot \vec{v}
+\uncover<2->{=
+\vec{v}^t\vec{v}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Flächeninhalt/Volumen}
+$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
+\\
+Volumen des Parallelepipeds: $\det V$
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-7pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Längenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\vec{x}^t\vec{y}
+&=
+(A\vec{x})
+\cdot
+(A\vec{y})
+\uncover<5->{=
+(A\vec{x})^t
+(A\vec{y})}
+\\
+\uncover<6->{
+\vec{x}^tI\vec{y}
+&=
+\vec{x}^tA^tA\vec{y}}
+\uncover<7->{
+\Rightarrow I=A^tA}
+\end{align*}
+\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<9->{%
+\begin{block}{Volumenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\det(V)
+&=
+\det(AV)
+\uncover<10->{=
+\det(A)\det(V)}
+\\
+\uncover<11->{
+1&=\det(A)}
+\end{align*}
+\uncover<10->{
+(Produktsatz für Determinante)
+}
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-3pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Orthogonale Matrizen}
+Längentreue Abbildungen = orthogonale Matrizen:
+\[
+O(n)
+=
+\{
+A \in \operatorname{GL}_n(\mathbb{R})
+\;|\;
+A^tA=I
+\}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<13->{%
+\begin{block}{``Spezielle'' Matrizen}
+Volumen-/Orientierungserhaltende Transformationen:
+\[
+\operatorname{SL}_n(\mathbb R)
+=
+\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 079cf16..0f14a9a 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -1,19 +1,19 @@ -% -% chapter.tex -% -% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi -% -\folie{7/symmetrien.tex} -\folie{7/algebraisch.tex} -\folie{7/parameter.tex} -\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} -\folie{7/ableitung.tex} -\folie{7/liealgebra.tex} -\folie{7/kommutator.tex} -\folie{7/dg.tex} +%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{7/symmetrien.tex}
+\folie{7/algebraisch.tex}
+\folie{7/parameter.tex}
+\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}
+\folie{7/ableitung.tex}
+\folie{7/liealgebra.tex}
+\folie{7/kommutator.tex}
+\folie{7/dg.tex}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 4447bac..446b2ab 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -1,92 +1,92 @@ -% -% dg.tex -- Differentialgleichung für die Exponentialabbildung -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Zurück zur Lie-Gruppe} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Tangentialvektor im Punkt $\gamma(t)$} -Ableitung von $\gamma(t)$ an der Stelle $t$: -\begin{align*} -\dot{\gamma}(t) -&\uncover<2->{= -\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} -} -\\ -&\uncover<3->{= -\frac{d}{ds} -\gamma(t+s) -\bigg|_{s=0} -} -\\ -&\uncover<4->{= -\frac{d}{ds} -\gamma(t)\gamma(s) -\bigg|_{s=0} -} -\\ -&\uncover<5->{= -\gamma(t) -\frac{d}{ds} -\gamma(s) -\bigg|_{s=0} -} -\uncover<6->{= -\gamma(t) \dot{\gamma}(0) -} -\end{align*} -\end{block} -\vspace{-10pt} -\uncover<7->{% -\begin{block}{Differentialgleichung} -\vspace{-10pt} -\[ -\dot{\gamma}(t) = \gamma(t) A -\quad -\text{mit} -\quad -A=\dot{\gamma}(0)\in LG -\] -\end{block}} -\end{column} -\begin{column}{0.50\textwidth} -\uncover<8->{% -\begin{block}{Lösung} -Exponentialfunktion -\[ -\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k -\] -\end{block}} -\vspace{-5pt} -\uncover<9->{% -\begin{block}{Kontrolle: Tangentialvektor berechnen} -\vspace{-10pt} -\begin{align*} -\frac{d}{dt}e^{At} -&\uncover<10->{= -\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!} -} -\\ -&\uncover<11->{= -\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A -} -\\ -&\uncover<12->{= -\sum_{k=0} A^k\frac{t^k}{k!} -A -} -\uncover<13->{= -e^{At} A -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% dg.tex -- Differentialgleichung für die Exponentialabbildung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zurück zur Lie-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
+Ableitung von $\gamma(t)$ an der Stelle $t$:
+\begin{align*}
+\dot{\gamma}(t)
+&\uncover<2->{=
+\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
+}
+\\
+&\uncover<3->{=
+\frac{d}{ds}
+\gamma(t+s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<4->{=
+\frac{d}{ds}
+\gamma(t)\gamma(s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<5->{=
+\gamma(t)
+\frac{d}{ds}
+\gamma(s)
+\bigg|_{s=0}
+}
+\uncover<6->{=
+\gamma(t) \dot{\gamma}(0)
+}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Differentialgleichung}
+\vspace{-10pt}
+\[
+\dot{\gamma}(t) = \gamma(t) A
+\quad
+\text{mit}
+\quad
+A=\dot{\gamma}(0)\in LG
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<8->{%
+\begin{block}{Lösung}
+Exponentialfunktion
+\[
+\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<9->{%
+\begin{block}{Kontrolle: Tangentialvektor berechnen}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}e^{At}
+&\uncover<10->{=
+\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!}
+}
+\\
+&\uncover<11->{=
+\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
+}
+\\
+&\uncover<12->{=
+\sum_{k=0} A^k\frac{t^k}{k!}
+A
+}
+\uncover<13->{=
+e^{At} A
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
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
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 2d7b317..e7b4a92 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -1,132 +1,132 @@ -% -% drehung.tex -- Drehung aus streckungen -% -% (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{Drehung aus Streckungen und Scherungen} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.38\textwidth} -\begin{block}{Drehung} -{\color{blue}Längen}, {\color<2->{blue}Winkel}, -{\color<2->{darkgreen}Orientierung} -erhalten -\uncover<2->{ -\[ -\operatorname{SO}(2) -= -{\color{blue}\operatorname{O}(2)} -\cap -{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})} -\]} -\vspace{-20pt} -\end{block} -\uncover<3->{% -\begin{block}{Zusammensetzung} -Eine Drehung muss als Zusammensetzung geschrieben werden können: -\[ -D_{\alpha} -= -\begin{pmatrix} -\cos\alpha & -\sin\alpha\\ -\sin\alpha &\phantom{-}\cos\alpha -\end{pmatrix} -= -DST -\] -\end{block}} -\vspace{-10pt} -\uncover<12->{% -\begin{block}{Beispiel} -\vspace{-12pt} -\[ -D_{60^\circ} -= -{\tiny -\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} -\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} -\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} -} -\] -\end{block}} -\end{column} -\begin{column}{0.58\textwidth} -\uncover<4->{% -\begin{block}{Ansatz} -\vspace{-12pt} -\begin{align*} -DST -&= -\begin{pmatrix} -c^{-1}&0\\ - 0 &c -\end{pmatrix} -\begin{pmatrix} -1&-s\\ -0&1 -\end{pmatrix} -\begin{pmatrix} -1&0\\ -t&1 -\end{pmatrix} -\\ -&\uncover<5->{= -\begin{pmatrix} -c^{-1}&0\\ - 0 &c -\end{pmatrix} -\begin{pmatrix} -1-st&-s\\ - t& 1 -\end{pmatrix} -} -\\ -&\uncover<6->{= -\begin{pmatrix} -{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ -{\color<9->{blue}ct}&{\color<8->{red}c} -\end{pmatrix}} -\uncover<7->{= -\begin{pmatrix} -{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\ -{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha} -\end{pmatrix}} -\end{align*} -\end{block}} -\vspace{-10pt} -\uncover<7->{% -\begin{block}{Koeffizientenvergleich} -\vspace{-15pt} -\begin{align*} -\uncover<8->{ -{\color{red} c} -&= -{\color{red}\cos\alpha }} -&& -& -\uncover<9->{ -{\color{blue} -t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ -\uncover<10->{ -{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} -& -&\Rightarrow& -{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha -} -\\ -\uncover<11->{ -{\color{orange} (1-st)c^{-t}} -&= -\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% drehung.tex -- Drehung aus streckungen
+%
+% (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{Drehung aus Streckungen und Scherungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Drehung}
+{\color{blue}Längen}, {\color<2->{blue}Winkel},
+{\color<2->{darkgreen}Orientierung}
+erhalten
+\uncover<2->{
+\[
+\operatorname{SO}(2)
+=
+{\color{blue}\operatorname{O}(2)}
+\cap
+{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
+\]}
+\vspace{-20pt}
+\end{block}
+\uncover<3->{%
+\begin{block}{Zusammensetzung}
+Eine Drehung muss als Zusammensetzung geschrieben werden können:
+\[
+D_{\alpha}
+=
+\begin{pmatrix}
+\cos\alpha & -\sin\alpha\\
+\sin\alpha &\phantom{-}\cos\alpha
+\end{pmatrix}
+=
+DST
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<12->{%
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
+}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\begin{block}{Ansatz}
+\vspace{-12pt}
+\begin{align*}
+DST
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+ 0 &c
+\end{pmatrix}
+\begin{pmatrix}
+1&-s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&0\\
+t&1
+\end{pmatrix}
+\\
+&\uncover<5->{=
+\begin{pmatrix}
+c^{-1}&0\\
+ 0 &c
+\end{pmatrix}
+\begin{pmatrix}
+1-st&-s\\
+ t& 1
+\end{pmatrix}
+}
+\\
+&\uncover<6->{=
+\begin{pmatrix}
+{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
+{\color<9->{blue}ct}&{\color<8->{red}c}
+\end{pmatrix}}
+\uncover<7->{=
+\begin{pmatrix}
+{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
+{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Koeffizientenvergleich}
+\vspace{-15pt}
+\begin{align*}
+\uncover<8->{
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }}
+&&
+&
+\uncover<9->{
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
+\uncover<10->{
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+}
+\\
+\uncover<11->{
+{\color{orange} (1-st)c^{-t}}
+&=
+\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex index 5171085..e9699a6 100644 --- a/vorlesungen/slides/7/einparameter.tex +++ b/vorlesungen/slides/7/einparameter.tex @@ -1,93 +1,93 @@ -% -% einparameter.tex -- Einparameter Untergruppen -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Einparameter-Untergruppen} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Definition} -Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, -die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit} -\[ -\uncover<3->{ -\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} -} -\] -\end{block} -\uncover<4->{% -\begin{block}{Drehungen} -Drehmatrizen bilden Einparameter- Untergruppen -\begin{align*} -t \mapsto D_{x,t} -&= -\begin{pmatrix} -1&0&0\\ -0&\cos t&-\sin t\\ -0&\sin t& \cos t -\end{pmatrix} -\\ -D_{x,t}D_{x,s} -&= -D_{x,t+s} -\end{align*} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<5->{% -\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} -\vspace{-12pt} -\[ -\begin{pmatrix} -1&s\\ -0&1 -\end{pmatrix} -\begin{pmatrix} -1&t\\ -0&1 -\end{pmatrix} -= -\begin{pmatrix} -1&s+t\\ -0&1 -\end{pmatrix} -\] -\end{block}} -\vspace{-12pt} -\uncover<6->{% -\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} -\vspace{-12pt} -\[ -\begin{pmatrix} -e^s&0\\0&e^{-s} -\end{pmatrix} -\begin{pmatrix} -e^t&0\\0&e^{-t} -\end{pmatrix} -= -\begin{pmatrix} -e^{t+s}&0\\0&e^{-(t+s)} -\end{pmatrix} -\] -\end{block}} -\vspace{-12pt} -\uncover<7->{% -\begin{block}{Gemischt} -\vspace{-12pt} -\begin{gather*} -A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t -\\ -\text{dank}\quad -\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 -=I -\end{gather*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% einparameter.tex -- Einparameter Untergruppen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Einparameter-Untergruppen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
+die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit}
+\[
+\uncover<3->{
+\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
+}
+\]
+\end{block}
+\uncover<4->{%
+\begin{block}{Drehungen}
+Drehmatrizen bilden Einparameter- Untergruppen
+\begin{align*}
+t \mapsto D_{x,t}
+&=
+\begin{pmatrix}
+1&0&0\\
+0&\cos t&-\sin t\\
+0&\sin t& \cos t
+\end{pmatrix}
+\\
+D_{x,t}D_{x,s}
+&=
+D_{x,t+s}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+1&s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&t\\
+0&1
+\end{pmatrix}
+=
+\begin{pmatrix}
+1&s+t\\
+0&1
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<6->{%
+\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+e^s&0\\0&e^{-s}
+\end{pmatrix}
+\begin{pmatrix}
+e^t&0\\0&e^{-t}
+\end{pmatrix}
+=
+\begin{pmatrix}
+e^{t+s}&0\\0&e^{-(t+s)}
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<7->{%
+\begin{block}{Gemischt}
+\vspace{-12pt}
+\begin{gather*}
+A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
+\\
+\text{dank}\quad
+\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
+=I
+\end{gather*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile index cc67c8a..9de1c34 100644 --- a/vorlesungen/slides/7/images/Makefile +++ b/vorlesungen/slides/7/images/Makefile @@ -1,19 +1,19 @@ -# -# Makefile -- Illustrationen zu -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -all: rodriguez.jpg - -rodriguez.png: rodriguez.pov - povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov - -rodriguez.jpg: rodriguez.png - convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg - -commutator: commutator.ini commutator.pov common.inc - povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini -jpg: - for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done - - +#
+# Makefile -- Illustrationen zu
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all: rodriguez.jpg
+
+rodriguez.png: rodriguez.pov
+ povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
+
+rodriguez.jpg: rodriguez.png
+ convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
+
+commutator: commutator.ini commutator.pov common.inc
+ povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
+jpg:
+ for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
+
+
diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc index 0e27c9a..b028956 100644 --- a/vorlesungen/slides/7/images/common.inc +++ b/vorlesungen/slides/7/images/common.inc @@ -1,70 +1,70 @@ -// -// common.inc -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#version 3.7; -#include "colors.inc" - -global_settings { - assumed_gamma 1 -} - -#declare imagescale = 0.025; -#declare O = <0, 0, 0>; -#declare at = 0.015; - -camera { - location <18, 15, -50> - look_at <0.0, 0.5, 0> - right 16/9 * x * imagescale - up y * imagescale -} - -light_source { - <-40, 30, -50> color White - area_light <1,0,0> <0,0,1>, 10, 10 - adaptive 1 - jitter -} - -sky_sphere { - pigment { - color rgb<1,1,1> - } -} - -#macro arrow(from, to, arrowthickness, c) -#declare arrowdirection = vnormalize(to - from); -#declare arrowlength = vlength(to - from); -union { - sphere { - from, 1.1 * arrowthickness - } - cylinder { - from, - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - arrowthickness - } - cone { - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - 2 * arrowthickness, - to, - 0 - } - pigment { - color c - } - finish { - specular 0.9 - metallic - } -} -#end - -#declare l = 1.2; - -arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) -arrow(< 0, 0, -l >, < 0, 0, l >, at, White) -arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) - +//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.025;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <18, 15, -50>
+ look_at <0.0, 0.5, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-40, 30, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare l = 1.2;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini index 8c2211e..44a5ac5 100644 --- a/vorlesungen/slides/7/images/commutator.ini +++ b/vorlesungen/slides/7/images/commutator.ini @@ -1,8 +1,8 @@ -Input_File_Name=commutator.pov -Initial_Frame=1 -Final_Frame=60 -Initial_Clock=1 -Final_Clock=60 -Cyclic_Animation=off -Pause_when_Done=off - +Input_File_Name=commutator.pov
+Initial_Frame=1
+Final_Frame=60
+Initial_Clock=1
+Final_Clock=60
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m index 5a448db..3f5ea17 100644 --- a/vorlesungen/slides/7/images/commutator.m +++ b/vorlesungen/slides/7/images/commutator.m @@ -1,111 +1,111 @@ -# -# commutator.m -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# - -X = [ - 0, 0, 0; - 0, 0, -1; - 0, 1, 0 -]; - -Y = [ - 0, 0, 1; - 0, 0, 0; - -1, 0, 0 -]; - -Z = [ - 0, -1, 0; - 1, 0, 0; - 0, 0, 0 -]; - -function retval = Dx(alpha) - retval = [ - 1, 0, 0 ; - 0, cos(alpha), -sin(alpha); - 0, sin(alpha), cos(alpha) - ]; -end - -function retval = Dy(beta) - retval = [ - cos(beta), 0, sin(beta); - 0, 1, 0 ; - -sin(beta), 0, cos(beta) - ]; -end - -t = 0.9; -P = Dx(t) * Dy(t) -Q = Dy(t) * Dx(t) -P - Q -(P - Q) * [0;0;1] - -function retval = kurven(filename, t) - retval = -1; - N = 20; - fn = fopen(filename, "w"); - fprintf(fn, "//\n"); - fprintf(fn, "// %s\n", filename); - fprintf(fn, "//\n"); - fprintf(fn, "#macro XYkurve()\n"); - for i = (0:N-1) - v1 = Dx(t * i / N) * [0;0;1]; - v2 = Dx(t * (i+1) / N) * [0;0;1]; - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1)); - fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); - end - for i = (0:N-1) - v1 = Dx(t) * Dy(t * i / N) * [0;0;1]; - v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1]; - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1)); - fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); - end - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v2(1,1), v2(3,1), v2(2,1)); - fprintf(fn, "#end\n"); - fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n", - v2(1,1), v2(3,1), v2(2,1)); - fprintf(fn, "#macro YXkurve()\n"); - for i = (0:N-1) - v1 = Dy(t * i / N) * [0;0;1]; - v2 = Dy(t * (i+1) / N) * [0;0;1]; - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1)); - fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); - end - for i = (0:N-1) - v1 = Dy(t) * Dx(t * i / N) * [0;0;1]; - v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1]; - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1)); - fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", - v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); - end - fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", - v2(1,1), v2(3,1), v2(2,1)); - fprintf(fn, "#end\n"); - fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n", - v2(1,1), v2(3,1), v2(2,1)); - - fclose(fn); - retval = 0; -end - -function retval = kurve(i) - n = pi / 180; - filename = sprintf("f/%04d.inc", i); - kurven(filename, n * i); -end - -for i = (1:60) - kurve(i); -end +#
+# commutator.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+X = [
+ 0, 0, 0;
+ 0, 0, -1;
+ 0, 1, 0
+];
+
+Y = [
+ 0, 0, 1;
+ 0, 0, 0;
+ -1, 0, 0
+];
+
+Z = [
+ 0, -1, 0;
+ 1, 0, 0;
+ 0, 0, 0
+];
+
+function retval = Dx(alpha)
+ retval = [
+ 1, 0, 0 ;
+ 0, cos(alpha), -sin(alpha);
+ 0, sin(alpha), cos(alpha)
+ ];
+end
+
+function retval = Dy(beta)
+ retval = [
+ cos(beta), 0, sin(beta);
+ 0, 1, 0 ;
+ -sin(beta), 0, cos(beta)
+ ];
+end
+
+t = 0.9;
+P = Dx(t) * Dy(t)
+Q = Dy(t) * Dx(t)
+P - Q
+(P - Q) * [0;0;1]
+
+function retval = kurven(filename, t)
+ retval = -1;
+ N = 20;
+ fn = fopen(filename, "w");
+ fprintf(fn, "//\n");
+ fprintf(fn, "// %s\n", filename);
+ fprintf(fn, "//\n");
+ fprintf(fn, "#macro XYkurve()\n");
+ for i = (0:N-1)
+ v1 = Dx(t * i / N) * [0;0;1];
+ v2 = Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
+ v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#macro YXkurve()\n");
+ for i = (0:N-1)
+ v1 = Dy(t * i / N) * [0;0;1];
+ v2 = Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
+ v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+
+ fclose(fn);
+ retval = 0;
+end
+
+function retval = kurve(i)
+ n = pi / 180;
+ filename = sprintf("f/%04d.inc", i);
+ kurven(filename, n * i);
+end
+
+for i = (1:60)
+ kurve(i);
+end
diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov index 9ae11b9..8229a06 100644 --- a/vorlesungen/slides/7/images/commutator.pov +++ b/vorlesungen/slides/7/images/commutator.pov @@ -1,59 +1,59 @@ -// -// commutator.pov -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#include "common.inc" - -sphere { O, 0.99 - pigment { - color rgbt<1,1,1,0.5> - } - finish { - specular 0.9 - metallic - } -} - -#declare filename = concat("f/", str(clock, -4, 0), ".inc"); - -#include filename - -#declare n1 = vcross(<0,1,0>, finalXY); -#declare n2 = vcross(<0,1,0>, finalYX); - -intersection { - sphere { O, 1 } - plane { -n1, 0 } - plane { n2, 0 } - pigment { - color rgb<0,0.4,0.1> - } - finish { - specular 0.9 - metallic - } -} - -union { - XYkurve() - pigment { - color Red - } - finish { - specular 0.9 - metallic - } -} - -union { - YXkurve() - pigment { - color Blue - } - finish { - specular 0.9 - metallic - } -} - +//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+sphere { O, 0.99
+ pigment {
+ color rgbt<1,1,1,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+#declare filename = concat("f/", str(clock, -4, 0), ".inc");
+
+#include filename
+
+#declare n1 = vcross(<0,1,0>, finalXY);
+#declare n2 = vcross(<0,1,0>, finalYX);
+
+intersection {
+ sphere { O, 1 }
+ plane { -n1, 0 }
+ plane { n2, 0 }
+ pigment {
+ color rgb<0,0.4,0.1>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ XYkurve()
+ pigment {
+ color Red
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ YXkurve()
+ pigment {
+ color Blue
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov index 07aec19..62306f8 100644 --- a/vorlesungen/slides/7/images/rodriguez.pov +++ b/vorlesungen/slides/7/images/rodriguez.pov @@ -1,118 +1,118 @@ -// -// rodriguez.pov -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#version 3.7; -#include "colors.inc" - -global_settings { - assumed_gamma 1 -} - -#declare imagescale = 0.020; -#declare O = <0, 0, 0>; -#declare at = 0.015; - -camera { - location <8, 15, -50> - look_at <0.1, 0.475, 0> - right 16/9 * x * imagescale - up y * imagescale -} - -light_source { - <-4, 20, -50> color White - area_light <1,0,0> <0,0,1>, 10, 10 - adaptive 1 - jitter -} - -sky_sphere { - pigment { - color rgb<1,1,1> - } -} - -#macro arrow(from, to, arrowthickness, c) -#declare arrowdirection = vnormalize(to - from); -#declare arrowlength = vlength(to - from); -union { - sphere { - from, 1.1 * arrowthickness - } - cylinder { - from, - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - arrowthickness - } - cone { - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - 2 * arrowthickness, - to, - 0 - } - pigment { - color c - } - finish { - specular 0.9 - metallic - } -} -#end - -#declare K = vnormalize(<0.2,1,0.1>); -#declare X = vnormalize(<1.1,1,-1.2>); -#declare O = <0,0,0>; - -#declare r = vlength(vcross(K, X)) / vlength(K); - -#declare l = 1.0; - -arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) -arrow(< 0, 0, -l >, < 0, 0, l >, at, White) -arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) - -arrow(O, X, at, Red) -arrow(O, K, at, Blue) - -#macro punkt(H,phi) - ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K) -#end - -cone { vdot(K, X) * K, r, O, 0 - pigment { - color rgbt<0.6,0.6,0.6,0.5> - } - finish { - specular 0.9 - metallic - } -} - - -union { - #declare phistep = pi / 100; - #declare phi = 0; - #while (phi < 2 * pi - phistep/2) - sphere { punkt(K, phi), at/2 } - cylinder { - punkt(X, phi), - punkt(X, phi + phistep), - at/2 - } - #declare phi = phi + phistep; - #end - pigment { - color Orange - } - finish { - specular 0.9 - metallic - } -} - -arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow) -#declare Darkgreen = rgb<0,0.5,0>; -arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen) +//
+// rodriguez.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.020;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <8, 15, -50>
+ look_at <0.1, 0.475, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-4, 20, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare K = vnormalize(<0.2,1,0.1>);
+#declare X = vnormalize(<1.1,1,-1.2>);
+#declare O = <0,0,0>;
+
+#declare r = vlength(vcross(K, X)) / vlength(K);
+
+#declare l = 1.0;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
+arrow(O, X, at, Red)
+arrow(O, K, at, Blue)
+
+#macro punkt(H,phi)
+ ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
+#end
+
+cone { vdot(K, X) * K, r, O, 0
+ pigment {
+ color rgbt<0.6,0.6,0.6,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+
+union {
+ #declare phistep = pi / 100;
+ #declare phi = 0;
+ #while (phi < 2 * pi - phistep/2)
+ sphere { punkt(K, phi), at/2 }
+ cylinder {
+ punkt(X, phi),
+ punkt(X, phi + phistep),
+ at/2
+ }
+ #declare phi = phi + phistep;
+ #end
+ pigment {
+ color Orange
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
+#declare Darkgreen = rgb<0,0.5,0>;
+arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index 84bf034..9000160 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -1,166 +1,166 @@ -% -% template.tex -- slide template -% -% (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{Kommutator in $\operatorname{SO}(3)$} -\vspace{-20pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\t{14.0cm} -\ifthenelse{\boolean{presentation}}{ -\only<1>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};} -\only<2>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};} -\only<3>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};} -\only<4>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};} 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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}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kommutator in $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\t{14.0cm}
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+ (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6);
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+\egroup
diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index e0690eb..bca8417 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -1,104 +1,104 @@ -% -% kurven.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Kurven und Tangenten} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Kurven} -Kurve in $\mathbb{R}^n$: -\vspace{-12pt} -\[ -\gamma -\colon -I=[a,b] \to \mathbb{R}^n -: -t\mapsto \gamma(t) -\uncover<2->{ -= -\begin{pmatrix} -x_1(t)\\ -x_2(t)\\ -\vdots\\ -x_n(t) -\end{pmatrix} -} -\] -\vspace{-15pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\coordinate (A) at (1,0.5); -\coordinate (B) at (4,0.5); -\coordinate (C) at (2,2.2); -\coordinate (D) at (5,2); -\coordinate (E) at ($(C)+(80:2)$); - -\draw[color=red,line width=1.4pt] - (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D); -\fill[color=red] (C) circle[radius=0.06]; -\node[color=red] at (C) [left] {$\gamma(t)$}; - -\uncover<4->{ - \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); - \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; -} - -\uncover<2->{ - \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; - \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; -} -\end{tikzpicture} -\end{center} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Tangenten} -Ableitung -\[ -\frac{d}{dt}\gamma(t) -= -\dot{\gamma}(t) -= -\begin{pmatrix} -\dot{x}_1(t)\\ -\dot{x}_2(t)\\ -\vdots\\ -\dot{x}_n(t) -\end{pmatrix} -\] -\uncover<5->{% -Lineare Approximation: -\[ -\gamma(t+h) -= -\gamma(t) -+ -\dot{\gamma}(t) \cdot h -+ -o(h) -\]}% -\vspace{-10pt} -\begin{itemize} -\item<6-> -Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ -in $\mathbb{R}^n$ liegen -\item<7-> -Gilt auch für -\[ -\operatorname{GL}_n(\mathbb{R}) -\uncover<8->{\subset M_n(\mathbb{R})} -\uncover<9->{ = \mathbb{R}^{n\times n}} -\] -\end{itemize} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% kurven.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kurven und Tangenten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Kurven}
+Kurve in $\mathbb{R}^n$:
+\vspace{-12pt}
+\[
+\gamma
+\colon
+I=[a,b] \to \mathbb{R}^n
+:
+t\mapsto \gamma(t)
+\uncover<2->{
+=
+\begin{pmatrix}
+x_1(t)\\
+x_2(t)\\
+\vdots\\
+x_n(t)
+\end{pmatrix}
+}
+\]
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (1,0.5);
+\coordinate (B) at (4,0.5);
+\coordinate (C) at (2,2.2);
+\coordinate (D) at (5,2);
+\coordinate (E) at ($(C)+(80:2)$);
+
+\draw[color=red,line width=1.4pt]
+ (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
+\fill[color=red] (C) circle[radius=0.06];
+\node[color=red] at (C) [left] {$\gamma(t)$};
+
+\uncover<4->{
+ \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
+ \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
+}
+
+\uncover<2->{
+ \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
+ \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Tangenten}
+Ableitung
+\[
+\frac{d}{dt}\gamma(t)
+=
+\dot{\gamma}(t)
+=
+\begin{pmatrix}
+\dot{x}_1(t)\\
+\dot{x}_2(t)\\
+\vdots\\
+\dot{x}_n(t)
+\end{pmatrix}
+\]
+\uncover<5->{%
+Lineare Approximation:
+\[
+\gamma(t+h)
+=
+\gamma(t)
++
+\dot{\gamma}(t) \cdot h
++
+o(h)
+\]}%
+\vspace{-10pt}
+\begin{itemize}
+\item<6->
+Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
+in $\mathbb{R}^n$ liegen
+\item<7->
+Gilt auch für
+\[
+\operatorname{GL}_n(\mathbb{R})
+\uncover<8->{\subset M_n(\mathbb{R})}
+\uncover<9->{ = \mathbb{R}^{n\times n}}
+\]
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 574467b..59c9121 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -1,85 +1,85 @@ -% -% liealgebra.tex -- Lie-Algebra -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Lie-Algebra} -\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} -\begin{block}{Vektorraum} -Tangentialvektoren im Punkt $I$: -\begin{center} -\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} -\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ -\hline -\uncover<2->{ -\operatorname{GL}_n(\mathbb{R}) -& beliebige Matrizen -& M_n(\mathbb{R}) -} -\\ -\uncover<3->{ -\operatorname{O(n)} -& antisymmetrische Matrizen -& \operatorname{o}(n) -} -\\ -\uncover<4->{ -\operatorname{SL}_n(\mathbb{R}) -& spurlose Matrizen -& \operatorname{sl}_2(\mathbb{R}) -} -\\ -\uncover<5->{ -\operatorname{U(n)} -& antihermitesche Matrizen -& \operatorname{u}(n) -} -\\ -\uncover<6->{ -\operatorname{SU(n)} -& spurlose, antihermitesche Matrizen -& \operatorname{su}(n) -} -\end{tabular} -\end{center} -\end{block} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.40\textwidth} -\uncover<7->{% -\begin{block}{Lie-Klammer} -Kommutator: $[A,B] = AB-BA$ -\end{block}} -\uncover<8->{% -\begin{block}{Nachprüfen} -$[A,B]\in LG$ -für $A,B\in LG$ -\end{block}} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<9->{% -\begin{block}{Algebraische Eigenschaften} -\begin{itemize} -\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ -\item<11-> Jacobi-Identität -\[ -[A,[B,C]]+ -[B,[C,A]]+ -[C,[A,B]] -= 0 -\] -\end{itemize} -\vspace{-13pt} -\uncover<12->{% -{\usebeamercolor[fg]{title} -Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% liealgebra.tex -- Lie-Algebra
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra}
+\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
+\begin{block}{Vektorraum}
+Tangentialvektoren im Punkt $I$:
+\begin{center}
+\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
+\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
+\hline
+\uncover<2->{
+\operatorname{GL}_n(\mathbb{R})
+& beliebige Matrizen
+& M_n(\mathbb{R})
+}
+\\
+\uncover<3->{
+\operatorname{O(n)}
+& antisymmetrische Matrizen
+& \operatorname{o}(n)
+}
+\\
+\uncover<4->{
+\operatorname{SL}_n(\mathbb{R})
+& spurlose Matrizen
+& \operatorname{sl}_2(\mathbb{R})
+}
+\\
+\uncover<5->{
+\operatorname{U(n)}
+& antihermitesche Matrizen
+& \operatorname{u}(n)
+}
+\\
+\uncover<6->{
+\operatorname{SU(n)}
+& spurlose, antihermitesche Matrizen
+& \operatorname{su}(n)
+}
+\end{tabular}
+\end{center}
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\uncover<7->{%
+\begin{block}{Lie-Klammer}
+Kommutator: $[A,B] = AB-BA$
+\end{block}}
+\uncover<8->{%
+\begin{block}{Nachprüfen}
+$[A,B]\in LG$
+für $A,B\in LG$
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<9->{%
+\begin{block}{Algebraische Eigenschaften}
+\begin{itemize}
+\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
+\item<11-> Jacobi-Identität
+\[
+[A,[B,C]]+
+[B,[C,A]]+
+[C,[A,B]]
+= 0
+\]
+\end{itemize}
+\vspace{-13pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}
+Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index 077dc9d..f88042a 100644 --- a/vorlesungen/slides/7/mannigfaltigkeit.tex +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -1,46 +1,46 @@ -% -% mannigfaltigkeit.tex -- Mannigfaltigkeit -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Mannigfaltigkeit} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf} -\end{center} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{Definition} -\begin{itemize} -\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ -\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet -\[ -\varphi_\beta\circ\varphi_\alpha^{-1} -\colon -\varphi_\alpha(U_\alpha\cap U_\beta) -\to -\varphi_\beta(U_\alpha\cap U_\beta) -\] -\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken -\end{itemize} -\end{block} -\vspace{-7pt} -\uncover<5->{% -\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} -Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus -\end{block}} -\vspace{-3pt} -\uncover<6->{% -\begin{block}{Lie-Gruppe} -Gruppe und Mannigfaltigkeit -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% mannigfaltigkeit.tex -- Mannigfaltigkeit
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Mannigfaltigkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
+\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
+\[
+\varphi_\beta\circ\varphi_\alpha^{-1}
+\colon
+\varphi_\alpha(U_\alpha\cap U_\beta)
+\to
+\varphi_\beta(U_\alpha\cap U_\beta)
+\]
+\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
+\end{itemize}
+\end{block}
+\vspace{-7pt}
+\uncover<5->{%
+\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
+Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
+\end{block}}
+\vspace{-3pt}
+\uncover<6->{%
+\begin{block}{Lie-Gruppe}
+Gruppe und Mannigfaltigkeit
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index 52c8e4a..afc67c5 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -1,107 +1,107 @@ -% -% parameter.tex -- Parametrisierung der Matrizen -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\definecolor{darkgreen}{rgb}{0,0.6,0} -\definecolor{darkyellow}{rgb}{1,0.8,0} -\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} -\vspace{-12pt} -\begin{align*} -\uncover<2->{ -D_{x,\alpha} -&= -\begin{pmatrix} -1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha -\end{pmatrix} -} -\\ -\uncover<3->{ -D_{y,\beta} -&= -\begin{pmatrix} -\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta -\end{pmatrix} -} -\\ -\uncover<4->{ -D_{z,\gamma} -&= -\begin{pmatrix} -\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 -\end{pmatrix} -} -\intertext{\uncover<5->{beliebige Drehung:}} -\uncover<5->{ -D -&= -D_{x,\alpha} -D_{y,\beta} -D_{z,\gamma} -} -\end{align*} -\end{block} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<6->{% -\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} -\uncover<7->{% -$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel -} -\\ -\uncover<8->{% -$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse -} -\[ -\uncover<9->{ -{\color{red}\vec{x}} -\mapsto -} -\uncover<10->{ -({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) -\cos\omega -+ -} -\uncover<11->{ -({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega -+ -} -\uncover<9->{ -{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) -} -\] -\vspace{-40pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\uncover<9->{ - \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) {$\vec{k}$}; -} -\uncover<11->{ - \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; -} -\uncover<10->{ - \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} -\uncover<5->{% -{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine -dreidimensionale Gruppe} -\end{frame} -\egroup +%
+% parameter.tex -- Parametrisierung der Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkyellow}{rgb}{1,0.8,0}
+\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}
+\vspace{-12pt}
+\begin{align*}
+\uncover<2->{
+D_{x,\alpha}
+&=
+\begin{pmatrix}
+1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
+\end{pmatrix}
+}
+\\
+\uncover<3->{
+D_{y,\beta}
+&=
+\begin{pmatrix}
+\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta
+\end{pmatrix}
+}
+\\
+\uncover<4->{
+D_{z,\gamma}
+&=
+\begin{pmatrix}
+\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
+\end{pmatrix}
+}
+\intertext{\uncover<5->{beliebige Drehung:}}
+\uncover<5->{
+D
+&=
+D_{x,\alpha}
+D_{y,\beta}
+D_{z,\gamma}
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<6->{%
+\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional}
+\uncover<7->{%
+$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
+}
+\\
+\uncover<8->{%
+$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
+}
+\[
+\uncover<9->{
+{\color{red}\vec{x}}
+\mapsto
+}
+\uncover<10->{
+({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
+\cos\omega
++
+}
+\uncover<11->{
+({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
++
+}
+\uncover<9->{
+{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x})
+}
+\]
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+ \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) {$\vec{k}$};
+}
+\uncover<11->{
+ \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$};
+}
+\uncover<10->{
+ \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}
+\uncover<5->{%
+{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
+dreidimensionale Gruppe}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 66b8d27..d74b7d0 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -1,117 +1,117 @@ -% -% semi.tex -- Beispiele: semidirekte Produkte -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Drehung/Skalierung und Verschiebung} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Skalierung und Verschiebung} -Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ -\\ -Wirkung auf $\mathbb{R}$: -\[ -x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t -\] -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<2->{% -\begin{block}{Drehung und Verschiebung} -Gruppe -$G= -\{ (\alpha,\vec{t}) -\;|\; -\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 -\}$ -Wirkung auf $\mathbb{R}^2$: -\[ -\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} -\] -\end{block}} -\end{column} -\end{columns} -\vspace{-15pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<3->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(e^{s_1},t_1)(e^{s_2},t_2)x -&\uncover<4->{= -(e^{s_1},t_1)(e^{s_2}x+t_2)} -\\ -&\uncover<5->{= -e^{s_1+s_2}x + e^{s_1}t_2+t_1} -\\ -\uncover<6->{ -(e^{s_1},t_1)(e^{s_2},t_2) -&= -(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} -\end{align*} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<7->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(\alpha_1,\vec{t}_1) -(\alpha_2,\vec{t}_2) -\vec{x} -&\uncover<8->{= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} -\\ -&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} -\\ -\uncover<10->{ -(\alpha_1,\vec{t}_1) -(\alpha_2,\vec{t}_2) -&= -(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\vspace{-10pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<11->{% -\begin{block}{Matrixschreibweise} -\vspace{-12pt} -\[ -g=(e^s,t) = -\begin{pmatrix} -e^s&t\\ -0&1 -\end{pmatrix} -\quad\text{auf}\quad -\begin{pmatrix}x\\1\end{pmatrix} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<12->{% -\begin{block}{Matrixschreibweise} -\vspace{-12pt} -\[ -g=(\alpha,\vec{t}) = -\begin{pmatrix} -D_{\alpha}&\vec{t}\\ -0&1 -\end{pmatrix} -\quad\text{auf}\quad -\begin{pmatrix}\vec{x}\\1\end{pmatrix} -\] -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% semi.tex -- Beispiele: semidirekte Produkte
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung/Skalierung und Verschiebung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalierung und Verschiebung}
+Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
+\\
+Wirkung auf $\mathbb{R}$:
+\[
+x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Drehung und Verschiebung}
+Gruppe
+$G=
+\{ (\alpha,\vec{t})
+\;|\;
+\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
+\}$
+Wirkung auf $\mathbb{R}^2$:
+\[
+\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(e^{s_1},t_1)(e^{s_2},t_2)x
+&\uncover<4->{=
+(e^{s_1},t_1)(e^{s_2}x+t_2)}
+\\
+&\uncover<5->{=
+e^{s_1+s_2}x + e^{s_1}t_2+t_1}
+\\
+\uncover<6->{
+(e^{s_1},t_1)(e^{s_2},t_2)
+&=
+(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+\vec{x}
+&\uncover<8->{=
+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
+\\
+&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
+\\
+\uncover<10->{
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+&=
+(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(e^s,t) =
+\begin{pmatrix}
+e^s&t\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}x\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(\alpha,\vec{t}) =
+\begin{pmatrix}
+D_{\alpha}&\vec{t}\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}\vec{x}\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex index a65b4f6..58e87a1 100644 --- a/vorlesungen/slides/7/sl2.tex +++ b/vorlesungen/slides/7/sl2.tex @@ -1,242 +1,242 @@ -% -% sl2.tex -- Beispiel: Parametrisierung von SL_2(R) -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t,fragile] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.44\textwidth} -\begin{block}{Determinante} -\[ -A=\begin{pmatrix} -a&b\\ -c&d -\end{pmatrix} -\;\Rightarrow\; -\det A = ad-bc -\] -\end{block} -\end{column} -\begin{column}{0.52\textwidth} -\begin{block}{Dimension} -\[ -4\; \text{Variablen} -- -1\; \text{Bedingung} -= -3\; \text{Dimensionen} -\] -\end{block} -\end{column} -\end{columns} -\vspace{-10pt} -\uncover<3->{% -\begin{columns}[t,onlytextwidth] -\def\s{0.94} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \fill[color=blue!20] (-1,-1) rectangle (1,1); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {4,...,10}{ - \only<\d>{ - \pgfmathparse{1+(\d-4)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - ({-\t},{-1/\t}) rectangle (\t,{1/\t}); - \foreach \x in {-2,...,2}{ - \draw[color=red,line width=0.3pt] - ({\x*\t},-3) -- ({\x*\t},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{\y/\t}) -- (3,{\y/\t}); - } - } - } - }{} - \uncover<11->{ - \xdef\t{1.6} - \fill[color=red!40,opacity=0.5] - ({-\t},{-1/\t}) rectangle (\t,{1/\t}); - \foreach \x in {-2,...,2}{ - \draw[color=red,line width=0.3pt] - ({\x*\t},-3) -- ({\x*\t},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{\y/\t}) -- (3,{\y/\t}); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<3->{% - \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - D - = - \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\fill[color=blue!20] (-1,-1) rectangle (1,1); -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {11,...,17}{ - \only<\d>{ - \pgfmathparse{(\d-11)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - ({-1+\t*(-1)},{-1}) - -- - ({1+\t*(-1)},{-1}) - -- - ({1+\t},{1}) - -- - ({-1+\t},{1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({-3+\t*\y},\y) -- ({3+\t*\y},\y); - } - } - } - }{} - \uncover<18->{ - \xdef\t{0.6} - \fill[color=red!40,opacity=0.5] - ({-1+\t*(-1)},{-1}) - -- - ({1+\t*(-1)},{-1}) - -- - ({1+\t},{1}) - -- - ({-1+\t},{1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({-3+\t*\y},\y) -- ({3+\t*\y},\y); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<11->{ - \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - S - = - \begin{pmatrix} 1&s\\ 0&1\end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\fill[color=blue!20] (-1,-1) rectangle (1,1); -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {18,...,24}{ - \only<\d>{ - \pgfmathparse{(\d-18)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - (-1,{\t*(-1)-1}) - -- - (1,{\t*1-1}) - -- - (1,{\t*1+1}) - -- - (-1,{\t*(-1)+1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); - } - } - } - }{} - \uncover<25->{ - \xdef\t{0.6} - \fill[color=red!40,opacity=0.5] - (-1,{\t*(-1)-1}) - -- - (1,{\t*1-1}) - -- - (1,{\t*1+1}) - -- - (-1,{\t*(-1)+1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<18->{% -\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - T - = - \begin{pmatrix} 1&0\\t&1\end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\end{columns}} -\end{frame} -\egroup +%
+% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t,fragile]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Determinante}
+\[
+A=\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\;\Rightarrow\;
+\det A = ad-bc
+\]
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Dimension}
+\[
+4\; \text{Variablen}
+-
+1\; \text{Bedingung}
+=
+3\; \text{Dimensionen}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{columns}[t,onlytextwidth]
+\def\s{0.94}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \fill[color=blue!20] (-1,-1) rectangle (1,1);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {4,...,10}{
+ \only<\d>{
+ \pgfmathparse{1+(\d-4)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+ }
+ }{}
+ \uncover<11->{
+ \xdef\t{1.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<3->{%
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ D
+ =
+ \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {11,...,17}{
+ \only<\d>{
+ \pgfmathparse{(\d-11)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+ }
+ }{}
+ \uncover<18->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<11->{
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ S
+ =
+ \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {18,...,24}{
+ \only<\d>{
+ \pgfmathparse{(\d-18)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+ }
+ }{}
+ \uncover<25->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<18->{%
+\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ T
+ =
+ \begin{pmatrix} 1&0\\t&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex index 35d62d8..8931a24 100644 --- a/vorlesungen/slides/7/symmetrien.tex +++ b/vorlesungen/slides/7/symmetrien.tex @@ -1,145 +1,145 @@ -% -% symmetrien.tex -- Symmetrien -% -% (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{Symmetrien} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Diskrete Symmetrien} -\begin{itemize} -\item<2-> -Ebenen-Spiegelung: -\[ -{\tiny -\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} -} -\mapsto -{\tiny -\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} -} -\uncover<4->{\!,\; -\vec{x} -\mapsto -\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} -} -\] -\vspace{-10pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\a{10} -\def\b{50} -\def\r{2} -\coordinate (O) at (0,0); -\coordinate (A) at (\b:\r); -\coordinate (B) at ({180+2*\a-\b}:\r); -\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); -\coordinate (N) at (\a:2); -\coordinate (D) at (\a:{\r*cos(\b-\a)}); -\uncover<3->{ -\clip (-2.5,-0.45) rectangle (2.5,1.95); - - \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) - -- cycle; - \draw[->,color=darkgreen] (O) -- (N); - \node[color=darkgreen] at (N) [above] {$\vec{n}$}; - - - \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) - -- cycle; - \fill[color=red] (O) circle[radius=0.06]; - \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); - \fill[color=blue] (C) circle[radius=0.06]; - \draw[color=blue,line width=0.1pt] (A) -- (D); - \node[color=darkgreen] at (D) [below,rotate=\a] - {$(\vec{n}\cdot\vec{x})\vec{n}$}; - \draw[color=blue,line width=0.5pt] (A)--(B); - - \node[color=blue] at (A) [above right] {$\vec{x}$}; - \node[color=blue] at (B) [above left] {$\vec{x}'$}; - - \node[color=red] at (O) [below left] {$O$}; - - \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); - \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); -} - -\end{tikzpicture} -\end{center} -\vspace{-5pt} -$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ -\item<5-> -Punkt-Spiegelung: -\[ -{\tiny -\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} -} -\mapsto -- -{\tiny -\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*} -} -\] -\end{itemize} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<6->{% -\begin{block}{Kontinuierliche Symmetrien} -\begin{itemize} -\item<7-> Translation: -\( -\vec{x} \mapsto \vec{x} + \vec{t} -\) -\item<8-> Drehung: -\vspace{-3pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\a{25} -\def\r{1.3} -\coordinate (O) at (0,0); -\begin{scope} -\clip (-1.1,-0.1) rectangle (2.3,2.3); -\draw[color=red] (O) circle[radius=2]; -\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle; -\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; -\node at ({0.5*\a}:1) {$\alpha$}; -\node at ({90+0.5*\a}:1) {$\alpha$}; -\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); -\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); -\end{scope} -\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; -\end{tikzpicture} -\end{center} -\[ -\uncover<9->{% -\begin{pmatrix}x\\y\end{pmatrix} -\mapsto -\begin{pmatrix} -{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\ -{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} -\end{pmatrix} -\begin{pmatrix}x\\y\end{pmatrix} -} -\] -\end{itemize} -\end{block}} -\vspace{-10pt} -\uncover<10->{% -\begin{block}{Definition} -Längen/Winkel bleiben erhalten -\\ -\uncover<11->{% -$\Rightarrow$ $\exists$ Erhaltungsgrösse} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% symmetrien.tex -- Symmetrien
+%
+% (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{Symmetrien}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diskrete Symmetrien}
+\begin{itemize}
+\item<2->
+Ebenen-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+{\tiny
+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\uncover<4->{\!,\;
+\vec{x}
+\mapsto
+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
+}
+\]
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{10}
+\def\b{50}
+\def\r{2}
+\coordinate (O) at (0,0);
+\coordinate (A) at (\b:\r);
+\coordinate (B) at ({180+2*\a-\b}:\r);
+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
+\coordinate (N) at (\a:2);
+\coordinate (D) at (\a:{\r*cos(\b-\a)});
+\uncover<3->{
+\clip (-2.5,-0.45) rectangle (2.5,1.95);
+
+ \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
+ -- cycle;
+ \draw[->,color=darkgreen] (O) -- (N);
+ \node[color=darkgreen] at (N) [above] {$\vec{n}$};
+
+
+ \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
+ -- cycle;
+ \fill[color=red] (O) circle[radius=0.06];
+ \draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
+ \fill[color=blue] (C) circle[radius=0.06];
+ \draw[color=blue,line width=0.1pt] (A) -- (D);
+ \node[color=darkgreen] at (D) [below,rotate=\a]
+ {$(\vec{n}\cdot\vec{x})\vec{n}$};
+ \draw[color=blue,line width=0.5pt] (A)--(B);
+
+ \node[color=blue] at (A) [above right] {$\vec{x}$};
+ \node[color=blue] at (B) [above left] {$\vec{x}'$};
+
+ \node[color=red] at (O) [below left] {$O$};
+
+ \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
+ \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-5pt}
+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
+\item<5->
+Punkt-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+-
+{\tiny
+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Kontinuierliche Symmetrien}
+\begin{itemize}
+\item<7-> Translation:
+\(
+\vec{x} \mapsto \vec{x} + \vec{t}
+\)
+\item<8-> Drehung:
+\vspace{-3pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{25}
+\def\r{1.3}
+\coordinate (O) at (0,0);
+\begin{scope}
+\clip (-1.1,-0.1) rectangle (2.3,2.3);
+\draw[color=red] (O) circle[radius=2];
+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
+\node at ({0.5*\a}:1) {$\alpha$};
+\node at ({90+0.5*\a}:1) {$\alpha$};
+\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
+\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
+\end{scope}
+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
+\end{tikzpicture}
+\end{center}
+\[
+\uncover<9->{%
+\begin{pmatrix}x\\y\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
+\end{pmatrix}
+\begin{pmatrix}x\\y\end{pmatrix}
+}
+\]
+\end{itemize}
+\end{block}}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Definition}
+Längen/Winkel bleiben erhalten
+\\
+\uncover<11->{%
+$\Rightarrow$ $\exists$ Erhaltungsgrösse}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
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