Visualisierungen für Perron-Frobenius-Theorie

17 files changed, 718 insertions, 1 deletions

diff --git a/buch/chapters/80-wahrscheinlichkeit/images/Makefile b/buch/chapters/80-wahrscheinlichkeit/images/Makefile
index 3c6a7f9..5511f14 100644
--- a/buch/chapters/80-wahrscheinlichkeit/images/Makefile
+++ b/buch/chapters/80-wahrscheinlichkeit/images/Makefile
@@ -3,8 +3,10 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschulen
 #
-all:	diffusion.png diffusion.pdf
+all:	dreieck.pdf trenn.pdf vergleich.pdf vergleich.jpg positiv.jpg \
+	diffusion.png diffusion.pdf
 
+# Visualisierung diffusion in einer primitiven Matrix
 diffusion.pdf:	diffusion.tex diffusion.jpg
 	pdflatex diffusion.tex
 
@@ -17,3 +19,34 @@ diffusion.jpg:	diffusion.png
 vektoren.inc:	diffusion.m
 	octave diffusion.m
 
+# Visualizierung positive Matrix
+positiv.png:	positiv.pov  quadrant.inc
+	povray +A0.1 +W1920 +H1080 -Opositiv.png positiv.pov
+
+positiv.jpg:	positiv.png
+	convert positiv.png -density 300 -units PixelsPerInch positiv.jpg
+
+quadrant.inc:	positiv.m
+	octave positiv.m
+
+# Visualiziserung Vergleichstrick
+vergleich.png:	vergleich.pov
+	povray +A0.1 +W1920 +H1080 -Overgleich.png vergleich.pov
+
+vergleich.jpg:	vergleich.png Makefile
+	convert -extract 1110x1080+180+0 vergleich.png \
+		-density 300 -units PixelsPerInch vergleich.jpg
+
+vergleich.pdf:	vergleich.tex vergleich.jpg
+	pdflatex vergleich.tex
+
+# Darstellung zum Trenntrick
+trenn.pdf:	trenn.tex
+	pdflatex trenn.tex
+
+# Darstellung zur Dreiecksungleichung
+dreieck.pdf:	dreieck.tex drei.inc
+	pdflatex dreieck.tex
+
+drei.inc:	dreieck.m
+	octave dreieck.m
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/diffusion.png b/buch/chapters/80-wahrscheinlichkeit/images/diffusion.png
new file mode 100644
index 0000000..f4c6294
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/diffusion.png
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/diffusion.pov b/buch/chapters/80-wahrscheinlichkeit/images/diffusion.pov
new file mode 100644
index 0000000..9b385da
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/diffusion.pov
@@ -0,0 +1,87 @@
+//
+// diffusion.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.270;
+#declare N = 30;
+#declare vscale = 10;
+#declare r = 0.08;
+
+camera {
+        location <43, 20, -50>
+        look_at <N/2+2, vscale*0.49, 3>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <-4, 20, -50> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+#macro saeule(xx,yy,h)
+box { <xx+0.1,0,yy+0.1>, <xx+0.9,vscale*h,yy+0.9> }
+#end
+
+#macro vektor(xx,a,b,c,d,e,f)
+	saeule(xx,5,a)
+	saeule(xx,4,b)
+	saeule(xx,3,c)
+	saeule(xx,2,d)
+	saeule(xx,1,e)
+	saeule(xx,0,f)
+#end
+
+union {
+#include "vektoren.inc"
+	pigment {
+		color rgb<0.8,1,1>*0.6
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+#declare xx = 1;
+#while (xx <= N+1)
+	cylinder { <xx, 0, 0>, <xx, 0, 6>, r }
+	#declare xx = xx + 1;
+#end
+#declare yy = 0;
+#while (yy <= 6)
+	cylinder { <1, 0, yy>, <N+1, 0, yy>, r }
+	#declare yy = yy + 1;
+#end
+	sphere { <1,0,0>, r }
+	sphere { <1,0,6>, r }
+	sphere { <N+1,0,0>, r }
+	sphere { <N+1,0,6>, r }
+	cylinder { <1,0,6>, <1,1.1*vscale,6>, r }
+	cylinder { <1,vscale-r/2,6>, <1,vscale+r/2,6>, 2*r }
+	cone { <1,1.1*vscale,6>, 2*r, <1,1.15*vscale,6>, 0 }
+	pigment {
+		color rgb<1,0.6,1>*0.6
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/dreieck.m b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.m
new file mode 100644
index 0000000..cc9661b
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.m
@@ -0,0 +1,41 @@
+#
+# dreieck.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+w = 12
+N = 10
+
+rand("seed", 1);
+
+angles = 80 * rand(1,N)
+radii = 2 * rand(1,N) + 0.4
+angle = 20
+
+radii = radii * w / (cosd(angle) * sum(radii))
+radius = sum(radii)
+radius * cosd(angle)
+
+points = zeros(2,N);
+ray = zeros(2,N);
+
+p = [ 0; 0 ];
+for i = (1:N)
+	p = p + radii(1,i) * [ cosd(angles(1,i)); sind(angles(1,i)) ];
+	points(:, i) = p;
+	ray(:, i) = sum(radii(1,1:i)) * [ cosd(angle); sind(angle) ];
+end
+
+points
+
+ray
+
+fn = fopen("drei.inc", "w");
+for i = (1:N)
+	fprintf(fn, "\\coordinate (A%d) at (%.4f,%.4f);\n", i,
+		points(1,i), points(2,i));
+	fprintf(fn, "\\coordinate (B%d) at (%.4f,%.4f);\n", i,
+		ray(1,i), ray(2,i));
+end
+fprintf(fn, "\\def\\r{%.4f}\n", radius);
+fclose(fn);
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf
new file mode 100644
index 0000000..0cca2e1
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/dreieck.tex b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.tex
new file mode 100644
index 0000000..0935992
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/dreieck.tex
@@ -0,0 +1,90 @@
+%
+% dreieck.tex -- verallgemeinerte Dreiecksungleichung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usepackage{pgf}
+\usetikzlibrary{arrows,intersections,math,calc,hobby}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\coordinate (O) at (0,0);
+
+\input{drei.inc}
+
+\begin{scope}
+\clip (O) rectangle (12.3,8);
+\draw[color=red!40] (O) circle[radius=\r];
+\end{scope}
+
+\draw[->] (-0.1,0) -- (12.3,0) coordinate[label={$\Re z$}];
+\draw[->] (0,-0.1) -- (0,8.3) coordinate[label={right:$\Im z$}];
+
+\fill[color=blue] (A1) circle[radius=0.05];
+\fill[color=blue] (A2) circle[radius=0.05];
+\fill[color=blue] (A3) circle[radius=0.05];
+\fill[color=blue] (A4) circle[radius=0.05];
+\fill[color=blue] (A5) circle[radius=0.05];
+\fill[color=blue] (A6) circle[radius=0.05];
+\fill[color=blue] (A7) circle[radius=0.05];
+\fill[color=blue] (A8) circle[radius=0.05];
+\fill[color=blue] (A9) circle[radius=0.05];
+\fill[color=blue] (A10) circle[radius=0.05];
+\draw[color=blue] (O) -- (A1);
+\draw[color=blue] (A1) -- (A2);
+\draw[color=blue] (A2) -- (A3);
+\draw[color=blue] (A3) -- (A4);
+\draw[color=blue] (A4) -- (A5);
+\draw[color=blue] (A5) -- (A6);
+\draw[color=blue] (A6) -- (A7);
+\draw[color=blue] (A7) -- (A8);
+\draw[color=blue] (A8) -- (A9);
+\draw[color=blue] (A9) -- (A10);
+\draw[->,color=blue!40] (O) -- (A10);
+\node[color=blue] at ($0.5*(A1)$) [left] {$z_1$};
+\node[color=blue] at ($0.5*(A1)+0.5*(A2)$) [left] {$z_2$};
+\node[color=blue] at ($0.5*(A2)+0.5*(A3)$) [above] {$z_3$};
+\node[color=blue] at ($0.5*(A3)+0.5*(A4)$) [above] {$z_4$};
+\node[color=blue] at ($0.5*(A4)+0.5*(A5)$) [below right] {$z_5$};
+\node[color=blue] at ($0.5*(A5)+0.5*(A6)$) [left] {$z_6$};
+\node[color=blue] at ($0.5*(A6)+0.5*(A7)$) [left] {$z_7$};
+\node[color=blue] at ($0.5*(A7)+0.5*(A8)$) [above] {$z_8$};
+\node[color=blue] at ($0.5*(A8)+0.5*(A9)$) [left] {$z_9$};
+\node[color=blue] at ($0.5*(A9)+0.5*(A10)$) [above] {$z_{10}$};
+\node[color=blue] at ($0.8*(A10)$) [rotate=35,below] {$\displaystyle\sum_{i=1}^n z_i$};
+
+\draw[->,color=red] (O) -- (B10);
+\fill[color=red] (B1) circle[radius=0.05];
+\fill[color=red] (B2) circle[radius=0.05];
+\fill[color=red] (B3) circle[radius=0.05];
+\fill[color=red] (B4) circle[radius=0.05];
+\fill[color=red] (B5) circle[radius=0.05];
+\fill[color=red] (B6) circle[radius=0.05];
+\fill[color=red] (B7) circle[radius=0.05];
+\fill[color=red] (B8) circle[radius=0.05];
+\fill[color=red] (B9) circle[radius=0.05];
+\fill[color=red] (B10) circle[radius=0.05];
+
+\node[color=red] at ($0.5*(B1)$) [above] {$|z_1|c$};
+\node[color=red] at ($0.5*(B1)+0.5*(B2)$) [above] {$|z_2|c$};
+\node[color=red] at ($0.5*(B2)+0.5*(B3)$) [above] {$|z_3|c$};
+\node[color=red] at ($0.5*(B3)+0.5*(B4)$) [above] {$|z_4|c$};
+\node[color=red] at ($0.5*(B4)+0.5*(B5)$) [above] {$|z_5|c$};
+\node[color=red] at ($0.5*(B5)+0.5*(B6)$) [above] {$|z_6|c$};
+\node[color=red] at ($0.5*(B6)+0.5*(B7)$) [above] {$|z_7|c$};
+\node[color=red] at ($0.5*(B7)+0.5*(B8)$) [above] {$|z_8|c$};
+\node[color=red] at ($0.5*(B8)+0.5*(B9)$) [above] {$|z_9|c$};
+\node[color=red] at ($0.5*(B9)+0.5*(B10)$) [above] {$|z_{10}|c$};
+
+\node[color=red] at ($0.8*(B10)$) [rotate=20,below] {$\displaystyle c\sum_{i=1}^n |z_i|$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/positiv.jpg b/buch/chapters/80-wahrscheinlichkeit/images/positiv.jpg
new file mode 100644
index 0000000..53544cb
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/positiv.jpg
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/positiv.m b/buch/chapters/80-wahrscheinlichkeit/images/positiv.m
new file mode 100644
index 0000000..4dca950
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/positiv.m
@@ -0,0 +1,36 @@
+#
+# positiv.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+N = 10;
+p = 0.2;
+
+A = eye(3) + p * rand(3,3);
+A = [
+ 1, 0.2, 0.2;
+ 0.1, 1, 0.1;
+ 0.05, 0.05, 1
+];
+B = eye(3);
+
+function retval = punkt(x)
+	retval = sprintf("<%.4f,%.4f,%.4f>", x(1), x(3), x(2));
+end
+
+fn = fopen("quadrant.inc", "w");
+for i = (1:N)
+	fprintf(fn, "quadrant(%s,%s,%s)\n",
+		punkt(B(:,1)), punkt(B(:,2)), punkt(B(:,3)))
+	B = B * A;
+end
+
+x = [ 1; 1; 1 ];
+for i = (1:100)
+	x = A * x;
+	x = x / norm(x);
+end
+fprintf(fn, "eigenvektor(<%.4f, %.4f, %.4f>)\n", x(1), x(3), x(2));
+
+
+fclose(fn);
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/positiv.png b/buch/chapters/80-wahrscheinlichkeit/images/positiv.png
new file mode 100644
index 0000000..a2bd9bf
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/positiv.png
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/positiv.pov b/buch/chapters/80-wahrscheinlichkeit/images/positiv.pov
new file mode 100644
index 0000000..9197498
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/positiv.pov
@@ -0,0 +1,137 @@
+//
+// diffusion.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.077;
+#declare N = 30;
+#declare vscale = 10;
+#declare r = 0.04;
+
+camera {
+        location <43, 20, -20>
+        look_at <1, 0.83, 2.5>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <40, 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>
+        }
+}
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * 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
+
+arrow(<0,0,0>, <3,0,0>, r, White)
+arrow(<0,0,0>, <0,3,0>, r, White)
+arrow(<0,0,0>, <0,0,3>, r, White)
+
+#macro quadrant(A, B, C)
+intersection {
+	sphere { <0, 0, 0>, 1 
+		matrix <A.x, A.y, A.z,
+			B.x, B.y, B.z,
+			C.x, C.y, C.z,
+			0,   0,   0    >
+	}
+	plane { vnormalize(vcross(A, B)), 0 }
+	plane { vnormalize(vcross(B, C)), 0 }
+	plane { vnormalize(vcross(C, A)), 0 }
+	pigment {
+		//color rgbf<0.8,1,1,0.7>
+		color rgb<0.8,1,1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro eigenvektor(E)
+union {
+	cylinder { -E, 8 * E, r }
+	#declare r0 = 0.7 * r;
+
+	sphere { 3 * <   0, E.y, E.z >, r0 }
+	sphere { 3 * < E.x,   0, E.z >, r0 }
+	sphere { 3 * < E.x, E.y,   0 >, r0 }
+	sphere { 3 * E, r0 }
+
+	cylinder { 3*< E.x,   0,   0 >, 3*< E.x,   0, E.z >, r0 }
+	cylinder { 3*< E.x,   0,   0 >, 3*< E.x, E.y,   0 >, r0 }
+	cylinder { 3*<   0, E.y,   0 >, 3*< E.x, E.y,   0 >, r0 }
+	cylinder { 3*<   0, E.y,   0 >, 3*<   0, E.y, E.z >, r0 }
+	cylinder { 3*<   0,   0, E.z >, 3*<   0, E.y, E.z >, r0 }
+	cylinder { 3*<   0,   0, E.z >, 3*< E.x,   0, E.z >, r0 }
+
+	cylinder { 3*< E.x, E.y,   0 >, 3*E, r0 }
+	cylinder { 3*<   0, E.y, E.z >, 3*E, r0 }
+	cylinder { 3*< E.x,   0, E.z >, 3*E, r0 }
+	pigment {
+		color rgb<1,0.6,1>*0.6
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#include "quadrant.inc"
+
+//union {
+//	pigment {
+//		color rgb<0.8,1,1>*0.6
+//	}
+//	finish {
+//		specular 0.9
+//		metallic
+//	}
+//}
+
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/trenn.pdf b/buch/chapters/80-wahrscheinlichkeit/images/trenn.pdf
new file mode 100644
index 0000000..f4fa58f
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/trenn.pdf
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/trenn.tex b/buch/chapters/80-wahrscheinlichkeit/images/trenn.tex
new file mode 100644
index 0000000..f34879c
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/trenn.tex
@@ -0,0 +1,44 @@
+%
+% trenn.tex -- Trenntrick graphische Darstellung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\d{6}
+
+\coordinate (u) at (5,3);
+\coordinate (v) at (3,1);
+\coordinate (ve) at (5,1.666);
+
+\fill[color=gray!40] (0,0) rectangle (u);
+
+\begin{scope}
+\clip (0,0) rectangle (6.1,4.1);
+\draw[color=red] (0,0) -- (9,3);
+\end{scope}
+
+\draw[->] (-0.1,0) -- (6.3,0) coordinate[label={$x_1$}];
+\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
+
+\fill (u) circle[radius=0.05];
+\node at (u) [above right] {$u$};
+
+\fill (v) circle[radius=0.05];
+\node at (v) [above right] {$v$};
+
+\fill[color=red] (ve) circle[radius=0.05];
+\node[color=red] at (ve) [above,rotate={atan(1/3)}] {$(1+\varepsilon)v$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.jpg b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.jpg
new file mode 100644
index 0000000..3274f42
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.jpg
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf
new file mode 100644
index 0000000..feb19a2
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.png b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.png
new file mode 100644
index 0000000..f20bd48
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.png
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pov b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pov
new file mode 100644
index 0000000..e696481
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pov
@@ -0,0 +1,203 @@
+//
+// diffusion.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "textures.inc"
+#include "transforms.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.077;
+#declare N = 30;
+#declare vscale = 10;
+#declare r = 0.04;
+
+camera {
+        location <43, 20, -20>
+        look_at <1, 0.83, 2.5>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <20, 60, -80> color White
+        area_light <1,0,0> <0,0,1>, 40, 40
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * 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 O = <0,0,0>;
+#declare Ex = <1,0,0>;
+#declare Ey = <0,1,0>;
+#declare Ez = <0,0,1>;
+#declare s = 3;
+
+#declare A_transformation = Matrix_Trans(<1.0300,0.2050,0.1050>,<0.4100,1.0250,0.1100>,<0.4200,0.2200,1.0150>,<0,0,0>);
+//#declare A_transformation = Matrix_Trans(<1.0300,0.2050,0.1050>,<0.4100,1.0250,0.1100>,<0.4200,0.2200,0.5150>,<0,0,0>);
+
+arrow(O, s * Ex, r, rgb<0.6,0.2,0.4>)
+arrow(O, s * Ez, r, rgb<0.2,0.4,0.2>)
+arrow(O, s * Ey, r, rgb<0.2,0.2,0.4>)
+
+#declare A = vtransform(Ex, A_transformation);
+#declare B = vtransform(Ey, A_transformation);
+#declare C = vtransform(Ez, A_transformation);
+
+#macro quadrant(rad)
+intersection {
+	sphere { <0, 0, 0>, rad
+		//A_transformation
+		matrix <A.x, A.y, A.z,
+			B.x, B.y, B.z,
+			C.x, C.y, C.z,
+			0,   0,   0    >
+	}
+	plane { vnormalize(-vcross(A, B)), 0 }
+	plane { vnormalize(-vcross(B, C)), 0 }
+	plane { vnormalize(-vcross(C, A)), 0 }
+	pigment {
+		color rgbf<0.8,1,1,0.5>
+		//color rgb<0.8,1,1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+union {
+	cylinder { O, s*A, 0.3*r }
+	sphere { s*A, 0.3*r }
+	cylinder { O, s*B, 0.3*r }
+	sphere { s*B, 0.3*r }
+	cylinder { O, s*C, 0.3*r }
+	sphere { s*C, 0.3*r }
+	pigment {
+		color White
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare d = 3;
+//union {
+//	plane { <0, 1, 0>, -d }
+//	plane { <1, 0, 0>, -d }
+//	pigment {
+//		color Gray
+//	}
+//	finish {
+//		specular 0.9
+//	}
+//}
+
+quadrant(s)
+
+#declare V = < 1, 1, 0 >;
+#declare U = < 1.3, 2.5, 0 >;
+
+#declare VV = vtransform(V, A_transformation);
+#declare Vx = vtransform(<V.x, 0, 0>, A_transformation);
+#declare Vy = vtransform(<0, V.y, 0>, A_transformation);
+#declare UU = vtransform(U, A_transformation);
+#declare Ux = vtransform(<U.x, 0, 0>, A_transformation);
+#declare Uy = vtransform(<0, U.y, 0>, A_transformation);
+
+union {
+	sphere { V, r }
+	sphere { U, r }
+	cylinder { U, V, 0.5*r }
+	pigment {
+		color Red
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+	cylinder { < U.x,   0, 0 >, < U.x, U.y,   0>, 0.3 * r }
+	cylinder { < V.x,   0, 0 >, < V.x, V.y,   0>, 0.3 * r }
+	cylinder { <   0, U.y, 0 >, < U.x, U.y,   0>, 0.3 * r }
+	cylinder { <   0, V.y, 0 >, < V.x, V.y,   0>, 0.3 * r }
+	pigment {
+		color rgb<1, 0.6, 1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+	sphere { VV, r }
+	sphere { UU, r }
+	cylinder { UU, VV, 0.5*r }
+	pigment {
+		color Yellow
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+	cylinder { Ux, UU, 0.3 * r }
+	cylinder { Uy, UU, 0.3 * r }
+	cylinder { Vx, VV, 0.3 * r }
+	cylinder { Vy, VV, 0.3 * r }
+	pigment {
+		color rgb<1, 1, 0.6>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.tex b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.tex
new file mode 100644
index 0000000..23d7d66
--- /dev/null
+++ b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.tex
@@ -0,0 +1,46 @@
+%
+% vergleich.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{5}
+\def\hoehe{5}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=10cm]{vergleich.jpg}};
+
+\node at (-1.3,-4.8) [right] {$x_1$};
+\node[opacity=0.5] at (1.9,-0.9) [right] {$x_2$};
+\node at (-4.6,4.7) [right] {$x_3$};
+
+\node at (-3.2,2.6) [above] {$u$};
+\node at (-3.5,-0.7) [below left] {$v$};
+\node at (-1,2.8) [above] {$Au$};
+\node at (-2.6,-0.5) [below] {$Av$};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\end{tikzpicture}
+
+\end{document}
+