Merge remote-tracking branch 'Upload/master'

12 files changed, 436 insertions, 4 deletions

diff --git a/vorlesungen/10_mseliealgebra/slides.tex b/vorlesungen/10_mseliealgebra/slides.tex
index e4493a3..936139a 100644
--- a/vorlesungen/10_mseliealgebra/slides.tex
+++ b/vorlesungen/10_mseliealgebra/slides.tex
@@ -19,6 +19,9 @@
 
 \section{Exponentialabbildung}
 \folie{7/dg.tex}
+\folie{7/logarithmus.tex}
+% Interpolation
+\folie{7/interpolation.tex}
 
 \section{Integration}
 \folie{7/haar.tex}
@@ -28,7 +31,5 @@
 \folie{7/integration.tex}
 
 % LOG Reihe
-% Interpolation
 % Mittelung auf einer Lie-Gruppe
-% Vektorprodukt als Lie-Gruppe
 
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 52c37d8..ffd5091 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -21,6 +21,9 @@ chapter5 =								\
 	../slides/7/kommutator.tex					\
 	../slides/7/bch.tex						\
 	../slides/7/dg.tex						\
+	../slides/7/interpolation.tex					\
+	../slides/7/exponentialreihe.tex				\
+	../slides/7/logarithmus.tex					\
 	../slides/7/zusammenhang.tex					\
 	../slides/7/quaternionen.tex					\
 	../slides/7/qdreh.tex						\
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
index 172b78a..3736e0f 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -20,6 +20,9 @@
 \folie{7/kommutator.tex}
 \folie{7/bch.tex}
 \folie{7/dg.tex}
+\folie{7/interpolation.tex}
+\folie{7/exponentialreihe.tex}
+\folie{7/logarithmus.tex}
 \folie{7/zusammenhang.tex}
 \folie{7/quaternionen.tex}
 \folie{7/qdreh.tex}
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
index cc67c8a..6f99bc3 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 # 
-all:	rodriguez.jpg 
+all:	rodriguez.jpg  test.png
 
 rodriguez.png:	rodriguez.pov
 	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
@@ -16,4 +16,14 @@ commutator:	commutator.ini commutator.pov common.inc
 jpg:
 	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
 
+dreibein/timestamp:	interpolation.m
+	octave interpolation.m
+	touch dreibein/timestamp
 
+test.png:	test.pov drehung.inc dreibein/d025.inc dreibein/timestamp
+	povray +A0.1 -W1080 -H1080 -Otest.png test.pov
+
+dreibein/d025.inc:	dreibein/timestamp
+
+animation:
+	povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini
diff --git a/vorlesungen/slides/7/images/drehung.inc b/vorlesungen/slides/7/images/drehung.inc
new file mode 100644
index 0000000..c9b4bb7
--- /dev/null
+++ b/vorlesungen/slides/7/images/drehung.inc
@@ -0,0 +1,142 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.23;
+#declare O = <0, 0, 0>;
+#declare at = 0.02;
+
+camera {
+	location <8.5, 2, 6.5>
+	look_at <0, 0, 0>
+	right x * imagescale
+	up y * imagescale
+}
+
+//light_source {
+//	<-14, 20, -50> color White
+//	area_light <1,0,0> <0,0,1>, 10, 10
+//	adaptive 1
+//	jitter
+//}
+
+light_source {
+	<41, 20, 10> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.0 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+#declare r = 1.0;
+
+arrow(< -r-0.2,    0.0,    0  >, <  r+0.2,    0.0,    0.0 >, at, Gray)
+arrow(<    0.0,    0.0, -r-0.2>, <    0.0,    0.0,  r+0.2 >, at, Gray)
+arrow(<    0.0, -r-0.2,    0  >, <    0.0,  r+0.2,    0.0 >, at, Gray)
+
+#declare farbeX = rgb<1.0,0.2,0.6>;
+#declare farbeY = rgb<0.0,0.8,0.4>;
+#declare farbeZ = rgb<0.4,0.6,1.0>;
+
+#declare farbex = rgb<1.0,0.0,0.0>;
+#declare farbey = rgb<0.0,0.6,0.0>;
+#declare farbez = rgb<0.0,0.0,1.0>;
+
+#macro quadrant(X, Y, Z)
+	intersection {
+		sphere { O, 0.5 }
+		plane { -X, 0 }
+		plane { -Y, 0 }
+		plane { -Z, 0 }
+		pigment {
+			color rgb<1.0,0.6,0.2>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+	arrow(O, X, 1.1*at, farbex)
+	arrow(O, Y, 1.1*at, farbey)
+	arrow(O, Z, 1.1*at, farbez)
+#end
+
+#macro drehung(X, Y, Z)
+//	intersection {
+//		sphere { O, 0.5 }
+//		plane { -X, 0 }
+//		plane { -Y, 0 }
+//		plane { -Z, 0 }
+//		pigment {
+//			color Gray
+//		}
+//		finish {
+//			specular 0.95
+//			metallic
+//		}
+//	}
+	arrow(O, 1.1*X, 0.9*at, farbeX)
+	arrow(O, 1.1*Y, 0.9*at, farbeY)
+	arrow(O, 1.1*Z, 0.9*at, farbeZ)
+#end
+
+#macro achse(H)
+	cylinder { H, -H, at
+		pigment {
+			color rgb<0.6,0.4,0.2>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+	cylinder { 0.003 * H, -0.003 * H, 1
+		pigment {
+			color rgbt<0.6,0.4,0.2,0.5>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+#end
diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini
new file mode 100644
index 0000000..f07c079
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.ini
@@ -0,0 +1,8 @@
+Input_File_Name=interpolation.pov
+Initial_Frame=0
+Final_Frame=50
+Initial_Clock=0
+Final_Clock=50
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m
new file mode 100644
index 0000000..31554e8
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.m
@@ -0,0 +1,54 @@
+#
+# interpolation.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 50;
+global A;
+global B;
+
+A = (pi / 2) * [
+  0, 0,  0;
+  0, 0, -1;
+  0, 1,  0
+];
+g0 = expm(A)
+
+B = (pi / 2) * [
+  0, 0, 1;
+  0, 0, 0;
+ -1, 0, 0
+];
+g1 = expm(B)
+
+function retval = g(t)
+	global A;
+	global B;
+	retval = expm((1-t)*A+t*B);
+endfunction
+
+function dreibein(fn, M, funktion)
+	fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n",
+		funktion,
+		M(1,1), M(3,1), M(2,1),
+		M(1,2), M(3,2), M(2,2),
+		M(1,3), M(3,3), M(2,3));
+endfunction
+
+G = g1 * inverse(g0);
+[V, lambda] = eig(G);
+H = real(V(:,3));
+
+D = logm(g1*inverse(g0));
+
+for i = (0:N)
+	filename = sprintf("dreibein/d%03d.inc", i);
+	fn = fopen(filename, "w");
+	t = i/N;
+	dreibein(fn, g(t), "quadrant");
+	dreibein(fn, expm(t*D)*g0, "drehung");
+	fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1));
+	fclose(fn);
+endfor
+
diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov
new file mode 100644
index 0000000..71e0257
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.pov
@@ -0,0 +1,10 @@
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+
+#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc");
+#include filename
+
diff --git a/vorlesungen/slides/7/images/test.pov b/vorlesungen/slides/7/images/test.pov
new file mode 100644
index 0000000..5707be1
--- /dev/null
+++ b/vorlesungen/slides/7/images/test.pov
@@ -0,0 +1,7 @@
+//
+// test.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+#include "dreibein/d025.inc"
diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex
new file mode 100644
index 0000000..249ee26
--- /dev/null
+++ b/vorlesungen/slides/7/interpolation.tex
@@ -0,0 +1,112 @@
+%
+% interpolation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Interpolation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen
+$g_0\in\operatorname{SO}(3)$
+und
+$g_1\in\operatorname{SO}(3)$:
+\[
+g_0=g(0)
+\quad\wedge\quad
+g_1=g(1)
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Lösung}
+$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$
+\begin{align*}
+\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)}
+\\
+\uncover<5->{A(t)^t
+&=(1-t)A_0^t + tA_1^t}
+\\
+&\uncover<6->{=
+-(1-t)A_0 - t A_1}
+\uncover<7->{=
+-A(t)}
+\\
+\uncover<9->{\Rightarrow
+g(t) &= \exp A(t) \in \operatorname{SO}(3)}
+\\
+&\uncover<10->{\ne 
+\exp (\log(g_1g_0^{-1})t) g_0}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Animation}
+\centering
+\ifthenelse{\boolean{presentation}}{
+\bild{12}{i00}
+\bild{13}{i01}
+\bild{14}{i02}
+\bild{15}{i03}
+\bild{16}{i04}
+\bild{17}{i05}
+\bild{18}{i06}
+\bild{19}{i07}
+\bild{20}{i08}
+\bild{21}{i09}
+\bild{22}{i10}
+\bild{23}{i11}
+\bild{24}{i12}
+\bild{25}{i13}
+\bild{26}{i14}
+\bild{27}{i15}
+\bild{28}{i16}
+\bild{29}{i17}
+\bild{30}{i18}
+\bild{31}{i19}
+\bild{32}{i20}
+\bild{33}{i21}
+\bild{34}{i22}
+\bild{35}{i23}
+\bild{36}{i24}
+\bild{37}{i25}
+\bild{38}{i26}
+\bild{39}{i27}
+\bild{40}{i28}
+\bild{41}{i29}
+\bild{42}{i30}
+\bild{43}{i31}
+\bild{44}{i32}
+\bild{45}{i33}
+\bild{46}{i34}
+\bild{47}{i35}
+\bild{48}{i36}
+\bild{49}{i37}
+\bild{50}{i38}
+\bild{51}{i39}
+\bild{52}{i40}
+\bild{53}{i41}
+\bild{54}{i42}
+\bild{55}{i43}
+\bild{56}{i44}
+\bild{57}{i45}
+\bild{58}{i46}
+\bild{59}{i47}
+\bild{60}{i48}
+\bild{61}{i49}
+\bild{62}{i50}
+}{
+\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png}
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex
new file mode 100644
index 0000000..58065d7
--- /dev/null
+++ b/vorlesungen/slides/7/logarithmus.tex
@@ -0,0 +1,82 @@
+%
+% logarithmus.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Logarithmus}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Taylor-Reihe}
+\begin{align*}
+\frac{d}{dx}\log(1+x)
+&= \frac{1}{1+x}
+\\
+\uncover<2->{
+\Rightarrow\quad
+\log (1+x)
+&=
+\int_0^x \frac{1}{1+t}\,dt}
+\end{align*}
+\begin{align*}
+\uncover<3->{\frac{1}{1+t}
+&=
+1-t+t^2-t^3+\dots}
+\\
+\uncover<4->{\log(1+x)
+&=\int_0^x
+1-t+t^2-t^3+\dots
+\,dt}
+\\
+&\only<5>{=
+x-\frac{x^2}{2}  + \frac{x^3}{3} - \frac{x^4}4 + \dots}
+\uncover<6->{=
+\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}}
+\\
+\uncover<7->{\log (I+A)
+&=
+\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Konvergenzradius}
+Polstelle bei $x=-1$
+\(
+\varrho =1
+\)
+\end{block}}
+\vspace{-5pt}
+\begin{block}{\uncover<9->{Alternative: Spektraltheorie}}
+\uncover<9->{
+Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$
+definiert:}
+\vspace{-15pt}
+\uncover<8->{
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+	\fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1);
+}
+\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}];
+\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}];
+\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1];
+\draw[color=blue] (1,0) circle[radius=1];
+\uncover<9->{
+	\draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0);
+}
+\fill (1,0) circle[radius=0.08];
+\node at (2.3,1.9) {$\mathbb{C}$};
+\node at (1,0) [below] {$1$};
+\end{tikzpicture}
+\end{center}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 35ebf03..7434b0b 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,5 +3,5 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\folie{7/integration.tex}
+\folie{7/logarithmus.tex}