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| -rw-r--r-- | vorlesungen/slides/fresnel/integrale.tex | 119 |
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diff --git a/vorlesungen/slides/fresnel/integrale.tex b/vorlesungen/slides/fresnel/integrale.tex new file mode 100644 index 0000000..906aec1 --- /dev/null +++ b/vorlesungen/slides/fresnel/integrale.tex @@ -0,0 +1,119 @@ +% +% integrale.tex -- Definition der Fresnel Integrale +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\input{../slides/fresnel/eulerpath.tex} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Fresnel-Integrale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Fresnel-Integrale: +\begin{align*} +\color{red}S(t) +&= +\int_0^t \sin\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau +\\ +\color{blue}C(t) +&= +\int_0^t \cos\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau +\end{align*} +\uncover<3->{% +Können nicht in geschlossener Form ausgewertet werden. +} +\end{block} +\uncover<4->{% +\begin{block}{Euler-Spirale} +\[ +\gamma_a(t) += +\begin{pmatrix} +C_a(t)\\S_a(t) +\end{pmatrix} += +\begin{pmatrix} +\displaystyle +\int_0^t \cos (a\tau^2)\,d\tau\\[8pt] +\displaystyle +\int_0^t \sin (a\tau^2)\,d\tau +\end{pmatrix} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\ifthenelse{\boolean{presentation}}{ +\only<2-4>{% +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=1] +\def\dx{0.6} +\def\dy{1.5} + +\begin{scope} + \draw[color=gray!50] (0,{0.5*\dy}) -- (3,{0.5*\dy}); + \draw[color=gray!50] (0,{-0.5*\dy}) -- (-3,{-0.5*\dy}); + \draw[->] (-3,0) -- (3.3,0) coordinate[label={$t$}]; + \draw[->] (0,-1.5) -- (0,1.5) coordinate[label={left:$S(t)$}]; + \draw (-0.1,{0.5*\dy}) -- (0.1,{0.5*\dy}); + \node at (-0.1,{0.5*\dy}) [left] {$\frac12$}; + \draw (-0.1,{-0.5*\dy}) -- (0.1,{-0.5*\dy}); + \node at (0.1,{-0.5*\dy}) [right] {$-\frac12$}; + \draw[color=red,line width=1.4pt] \Splotright; + \draw[color=red,line width=1.4pt] \Splotleft; +\end{scope} + +\begin{scope}[yshift=-3.4cm] + \draw[color=gray!50] (0,{0.5*\dy}) -- (3,{0.5*\dy}); + \draw[color=gray!50] (0,{-0.5*\dy}) -- (-3,{-0.5*\dy}); + \draw[->] (-3,0) -- (3.3,0) coordinate[label={$t$}]; + \draw[->] (0,-1.5) -- (0,1.5) coordinate[label={left:$C(t)$}]; + \draw (-0.1,{0.5*\dy}) -- (0.1,{0.5*\dy}); + \node at (-0.1,{0.5*\dy}) [left] {$\frac12$}; + \draw (-0.1,{-0.5*\dy}) -- (0.1,{-0.5*\dy}); + \node at (0.1,{-0.5*\dy}) [right] {$-\frac12$}; + \draw[color=blue,line width=1.4pt] \Cplotright; + \draw[color=blue,line width=1.4pt] \Cplotleft; +\end{scope} + +\end{tikzpicture} +\end{center} +}}{} +\uncover<5->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=3.5] + +\draw[color=gray!50] (-0.5,-0.5) rectangle (0.5,0.5); + +\draw[->] (-0.8,0) -- (0.9,0) coordinate[label={$\color{blue}C(t)$}]; +\draw[->] (0,-0.8) -- (0,0.9) coordinate[label={right:$\color{red}S(t)$}]; + +\draw[color=darkgreen,line width=1.0pt] \fresnela; +\draw[color=darkgreen,line width=1.0pt] \fresnelb; + +\fill[color=orange] (0.5,0.5) circle[radius=0.02]; +\fill[color=orange] (-0.5,-0.5) circle[radius=0.02]; + +\draw (0.5,-0.02) -- (0.5,0.02); +\node at (0.5,-0.02) [below right] {$\frac12$}; + +\draw (-0.5,-0.02) -- (-0.5,0.02); +\node at (-0.5,0.02) [above left] {$-\frac12$}; + +\draw (-0.01,0.5) -- (0.02,0.5); +\node at (-0.02,0.5) [above left] {$\frac12$}; + +\draw (-0.02,-0.5) -- (0.02,-0.5); +\node at (0.02,-0.5) [below right] {$-\frac12$}; + +\end{tikzpicture} +\end{center} +} +\end{column} +\end{columns} +\end{frame} +\egroup |