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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/5/verzerrung.tex | 121 | ||||
| -rw-r--r-- | vorlesungen/slides/5/verzerrung/verzerrung.m | 13 |
2 files changed, 134 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/verzerrung.tex b/vorlesungen/slides/5/verzerrung.tex new file mode 100644 index 0000000..8d6514c --- /dev/null +++ b/vorlesungen/slides/5/verzerrung.tex @@ -0,0 +1,121 @@ +% +% verzerrung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\r{1.10} +\def\s{1.12} +\def\q{1.23} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Verzerrung} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\begin{block}{Abbildung $A\colon v\mapsto Av$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.974,0.171); +\coordinate (a2) at (0.037,1.018); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\foreach \a in {0,5,...,355}{ + \draw[color=red,line width=1.2pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {1,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$Av$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<73->{% +\begin{block}{Abbildung $A\colon v\mapsto (A-\lambda)v$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.121,0.343); +\coordinate (a2) at (0.074,0.209); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\begin{scope} +\clip (-1.2,-1.2) rectangle (1.2,1.2); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\end{scope} + +\foreach \a in {0,5,...,355}{ + \draw[color=red!60,line width=4pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {73,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$(A-\lambda)v$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} + +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/verzerrung/verzerrung.m b/vorlesungen/slides/5/verzerrung/verzerrung.m new file mode 100644 index 0000000..028e7f9 --- /dev/null +++ b/vorlesungen/slides/5/verzerrung/verzerrung.m @@ -0,0 +1,13 @@ +# +# verzerrung.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +rand("seed", 4712); + +A = eye(2) + 1.0 * (rand(2,2) - 0.5 * ones(2,2)) + +[V, lambda] = eig(A) + +B = A - lambda(1,1) * eye(2) |