add Fresnel presentation

8 files changed, 358 insertions, 2 deletions

diff --git a/vorlesungen/slides/fresnel/Makefile b/vorlesungen/slides/fresnel/Makefile
new file mode 100644
index 0000000..77ad9a2
--- /dev/null
+++ b/vorlesungen/slides/fresnel/Makefile
@@ -0,0 +1,9 @@
+#
+# Makefile
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	eulerpath.tex
+
+eulerpath.tex:	eulerspirale.m
+	octave eulerspirale.m
diff --git a/vorlesungen/slides/fresnel/Makefile.inc b/vorlesungen/slides/fresnel/Makefile.inc
index c17b654..4ab3b2f 100644
--- a/vorlesungen/slides/fresnel/Makefile.inc
+++ b/vorlesungen/slides/fresnel/Makefile.inc
@@ -4,4 +4,7 @@
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
 chapterfresnel =							\
-	../slides/fresnel/test.tex
+	../slides/fresnel/integrale.tex					\
+	../slides/fresnel/kruemmung.tex					\
+	../slides/fresnel/klothoide.tex					\
+	../slides/fresnel/numerik.tex
diff --git a/vorlesungen/slides/fresnel/chapter.tex b/vorlesungen/slides/fresnel/chapter.tex
index dc5d031..ad0c011 100644
--- a/vorlesungen/slides/fresnel/chapter.tex
+++ b/vorlesungen/slides/fresnel/chapter.tex
@@ -3,4 +3,7 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
-\folie{fresnel/test.tex}
+\folie{fresnel/integrale.tex}
+\folie{fresnel/kruemmung.tex}
+\folie{fresnel/klothoide.tex}
+\folie{fresnel/numerik.tex}
diff --git a/vorlesungen/slides/fresnel/eulerspirale.m b/vorlesungen/slides/fresnel/eulerspirale.m
new file mode 100644
index 0000000..312541a
--- /dev/null
+++ b/vorlesungen/slides/fresnel/eulerspirale.m
@@ -0,0 +1,34 @@
+#
+# eulerspirale.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue
+#
+global n;
+n = 10000;
+global tmax;
+tmax = 10;
+
+function retval = f(x, t)
+	retval = [ cos(t*t); sin(t*t) ];
+endfunction
+
+x0 = [ 0; 0 ];
+t = tmax * (0:n) / n;
+
+c = lsode(@f, x0, t);
+
+fn = fopen("eulerpath.tex", "w");
+
+fprintf(fn, "\\def\\fresnela{ (0,0)");
+for i = (2:n)
+	fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2));
+end
+fprintf(fn, "\n}\n");
+
+fprintf(fn, "\\def\\fresnelb{ (0,0)");
+for i = (2:n)
+	fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2));
+end
+fprintf(fn, "\n}\n");
+
+fclose(fn);
diff --git a/vorlesungen/slides/fresnel/integrale.tex b/vorlesungen/slides/fresnel/integrale.tex
new file mode 100644
index 0000000..7798932
--- /dev/null
+++ b/vorlesungen/slides/fresnel/integrale.tex
@@ -0,0 +1,72 @@
+%
+% 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
diff --git a/vorlesungen/slides/fresnel/klothoide.tex b/vorlesungen/slides/fresnel/klothoide.tex
new file mode 100644
index 0000000..dcf52be
--- /dev/null
+++ b/vorlesungen/slides/fresnel/klothoide.tex
@@ -0,0 +1,61 @@
+%
+% klothoide.tex -- Klothoide
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Klothoide}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Krümmung der Euler-Spirale}
+\begin{align*}
+\frac{d}{dt}\gamma(t)
+&=
+\begin{pmatrix}
+\cos t^2\\
+\sin t^2
+\end{pmatrix}
+\intertext{\uncover<2->{Bogenlänge:}}
+\uncover<2->{
+|\dot{\gamma}(t)|
+&=
+\sqrt{\cos^2 t^2 + \sin^2 t^2}
+=
+1
+}
+\intertext{\uncover<3->{Polarwinkel:}}
+\uncover<3->{
+\varphi&=t^2
+\intertext{\uncover<4->{Krümmung:}}
+\uncover<4->{
+\frac{d\varphi}{dt}
+&=
+2t
+}
+}
+\end{align*}
+\uncover<5->{%
+$\Rightarrow$ Krümmung ist proportional zur Bogenlänge
+}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Definition}
+Eine Kurve, deren Krümmung proportional zur Bogenlänge ist, heisst
+{\em Klothoid}
+\end{block}}
+\uncover<7->{%
+\begin{block}{Anwendung}
+Strassenbau: Um mit konstanter Geschwindigkeit auf einer
+Klothoidenkurve zu fahren, muss man das Lenkrad mit konstanter Geschwindigkeit
+drehen
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/fresnel/kruemmung.tex b/vorlesungen/slides/fresnel/kruemmung.tex
new file mode 100644
index 0000000..e75611b
--- /dev/null
+++ b/vorlesungen/slides/fresnel/kruemmung.tex
@@ -0,0 +1,91 @@
+%
+% kruemmung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Krümmung einer Kurve}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Krümmungsradius}
+Bogen und Radius:
+\[
+s=r\cdot\Delta\varphi
+\uncover<2->{
+\quad
+\Rightarrow
+\quad
+r
+=
+\frac{s}{\Delta\varphi}
+}
+\]
+\end{block}
+\vspace*{-12pt}
+\uncover<3->{
+\begin{block}{Krümmung}
+Je grösser der Krümmungsradius, desto kleiner die Krümmung:
+\[
+\kappa = \frac{1}{r}
+\]
+\end{block}}
+\vspace*{-12pt}
+\uncover<5->{%
+\begin{block}{Definition}
+Änderungsgeschwindigkeit des Polarwinkels der Tangente
+\[
+\kappa
+=
+\frac{1}{r}
+\uncover<6->{=
+\frac{\Delta\varphi}{s}}
+\uncover<7->{=
+\frac{d\varphi}{dt}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\begin{scope}
+\clip (-1,-1) rectangle (4,4);
+
+\def\r{3}
+\def\winkel{30}
+
+\fill[color=blue!20] (0,0) -- (0:\r) arc (0:\winkel:\r) -- cycle;
+\node[color=blue] at ({0.5*\winkel}:{0.5*\r}) {$\Delta\varphi$};
+
+\draw[line width=0.3pt] (0,0) circle[radius=\r];
+
+\draw[->] (0,0) -- (0:\r);
+\draw[->] (0,0) -- (\winkel:\r);
+
+\uncover<4->{
+\draw[->] (0:\r) -- ($(0:\r)+(90:0.7*\r)$);
+\draw[->] (\winkel:\r) -- ($(\winkel:\r)+({90+\winkel}:0.7*\r)$);
+}
+
+\draw[color=red,line width=1.4pt] (0:\r) arc (0:\winkel:\r);
+\node[color=red] at ({0.5*\winkel}:\r) [left] {$s$};
+\fill[color=red] (0:\r) circle[radius=0.05];
+\fill[color=red] (\winkel:\r) circle[radius=0.05];
+
+\node at (\winkel:{0.5*\r}) [above] {$r$};
+\node at (0:{0.5*\r}) [below] {$r$};
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\uncover<4->{%
+Für $\varphi$ kann man auch den Polarwinkel des Tangentialvektors nehmen
+}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/fresnel/numerik.tex b/vorlesungen/slides/fresnel/numerik.tex
new file mode 100644
index 0000000..5c6f96d
--- /dev/null
+++ b/vorlesungen/slides/fresnel/numerik.tex
@@ -0,0 +1,83 @@
+%
+% numerik.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Numerik}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Taylor-Reihe}
+\begin{align*}
+\sin t^2
+&=
+\sum_{k=0}^\infty
+(-1)^k \frac{t^{2k+1}}{(2k+1)!}
+\\
+%\int \sin t^2\,dt
+\uncover<2->{
+S(t)
+&=
+\sum_{k=0}^\infty
+(-1)^k \frac{t^{4k+3}}{(2k+1)!(4n+3)}
+}
+\\
+\cos t^2
+&=
+\sum_{k=0}^\infty
+(-1)^k \frac{t^{2k}}{(2k)!}
+\\
+%\int \sin t^2\,dt
+\uncover<3->{
+C(t)
+&=
+\sum_{k=0}^\infty
+(-1)^k \frac{t^{4k+1}}{(2k)!(4k+1)}
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{
+\begin{block}{Differentialgleichung}
+\[
+\dot{\gamma}(t)
+=
+\begin{pmatrix}
+\sin t^2\\ \cos t^2
+\end{pmatrix}
+\]
+\end{block}}
+\uncover<5->{%
+\begin{block}{Hypergeometrische Reihen}
+\begin{align*}
+\uncover<6->{%
+S(t)
+&=
+\frac{\pi z^3}{6}\,
+\mathstrut_1F_2\biggl(
+\begin{matrix}\frac34\\\frac32,\frac74\end{matrix}
+;
+-\frac{\pi^2z^4}{16}
+\biggr)
+}
+\\
+\uncover<7->{
+C(t)
+&=
+z\,
+\mathstrut_1F_2\biggl(
+\begin{matrix}\frac14\\\frac12,\frac54\end{matrix}
+;
+-\frac{\pi^2z^4}{16}
+\biggr)}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup