From a866d7cd6672474e9376617aadc91424b9ba3506 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 8 May 2022 22:37:45 +0200 Subject: add Fresnel presentation --- vorlesungen/slides/fresnel/Makefile | 9 +++ vorlesungen/slides/fresnel/Makefile.inc | 5 +- vorlesungen/slides/fresnel/chapter.tex | 5 +- vorlesungen/slides/fresnel/eulerspirale.m | 34 ++++++++++++ vorlesungen/slides/fresnel/integrale.tex | 72 ++++++++++++++++++++++++ vorlesungen/slides/fresnel/klothoide.tex | 61 +++++++++++++++++++++ vorlesungen/slides/fresnel/kruemmung.tex | 91 +++++++++++++++++++++++++++++++ vorlesungen/slides/fresnel/numerik.tex | 83 ++++++++++++++++++++++++++++ 8 files changed, 358 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/fresnel/Makefile create mode 100644 vorlesungen/slides/fresnel/eulerspirale.m create mode 100644 vorlesungen/slides/fresnel/integrale.tex create mode 100644 vorlesungen/slides/fresnel/klothoide.tex create mode 100644 vorlesungen/slides/fresnel/kruemmung.tex create mode 100644 vorlesungen/slides/fresnel/numerik.tex (limited to 'vorlesungen/slides') 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 -- cgit v1.2.1