5 files changed, 418 insertions, 0 deletions
diff --git a/buch/chapters/95-homologie/torus/Makefile b/buch/chapters/95-homologie/torus/Makefile
new file mode 100644
index 0000000..4eb1381
--- /dev/null
+++ b/buch/chapters/95-homologie/torus/Makefile
@@ -0,0 +1,15 @@
+#
+# Makefile -- torus homology
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	torus.jpg
+
+torus.inc:	torus.m
+	octave torus.m
+
+torus.png:	torus.pov torus.inc
+	povray +A0.1 -W1920 -H1080 -Otorus.png torus.pov
+
+torus.jpg:	torus.png Makefile
+	convert -extract 1624x970+150+78 torus.png torus.jpg
diff --git a/buch/chapters/95-homologie/torus/torus.jpg b/buch/chapters/95-homologie/torus/torus.jpg
new file mode 100644
index 0000000..eaa92d9
--- /dev/null
+++ b/buch/chapters/95-homologie/torus/torus.jpg
diff --git a/buch/chapters/95-homologie/torus/torus.m b/buch/chapters/95-homologie/torus/torus.m
new file mode 100644
index 0000000..00134d7
--- /dev/null
+++ b/buch/chapters/95-homologie/torus/torus.m
@@ -0,0 +1,334 @@
+#
+# torus.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+global n;
+n = 24;
+global m;
+m = 12;
+global R;
+R = 3;
+global r;
+r = 1;
+
+# Knoten, Kanten, Dreiecke
+global nvertices;
+nvertices = n * m;
+global nedges;
+nedges = 3 * nvertices;
+global ntriangles;
+ntriangles = 2 * nvertices;
+
+edges = zeros(nedges, 2);
+global edges;
+triangles = zeros(ntriangles, 3);
+global triangles;
+
+function retval = torus(x, y)
+	global n;
+	global m;
+	global r;
+	global R;
+	phi = x * 2 * pi / n;
+	theta = y * 2 * pi / m;
+	z = -r * sin(theta);
+	x = (R + r * cos(theta)) * cos(phi);
+	y = -(R + r * cos(theta)) * sin(phi);
+	retval = [x, y, z];
+endfunction
+
+coordinates = zeros(nvertices, 3);
+for x = (0:n-1)
+	for y = (0:m-1)
+		k = x + n * y;
+		coordinates(k+1,:) = torus(x, y);
+	endfor
+endfor
+coordinates
+
+function insert_edges(ll, lr, ul, ur)
+	global edges;
+	k = 3 * ll + 1;
+	edges(k,1) = ll;
+	edges(k,2) = lr;
+	edges(k+1,1) = ll;
+	edges(k+1,2) = ur;
+	edges(k+2,1) = ll;
+	edges(k+2,2) = ul;
+endfunction
+
+function insert_triangles(ll, lr, ul, ur)
+	global triangles;
+	k = 2 * ll;
+printf("ll=%d, k=%d\n", ll, k);
+	triangles(k+1,1) = ll;
+	triangles(k+1,2) = lr;
+	triangles(k+1,3) = ur;
+	triangles(k+2,1) = ll;
+	triangles(k+2,2) = ur;
+	triangles(k+2,3) = ul;
+endfunction
+
+# normal squares
+for x = (0:n-2)
+	for y = (0:m-2)
+		ll = x + n * y;
+		lr = ll + 1;
+		ul = ll + n;
+		ur = ul + 1;
+		insert_edges(ll, lr, ul, ur);
+		insert_triangles(ll, lr, ul, ur);
+	endfor
+endfor
+
+# right border
+x = n-1;
+for y = (0:m-2)
+	ll = x + n * y;
+	lr = ll - (n-1);
+	ul = ll + n;
+	ur = lr + n;
+	insert_edges(ll, lr, ul, ur);
+	insert_triangles(ll, lr, ul, ur);
+endfor
+
+# top border
+y = m-1;
+for x = (0:n-2)
+	ll = x + n * y;
+	lr = ll + 1;
+	ul = x;
+	ur = ul + 1;
+	insert_edges(ll, lr, ul, ur);
+	insert_triangles(ll, lr, ul, ur);
+endfor
+
+# upper right corner
+x = n-1;
+y = m-1;
+ll = n * m - 1;
+lr = n * (m-1);
+ur = 0;
+ul = n - 1;
+insert_edges(ll, lr, ul, ur);
+insert_triangles(ll, lr, ul, ur);
+
+edges;
+triangles;
+
+d1 = zeros(nvertices, nedges);
+for i = (1:nedges)
+	d1(edges(i,1) + 1, i) = -1;
+	d1(edges(i,2) + 1, i) = +1;
+endfor
+
+function retval = find_edge(from, to)
+	global edges;
+	global nedges;
+	retval = 0;
+	for i = (1:nedges)
+		if ((edges(i,1) == from) && (edges(i,2) == to))
+			retval = i;
+			printf("    test (%d,%d) == (%d,%d) -> %d\n", from, to,
+				edges(i,1), edges(i,2), retval);
+		elseif ((edges(i,1) == to) && (edges(i,2) == from))
+			retval = -i;
+			printf("    test (%d,%d) == (%d,%d) -> %d\n", from, to,
+				edges(i,1), edges(i,2), retval);
+		endif
+	endfor
+endfunction
+
+global d2;
+d2 = zeros(nedges, ntriangles);
+function triangle_edge(i, pointa, pointb)
+	global d2;
+	edge = find_edge(pointa, pointb);
+	if (edge > 0)
+		d2(edge, i) = +1;
+	elseif (edge < 0)
+		d2(-edge, i) = -1;
+	endif
+endfunction
+
+for i = (1:ntriangles)
+	point1 = triangles(i,1);
+	point2 = triangles(i,2);
+	point3 = triangles(i,3);
+	#printf("triangle %d: %d, %d, %d\n", i-1, point1, point2, point3);
+	triangle_edge(i, point1, point2);
+	triangle_edge(i, point2, point3);
+	triangle_edge(i, point3, point1);
+endfor
+
+B = d2';
+
+global zyklentableau
+zyklentableau = rref(d1)
+i=((zyklentableau != 0) .* (1:nedges));
+global determined;
+determined = min((i+nedges*(i==0))')';
+determined = determined(1:nvertices-1,1);
+
+function retval = is_determined(i)
+	global nvertices;
+	global determined;
+#	printf("test i = %d\n", i);
+	retval = 0;
+	for k = (1:nvertices-1)
+#printf("check k=%d, i=%d\n", k, i);
+		if (determined(k,1) == i)
+			retval = 1;
+		endif
+	endfor
+endfunction
+
+function retval = zyklus(i)
+	global zyklentableau
+	global nedges
+	global nvertices
+	global determined
+	retval = zeros(nedges,1);
+	retval(i,1) = 1;
+	for j = (1:nvertices-1)
+		retval(determined(j,1),1) = -zyklentableau(j,i);
+	endfor
+endfunction
+
+Z = zeros(nedges, nedges - nvertices)
+current = 1;
+for i = (1:nedges)
+	if 1 == is_determined(i)
+		# printf("skipping %d\n", i);
+	else
+		Z(:,current) = zyklus(i);
+		current = current + 1;
+	endif
+endfor
+
+Z;
+
+zh = size(Z)(2);
+H = zeros(ntriangles + zh,nedges);
+H(1:ntriangles,:) = B;
+H((ntriangles+1):(ntriangles+zh),:) = Z';
+
+H;
+size(H)
+
+rref(B');
+
+function retval = pivotindex(zeile)
+	w = size(zeile)(2);
+	for k = (1:w)
+		if (zeile(1,k) != 0)
+			retval = k;
+			return;
+		endif
+	endfor
+	retval = 0;
+endfunction
+
+h = size(H)(1)
+w = size(H)(2)
+j = 1;
+
+for i = (1:h-1)
+	j = pivotindex(H(i,:));
+	printf("reduction for i = %d, j = %d\n", i, j);
+	if (j != 0) 
+		# find pivot index
+		pivot = H(i,j);
+		if (pivot != 0)
+			H(i,:) = H(i,:) / pivot;
+			for k = ((i+1):h)
+				H(k,:) = H(k,:) - H(k,j) * H(i,:);
+			endfor
+		endif
+	endif
+endfor
+H;
+
+Hbasis = zeros(2,nedges);
+current = 1;
+for i = ((ntriangles+1):h)
+	if norm(H(i,:)) > 0
+		Hbasis(current,:) = H(i,:)
+		current = current + 1;
+	endif
+endfor
+#size(Hbasis)
+#nedges
+#Hbasis
+#Hbasis(1,nedges-2)=0;
+#Hbasis(1,nedges-1)=1;
+Hbasis
+
+edges(71,:)
+coordinates
+
+fn = fopen("torus.inc", "w");
+
+# torusflaeche
+
+fprintf(fn, "#macro torusflaeche()\n");
+fprintf(fn, "    mesh {\n");
+for k = (1:ntriangles)
+	fprintf(fn, "\ttriangle { ");
+	punkt1 = coordinates(triangles(k,1)+1,:);
+	punkt2 = coordinates(triangles(k,2)+1,:);
+	punkt3 = coordinates(triangles(k,3)+1,:);
+	fprintf(fn, "<%.4f,%.4f,%.4f>, ", punkt1(1,1), punkt1(1,3), punkt1(1,2));
+	fprintf(fn, "<%.4f,%.4f,%.4f>, ", punkt2(1,1), punkt2(1,3), punkt2(1,2));
+	fprintf(fn, "<%.4f,%.4f,%.4f> }\n", punkt3(1,1), punkt3(1,3), punkt3(1,2));
+endfor
+fprintf(fn, "    }\n")
+fprintf(fn, "#end\n");
+
+fprintf(fn, "#macro zyklus1()\n");
+fprintf(fn, "    union {\n");
+for i = (1:nedges)
+	h = Hbasis(1,i);
+	if h != 0
+		if h > 0
+			punkt1 = coordinates(edges(i,1)+1,:);
+printf("i=%d\n", i);
+			punkt2 = coordinates(edges(i,2)+1,:);
+		else
+			punkt2 = coordinates(edges(i,1)+1,:);
+			punkt1 = coordinates(edges(i,2)+1,:);
+		endif
+		fprintf(fn, "\tcylinder { ");
+		fprintf(fn, "<%.4f,%.4f,%.4f>, ", punkt1(1,1), punkt1(1,3), punkt1(1,2));
+		fprintf(fn, "<%.4f,%.4f,%.4f>, r }\n", punkt2(1,1), punkt2(1,3), punkt2(1,2));
+		fprintf(fn, "\tsphere { <%.4f,%.4f,%.4f>, r }\n", punkt1(1,1), punkt1(1,3), punkt1(1,2));
+	endif
+endfor
+fprintf(fn, "    }\n");
+fprintf(fn, "#end\n");
+
+fprintf(fn, "#macro zyklus2()\n");
+fprintf(fn, "    union {\n");
+for i = (1:nedges)
+	h = Hbasis(2,i);
+	if h != 0
+		if h > 0
+			punkt1 = coordinates(edges(i,1)+1,:);
+			punkt2 = coordinates(edges(i,2)+1,:);
+		else
+			punkt2 = coordinates(edges(i,1)+1,:);
+			punkt1 = coordinates(edges(i,2)+1,:);
+		endif
+		fprintf(fn, "\tcylinder { ");
+		fprintf(fn, "<%.4f,%.4f,%.4f>, ", punkt1(1,1), punkt1(1,3), punkt1(1,2));
+		fprintf(fn, "<%.4f,%.4f,%.4f>, r }\n", punkt2(1,1), punkt2(1,3), punkt2(1,2));
+		fprintf(fn, "\tsphere { <%.4f,%.4f,%.4f>, r }\n", punkt1(1,1), punkt1(1,3), punkt1(1,2));
+	endif
+endfor
+fprintf(fn, "    }\n");
+fprintf(fn, "#end\n");
+
+fclose(fn);
+
diff --git a/buch/chapters/95-homologie/torus/torus.png b/buch/chapters/95-homologie/torus/torus.png
new file mode 100644
index 0000000..814c466
--- /dev/null
+++ b/buch/chapters/95-homologie/torus/torus.png
diff --git a/buch/chapters/95-homologie/torus/torus.pov b/buch/chapters/95-homologie/torus/torus.pov
new file mode 100644
index 0000000..292ad00
--- /dev/null
+++ b/buch/chapters/95-homologie/torus/torus.pov
@@ -0,0 +1,69 @@
+//
+// torus.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.1;
+#declare r = 0.03;
+
+camera {
+        location <43, 25, -20>
+        look_at <0, 0, 0>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <10, 20, -40> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+#include "torus.inc"
+
+object {
+	torusflaeche()
+	pigment {
+		color rgbt<0.8,0.8,0.8,0.5>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+object {
+	zyklus1()
+	pigment {
+		color Red
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+object {
+	zyklus2()
+	pigment {
+		color Blue
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}