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-rw-r--r--buch/chapters/95-homologie/torus/Makefile15
-rw-r--r--buch/chapters/95-homologie/torus/torus.jpgbin0 -> 168494 bytes
-rw-r--r--buch/chapters/95-homologie/torus/torus.m334
-rw-r--r--buch/chapters/95-homologie/torus/torus.pngbin0 -> 272544 bytes
-rw-r--r--buch/chapters/95-homologie/torus/torus.pov69
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
Binary files differ
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
Binary files differ
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
+ }
+}