new spherical graph

4 files changed, 257 insertions, 1 deletions

diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile
index e8bf919..6187fed 100644
--- a/buch/papers/kugel/images/Makefile
+++ b/buch/papers/kugel/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all: curvature.jpg
+all: curvature.jpg spherecurve.jpg
 
 curvature.inc:	curvgraph.m
 	octave curvgraph.m
@@ -13,3 +13,13 @@ curvature.png:	curvature.pov curvature.inc
 
 curvature.jpg:	curvature.png
 	convert curvature.png -density 300 -units PixelsPerInch curvature.jpg
+
+spherecurve.inc:	spherecurve.m
+	octave spherecurve.m
+
+spherecurve.png:	spherecurve.pov spherecurve.inc
+	povray +A0.1 +W1920 +H1080 +Ospherecurve.png spherecurve.pov
+
+spherecurve.jpg:	spherecurve.png
+	convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg
+
diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m
new file mode 100644
index 0000000..ea9c901
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.m
@@ -0,0 +1,160 @@
+#
+# spherecurv.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global a;
+a = 5;
+global A;
+A = 10;
+
+phisteps = 400;
+hphi = 2 * pi / phisteps;
+thetasteps = 200;
+htheta = pi / thetasteps;
+
+function retval = f(z)
+	global a;
+	global A;
+	retval = A * exp(a * (z^2 - 1));
+endfunction
+
+function retval = g(z)
+	global a;
+	retval = -f(z) * 2 * a * (2 * a * z^4 + (3 - 2*a) * z^2 - 1);
+	#                                                              2
+	#                   - a     2  4             2   2          a z
+	#(%o6)          - %e    (4 a  z  + (6 a - 4 a ) z  - 2 a) %e
+endfunction
+
+phi = (1 + sqrt(5)) / 2;
+
+global axes;
+axes = [
+     0,   0,   1,  -1, phi, -phi;
+     1,  -1, phi, phi,   0,    0;
+   phi, phi,   0,   0,   1,    1;
+];
+axes = axes / (sqrt(phi^2+1));
+
+function retval = kugel(theta, phi)
+	retval = [
+	    cos(phi) * sin(theta);
+	    sin(phi) * sin(theta);
+	               cos(theta)
+	];
+endfunction
+
+function retval = F(v)
+	global axes;
+	s = 0;
+	for i = (1:6)
+		z = axes(:,i)' * v;
+		s = s + f(z);
+	endfor
+	retval = s / 6;
+endfunction
+
+function retval = F2(theta, phi)
+	v = kugel(theta, phi);
+	retval = F(v);
+endfunction
+
+function retval = G(v)
+	global axes;
+	s = 0;
+	for i = (1:6)
+		s = s + g(axes(:,i)' * v);
+	endfor
+	retval = s / 6;
+endfunction
+
+function retval = G2(theta, phi)
+	v = kugel(theta, phi);
+	retval = G(v);
+endfunction
+
+function retval = cnormalize(u)
+	utop = 11;
+	ubottom = -30;
+	retval = (u - ubottom)  / (utop - ubottom);
+	if (retval > 1) 
+		retval = 1;
+	endif
+	if (retval < 0)
+		retval = 0;
+	endif
+endfunction
+
+global umin;
+umin = 0;
+global umax;
+umax = 0;
+
+function color = farbe(v)
+	global umin;
+	global umax;
+	u = G(v);
+	if (u < umin) 
+		umin = u;
+	endif
+	if (u > umax)
+		umax = u;
+	endif
+	u = cnormalize(u);
+	color = [ u, 0.5, 1-u ];
+	color = color/max(color);
+endfunction
+
+function dreieck(fn, v0, v1, v2)
+	fprintf(fn, "    mesh {\n");
+	c = (v0 + v1 + v2) / 3;
+	c = c / norm(c);
+	color = farbe(c);
+	v0 = v0 * (1 + F(v0));
+	v1 = v1 * (1 + F(v1));
+	v2 = v2 * (1 + F(v2));
+	fprintf(fn, "\ttriangle {\n");
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>,\n", v0(1,1), v0(3,1), v0(2,1));
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>,\n", v1(1,1), v1(3,1), v1(2,1));
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>\n",  v2(1,1), v2(3,1), v2(2,1));
+	fprintf(fn, "\t}\n");
+	fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n",
+		color(1,1), color(1,2), color(1,3));
+	fprintf(fn, "\tfinish { metallic specular 0.5 }\n");
+	fprintf(fn, "    }\n");
+endfunction
+
+fn = fopen("spherecurve.inc", "w");
+
+	for i = (1:phisteps)
+		# Polkappe nord
+		v0 = [ 0; 0; 1 ];
+		v1 = kugel(htheta, (i-1) * hphi);
+		v2 = kugel(htheta,  i    * hphi);
+		fprintf(fn, "    // i = %d\n", i);
+		dreieck(fn, v0, v1, v2);
+
+		# Polkappe sued
+		v0 = [ 0; 0; -1 ];
+		v1 = kugel(pi-htheta, (i-1) * hphi);
+		v2 = kugel(pi-htheta,  i    * hphi);
+		dreieck(fn, v0, v1, v2);
+	endfor
+
+	for j = (1:thetasteps-2)
+		for i = (1:phisteps)
+			v0 = kugel( j    * htheta, (i-1) * hphi);
+			v1 = kugel((j+1) * htheta, (i-1) * hphi);
+			v2 = kugel( j    * htheta,  i    * hphi);
+			v3 = kugel((j+1) * htheta,  i    * hphi);
+			fprintf(fn, "    // i = %d, j = %d\n", i, j);
+			dreieck(fn, v0, v1, v2);
+			dreieck(fn, v1, v2, v3);
+		endfor
+	endfor
+
+fclose(fn);
+
+umin
+umax
diff --git a/buch/papers/kugel/images/spherecurve.maxima b/buch/papers/kugel/images/spherecurve.maxima
new file mode 100644
index 0000000..1e9077c
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.maxima
@@ -0,0 +1,13 @@
+/*
+ * spherecurv.maxima
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+f: exp(-a * sin(theta)^2);
+
+g: ratsimp(diff(sin(theta) * diff(f, theta), theta)/sin(theta));
+g: subst(z, cos(theta), g);
+g: subst(sqrt(1-z^2), sin(theta), g);
+ratsimp(g);
+
+f: ratsimp(subst(sqrt(1-z^2), sin(theta), f));
diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov
new file mode 100644
index 0000000..86c3745
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.pov
@@ -0,0 +1,73 @@
+//
+// curvature.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.14;
+
+camera {
+	location <10, 10, -40>
+	look_at <0, 0, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-10, 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(<-2.7,0,0>, <2.7,0,0>, 0.03, White)
+arrow(<0,-2.7,0>, <0,2.7,0>, 0.03, White)
+arrow(<0,0,-2.7>, <0,0,2.7>, 0.03, White)
+
+#include "spherecurve.inc"
+