%
% 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
