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Diffstat (limited to 'vorlesungen/slides/fresnel/integrale.tex')
| -rw-r--r-- | vorlesungen/slides/fresnel/integrale.tex | 105 |
1 files changed, 76 insertions, 29 deletions
diff --git a/vorlesungen/slides/fresnel/integrale.tex b/vorlesungen/slides/fresnel/integrale.tex index 7798932..906aec1 100644 --- a/vorlesungen/slides/fresnel/integrale.tex +++ b/vorlesungen/slides/fresnel/integrale.tex @@ -1,10 +1,11 @@ % -% fresnel.tex -- slide template +% 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} @@ -15,57 +16,103 @@ \begin{block}{Definition} Fresnel-Integrale: \begin{align*} -S(t) +\color{red}S(t) &= -\int_0^t \sin(\tau^2)\,d\tau +\int_0^t \sin\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau \\ -C(t) +\color{blue}C(t) &= -\int_0^t \cos(\tau^2)\,d\tau +\int_0^t \cos\biggl(\frac{\pi\tau^2}2\biggr)\,d\tau \end{align*} -\uncover<2->{% +\uncover<3->{% Können nicht in geschlossener Form ausgewertet werden. } \end{block} -\uncover<3->{% -\begin{block}{Kurve} +\uncover<4->{% +\begin{block}{Euler-Spirale} \[ -\gamma(t) +\gamma_a(t) = \begin{pmatrix} -S(t)\\C(t) +C_a(t)\\S_a(t) +\end{pmatrix} += +\begin{pmatrix} +\displaystyle +\int_0^t \cos (a\tau^2)\,d\tau\\[8pt] +\displaystyle +\int_0^t \sin (a\tau^2)\,d\tau \end{pmatrix} \] \end{block}} \end{column} \begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Euler-Spirale} +\ifthenelse{\boolean{presentation}}{ +\only<2-4>{% \begin{center} -\begin{tikzpicture}[>=latex,thick,scale=2.7] +\begin{tikzpicture}[>=latex,thick,scale=1] +\def\dx{0.6} +\def\dy{1.5} -\draw[->] (-1.05,0) -- (1.05,0) coordinate[label={$C(t)$}]; -\draw[->] (0,-1.05) -- (0,1.05) coordinate[label={right:$S(t)$}]; +\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} -\draw[color=red,line width=1.4pt] \fresnela; -\draw[color=red,line width=1.4pt] \fresnelb; +\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} -\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]; +\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 (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] {$0$}; +\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{block}} +} \end{column} \end{columns} \end{frame} |