Merge branch 'master' of github.com:AndreasFMueller/SeminarSpezielleFunktionen

8 files changed, 701 insertions, 1 deletions

diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index a7c9e74..c8f98cb 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -5,7 +5,7 @@
 #
 all:	lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf \
 	ellipse.pdf pendel.pdf jacobiplots.pdf jacobidef.pdf jacobi12.pdf \
-	sncnlimit.pdf slcl.pdf
+	sncnlimit.pdf slcl.pdf torusschnitt.pdf kegelpara.pdf
 
 lemniskate.pdf:	lemniskate.tex
 	pdflatex lemniskate.tex
@@ -78,3 +78,27 @@ slcldata.tex:	slcl
 	./slcl --outfile=slcldata.tex --a=0 --b=13.4 --steps=200
 slcl.pdf:	slcl.tex slcldata.tex
 	pdflatex slcl.tex
+
+KEGELSIZE = -W256 -H256
+KEGELSIZE = -W128 -H128
+KEGELSIZE = -W1080 -H1080
+kegelpara.png:	kegelpara.pov
+	povray +A0.1 $(KEGELSIZE) -Okegelpara.png kegelpara.pov
+
+kegelpara.jpg:	kegelpara.png Makefile
+	convert -extract 1080x1040+0+0 kegelpara.png \
+		-density 300 -units PixelsPerInch kegelpara.jpg
+
+kegelpara.pdf:	kegelpara.tex kegelpara.jpg
+	pdflatex kegelpara.tex
+
+torusschnitt.png:	torusschnitt.pov
+	povray +A0.1 -W1920 -H1080 -Otorusschnitt.png torusschnitt.pov
+
+torusschnitt.jpg:	torusschnitt.png Makefile
+	convert -extract 1560x1080+180+0 torusschnitt.png \
+		-density 300 -units PixelsPerInch torusschnitt.jpg
+
+torusschnitt.pdf:	torusschnitt.tex torusschnitt.jpg
+	pdflatex torusschnitt.tex
+
diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
index f0e6e78..c11affc 100644
--- a/buch/chapters/110-elliptisch/images/jacobiplots.pdf
+++ b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pdf b/buch/chapters/110-elliptisch/images/kegelpara.pdf
new file mode 100644
index 0000000..2f76593
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pdf
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pov b/buch/chapters/110-elliptisch/images/kegelpara.pov
new file mode 100644
index 0000000..13b66cc
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pov
@@ -0,0 +1,329 @@
+//
+// kegelpara.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+#declare O = <0,0,0>;
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.08;
+
+camera {
+        location <28, 20, -40>
+        look_at <0, 0.1, 0>
+        right x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <30, 10, -40> 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(<-2.6,0,0>,<2.5,0,0>,0.02,White)
+arrow(<0,-2,0>,<0,2.3,0>,0.02,White)
+arrow(<0,0,-3.2>,<0,0,3.7>,0.02,White)
+
+#declare epsilon = 0.0001;
+#declare l = 1.5;
+
+#macro Kegel(farbe)
+union {
+	difference {
+		cone { O, 0, <l, 0, 0>, l }
+		cone { O + <epsilon, 0,0>, 0, <l+epsilon, 0, 0>, l }
+	}
+	difference {
+		cone { O, 0, <-l, 0, 0>, l }
+		cone { O + <-epsilon, 0, 0>, 0, <-l-epsilon, 0, 0>, l }
+	}
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Kegelpunkt(xx, phi)
+	< xx, xx * sin(phi), xx * cos(phi) >
+#end
+
+#macro Kegelgitter(farbe, r)
+union {
+	#declare s = 0;
+	#declare smax = 2 * pi;
+	#declare sstep = pi / 6;
+	#while (s < smax - sstep/2)
+		cylinder { Kegelpunkt(l, s), Kegelpunkt(-l, s), r }
+		#declare s = s + sstep;
+	#end
+	#declare phimax = 2 * pi;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#declare xxstep = 0.5;
+	#declare xxmax = 2;
+	#declare xx = xxstep;
+	#while (xx < xxmax - xxstep/2)
+		#declare phi = 0;
+		#while (phi < phimax - phistep/2)
+			cylinder {
+				Kegelpunkt(xx, phi),
+				Kegelpunkt(xx, phi + phistep),
+				r
+			}
+			sphere { Kegelpunkt(xx, phi), r }
+			cylinder {
+				Kegelpunkt(-xx, phi),
+				Kegelpunkt(-xx, phi + phistep),
+				r
+			}
+			sphere { Kegelpunkt(-xx, phi), r }
+			#declare phi = phi + phistep;
+		#end
+		#declare xx = xx + xxstep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro F(w, r)
+	<r * cos(w), r * r/sqrt(2), r * sin(w) >
+#end
+
+#macro Paraboloid(farbe)
+mesh {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 100;
+	#declare phistep = pi / phisteps;
+	#declare rsteps = 100;
+	#declare rmax = 1.5;
+	#declare rstep = rmax / rsteps;
+	#while (phi < phimax - phistep/2)
+		#declare r = rstep;
+		#declare h = r * r / sqrt(2);
+		triangle {
+			O, F(phi, r), F(phi + phistep, r)
+		}
+		#while (r < rmax - rstep/2)
+			// ring
+			triangle {
+				F(phi, r),
+				F(phi + phistep, r),
+				F(phi + phistep, r + rstep)
+			}
+			triangle {
+				F(phi, r),
+				F(phi + phistep, r + rstep),
+				F(phi, r + rstep)
+			}
+			#declare r = r + rstep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Paraboloidgitter(farbe, gr)
+union {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phistep = pi / 6;
+
+	#declare rmax = 1.5;
+	#declare rsteps = 100;
+	#declare rstep = rmax / rsteps;
+
+	#while (phi < phimax - phistep/2)
+		#declare r = rstep;
+		#while (r < rmax - rstep/2)
+			cylinder { F(phi, r), F(phi, r + rstep), gr }
+			sphere { F(phi, r), gr }
+			#declare r = r + rstep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+
+	#declare rstep = 0.2;
+	#declare r = rstep;
+
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (r < rmax)
+		#declare phi = 0;
+		#while (phi < phimax - phistep/2)
+			cylinder { F(phi, r), F(phi + phistep, r), gr }
+			sphere { F(phi, r), gr }
+			#declare phi = phi + phistep;
+		#end
+		#declare r = r + rstep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare a = sqrt(2);
+#macro G(phi,sg)
+	< a*sg*sqrt(cos(2*phi))*cos(phi), a*cos(2*phi), a*sqrt(cos(2*phi))*sin(phi)>
+#end
+
+#macro Lemniskate3D(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G(phi,1), s }
+		cylinder { G(phi,1), G(phi+phistep,1), s }
+		sphere { G(phi,-1), s }
+		cylinder { G(phi,-1), G(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare a = sqrt(2);
+#macro G2(phi,sg)
+	a * sqrt(cos(2*phi)) * < sg * cos(phi), 0, sin(phi)>
+#end
+
+#macro Lemniskate(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G2(phi,1), s }
+		cylinder { G2(phi,1), G2(phi+phistep,1), s }
+		sphere { G2(phi,-1), s }
+		cylinder { G2(phi,-1), G2(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Projektion(s, farbe)
+union {
+	#declare phistep = pi / 16;
+	#declare phi = -pi / 4 + phistep;
+	#declare phimax = pi / 4;
+	#while (phi < phimax - phistep/2)
+		cylinder { G(phi,  1), G2(phi,  1), s }
+		cylinder { G(phi, -1), G2(phi, -1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare kegelfarbe = rgbf<0.2,0.6,0.2,0.2>;
+#declare kegelgitterfarbe = rgb<0.2,0.8,0.2>;
+#declare paraboloidfarbe = rgbf<0.2,0.6,1.0,0.2>;
+#declare paraboloidgitterfarbe = rgb<0.4,1,1>;
+
+//intersection {
+//	union {
+		Paraboloid(paraboloidfarbe)
+		Paraboloidgitter(paraboloidgitterfarbe, 0.004)
+
+		Kegel(kegelfarbe)
+		Kegelgitter(kegelgitterfarbe, 0.004)
+//	}
+//	plane { <0, 0, -1>, 0.6 }
+//}
+
+
+Lemniskate3D(0.02, rgb<0.8,0.0,0.8>)
+Lemniskate(0.02, Red)
+Projektion(0.01, Yellow)
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.tex b/buch/chapters/110-elliptisch/images/kegelpara.tex
new file mode 100644
index 0000000..8fcefbf
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.tex
@@ -0,0 +1,41 @@
+%
+% kegelpara.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{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{kegelpara.jpg}};
+
+% 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];
+}{}
+
+\node at (4.1,-1.4) {$X$};
+\node at (0.2,3.8) {$Z$};
+\node at (4.0,1.8) {$Y$};
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pdf b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
new file mode 100644
index 0000000..11bd353
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pov b/buch/chapters/110-elliptisch/images/torusschnitt.pov
new file mode 100644
index 0000000..43d50c6
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pov
@@ -0,0 +1,265 @@
+//
+// kegelpara.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+#declare O = <0,0,0>;
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.060;
+
+camera {
+        location <28, 20, -40>
+        look_at <0, 0.55, 0>
+        right (16/9) * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <30, 10, -40> 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(<-2,0,0>,<2,0,0>,0.02,White)
+arrow(<0,-1.1,0>,<0,2.2,0>,0.02,White)
+arrow(<0,0,-1.6>,<0,0,2.4>,0.02,White)
+
+#declare epsilon = 0.001;
+#declare l = 1.5;
+
+
+#declare a = sqrt(2);
+#macro G2(phi,sg)
+	a * sqrt(cos(2*phi)) * < sg * cos(phi), 0, sin(phi)>
+#end
+
+#macro Lemniskate(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G2(phi,1), s }
+		cylinder { G2(phi,1), G2(phi+phistep,1), s }
+		sphere { G2(phi,-1), s }
+		cylinder { G2(phi,-1), G2(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Projektion(s, farbe)
+union {
+	#declare phistep = pi / 16;
+	#declare phi = -pi / 4 + phistep;
+	#declare phimax = pi / 4;
+	#while (phi < phimax - phistep/2)
+		cylinder { G(phi,  1), G2(phi,  1), s }
+		cylinder { G(phi, -1), G2(phi, -1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Ebene(l, b, farbe)
+mesh {
+	triangle { <-l, 0, -b>, < l, 0, -b>, < l, 0,  b> }
+	triangle { <-l, 0, -b>, < l, 0,  b>, <-l, 0,  b> }
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Ebenengitter(l, b, s, r, farbe)
+union {
+	#declare lmax = floor(l / s);
+	#declare ll = -lmax;
+	#while (ll <= lmax)
+		cylinder { <ll * s, 0, -b>, <ll * s, 0, b>, r }
+		#declare ll = ll + 1;
+	#end
+	#declare bmax = floor(b / s);
+	#declare bb = -bmax;
+	#while (bb <= bmax)
+		cylinder { <-l, 0, bb * s>, <l, 0, bb * s>, r }
+		#declare bb = bb + 1;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare b = 0.5;
+#macro T(phi, theta)
+	b * < (2 + cos(theta)) * cos(phi), (2 + cos(theta)) * sin(phi) + 1, sin(theta) >
+#end
+
+#macro breitenkreis(theta, r)
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 200;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		cylinder { T(phi, theta), T(phi + phistep, theta), r }
+		sphere { T(phi, theta), r }
+		#declare phi = phi + phistep;
+	#end
+#end
+
+#macro laengenkreis(phi, r)
+	#declare theta = 0;
+	#declare thetamax = 2 * pi;
+	#declare thetasteps = 200;
+	#declare thetastep = thetamax / thetasteps;
+	#while (theta < thetamax - thetastep/2)
+		cylinder { T(phi, theta), T(phi, theta + thetastep), r }
+		sphere { T(phi, theta), r }
+		#declare theta = theta + thetastep;
+	#end
+#end
+
+#macro Torusgitter(farbe, r)
+union {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phistep = pi / 6;
+	#while (phi < phimax - phistep/2)
+		laengenkreis(phi, r)
+		#declare phi = phi + phistep;
+	#end
+	#declare thetamax = pi;
+	#declare thetastep = pi / 6;
+	#declare theta = thetastep;
+	#while (theta < thetamax - thetastep/2)
+		breitenkreis(theta, r)
+		breitenkreis(thetamax + theta, r)
+		#declare theta = theta + thetastep;
+	#end
+	breitenkreis(0, 1.5 * r)
+	breitenkreis(pi, 1.5 * r)
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Torus(farbe)
+mesh {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 200;
+	#declare phistep = phimax/phisteps;
+	#while (phi < phimax - phistep/2)
+		#declare theta = 0;
+		#declare thetamax = 2 * pi;
+		#declare thetasteps = 200;
+		#declare thetastep = thetamax / thetasteps;
+		#while (theta < thetamax - thetastep/2)
+			triangle {
+				T(phi,           theta),
+				T(phi + phistep, theta),
+				T(phi + phistep, theta + thetastep)
+			}
+			triangle {
+				T(phi,           theta),
+				T(phi + phistep, theta + thetastep),
+				T(phi,           theta + thetastep)
+			}
+			#declare theta = theta + thetastep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare torusfarbe = rgbt<0.2,0.6,0.2,0.2>;
+#declare ebenenfarbe = rgbt<0.2,0.6,1.0,0.2>;
+
+Lemniskate(0.02, Red)
+Ebene(1.8, 1.4, ebenenfarbe)
+Ebenengitter(1.8, 1.4, 0.5, 0.005, rgb<0.4,1,1>)
+Torus(torusfarbe)
+Torusgitter(Yellow, 0.005)
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.tex b/buch/chapters/110-elliptisch/images/torusschnitt.tex
new file mode 100644
index 0000000..3053ac5
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.tex
@@ -0,0 +1,41 @@
+%
+% torusschnitt.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{6}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=11.4cm]{torusschnitt.jpg}};
+
+% 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];
+}{}
+
+\node at (4.4,-2.4) {$X$};
+\node at (3.5,0.6) {$Y$};
+\node at (0.3,3.8) {$Z$};
+
+\end{tikzpicture}
+
+\end{document}
+