From 1fb743df08b0734932d510c6b11405d0a2dbbe47 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 19:57:13 +0200 Subject: new slides --- vorlesungen/slides/7/drehung.tex | 113 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 113 insertions(+) create mode 100644 vorlesungen/slides/7/drehung.tex (limited to 'vorlesungen/slides/7/drehung.tex') diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex new file mode 100644 index 0000000..ae0dbe3 --- /dev/null +++ b/vorlesungen/slides/7/drehung.tex @@ -0,0 +1,113 @@ +% +% 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} +\[ +\operatorname{SO}(2) += +\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2) +\] +\end{block} +\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} +\begin{block}{Beispiel} +\vspace{-12pt} +\[ +D_{60^\circ} += +{\tiny +\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} +\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} +} +\] +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\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} +\\ +&= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +-st&-s\\ + t& 1 +\end{pmatrix} +\\ +&= +\begin{pmatrix} +-stc^{-1}&{\color{darkgreen}sc^{-1}}\\ +{\color{blue}ct}&{\color{red}c} +\end{pmatrix} += +\begin{pmatrix} +\cos\alpha & {\color{darkgreen}- \sin\alpha} \\ +{\color{blue}\sin\alpha} & \phantom{-} {\color{red}\cos\alpha} +\end{pmatrix} +\end{align*} +\end{block} +\vspace{-10pt} +\begin{block}{Koeffizientenvergleich} +\vspace{-15pt} +\begin{align*} +{\color{red} c} +&= +{\color{red}\cos\alpha } +&& +& +{\color{blue} +t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\ +{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} +& +&\Rightarrow& +{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha +\\ +{\color{orange} -stc^{-t}} +&= +\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup -- cgit v1.2.1 From 15881729aa3f1293d546a1692a02094ed3f24e2b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 10:30:05 +0200 Subject: phases --- vorlesungen/slides/7/drehung.tex | 53 +++++++++++++++++++++++++++------------- 1 file changed, 36 insertions(+), 17 deletions(-) (limited to 'vorlesungen/slides/7/drehung.tex') diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index ae0dbe3..7744e99 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -13,12 +13,20 @@ \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->{ \[ \operatorname{SO}(2) = -\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2) -\] +{\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: \[ @@ -31,7 +39,9 @@ D_{\alpha} = DST \] -\end{block} +\end{block}} +\vspace{-10pt} +\uncover<12->{% \begin{block}{Beispiel} \vspace{-12pt} \[ @@ -43,9 +53,10 @@ D_{60^\circ} \begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} } \] -\end{block} +\end{block}} \end{column} \begin{column}{0.58\textwidth} +\uncover<4->{% \begin{block}{Ansatz} \vspace{-12pt} \begin{align*} @@ -64,7 +75,7 @@ c^{-1}&0\\ t&1 \end{pmatrix} \\ -&= +&\uncover<5->{= \begin{pmatrix} c^{-1}&0\\ 0 &c @@ -73,40 +84,48 @@ c^{-1}&0\\ -st&-s\\ t& 1 \end{pmatrix} +} \\ -&= +&\uncover<6->{= \begin{pmatrix} --stc^{-1}&{\color{darkgreen}sc^{-1}}\\ -{\color{blue}ct}&{\color{red}c} -\end{pmatrix} -= +{\color<11->{orange}-stc^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<9->{blue}ct}&{\color<8->{red}c} +\end{pmatrix}} +\uncover<7->{= \begin{pmatrix} -\cos\alpha & {\color{darkgreen}- \sin\alpha} \\ -{\color{blue}\sin\alpha} & \phantom{-} {\color{red}\cos\alpha} -\end{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} +\end{block}} \vspace{-10pt} +\uncover<7->{% \begin{block}{Koeffizientenvergleich} \vspace{-15pt} \begin{align*} +\uncover<8->{ {\color{red} c} &= -{\color{red}\cos\alpha } +{\color{red}\cos\alpha }} && & +\uncover<9->{ {\color{blue} -t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\ +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} -stc^{-t}} &= \rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} -- cgit v1.2.1 From eb8f7d58101bd19006019dfbfafca92dd080fe41 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 12 Apr 2021 13:30:14 +0200 Subject: typos, new slides --- vorlesungen/slides/7/drehung.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) (limited to 'vorlesungen/slides/7/drehung.tex') 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}} -- cgit v1.2.1 From 3fe0e1a234175903de4a161444b9db9810956f9d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 29 Apr 2021 09:03:37 +0200 Subject: add new slides --- vorlesungen/slides/7/drehung.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'vorlesungen/slides/7/drehung.tex') diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 2d7b317..02201d4 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -58,7 +58,7 @@ D_{60^\circ} \begin{column}{0.58\textwidth} \uncover<4->{% \begin{block}{Ansatz} -\vspace{-12pt} +%\vspace{-12pt} \begin{align*} DST &= @@ -101,7 +101,7 @@ c^{-1}&0\\ \vspace{-10pt} \uncover<7->{% \begin{block}{Koeffizientenvergleich} -\vspace{-15pt} +%\vspace{-15pt} \begin{align*} \uncover<8->{ {\color{red} c} -- cgit v1.2.1