%
% fresnel.tex -- slide template
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\input{../slides/fresnel/eulerpath.tex}
\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*}
S(t)
&=
\int_0^t \sin(\tau^2)\,d\tau
\\
C(t)
&=
\int_0^t \cos(\tau^2)\,d\tau
\end{align*}
\uncover<2->{%
Können nicht in geschlossener Form ausgewertet werden.
}
\end{block}
\uncover<3->{%
\begin{block}{Kurve}
\[
\gamma(t)
=
\begin{pmatrix}
S(t)\\C(t)
\end{pmatrix}
\]
\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
\uncover<4->{%
\begin{block}{Euler-Spirale}
\begin{center}
\begin{tikzpicture}[>=latex,thick,scale=2.7]

\draw[->] (-1.05,0) -- (1.05,0) coordinate[label={$C(t)$}];
\draw[->] (0,-1.05) -- (0,1.05) coordinate[label={right:$S(t)$}];

\draw[color=red,line width=1.4pt] \fresnela;
\draw[color=red,line width=1.4pt] \fresnelb;

\fill[color=blue] ({sqrt(3.14159/8)},{sqrt(3.14159/8)}) circle[radius=0.02];
\fill[color=blue] ({-sqrt(3.14159/8)},{-sqrt(3.14159/8)}) circle[radius=0.02];

\draw (1,-0.03) -- (1,0.03);
\node at (1,-0.03) [below] {$1$};
\draw (-1,-0.03) -- (-1,0.03);
\node at (-1,0.03) [above] {$-1$};
\draw (-0.03,1) -- (0.03,1);
\node at (-0.03,1) [left] {$1$};
\draw (-0.03,-1) -- (0.03,-1);
\node at (0.03,-1) [right] {$-1$};
\node at (0,0) [below right] {$0$};

\end{tikzpicture}
\end{center}
\end{block}}
\end{column}
\end{columns}
\end{frame}
\egroup