euler characteristic - oregon state...
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Euler Characteristic
Face Classificationset_view(GL_RENDER);set_scene(GL_RENDER);
glGetDoublev(GL_MODELVIEW_MATRIX, modelview_matrix1);glGetDoublev(GL_PROJECTION_MATRIX, projection_matrix1);glGetIntegerv(GL_VIEWPORT, viewport1);
gluProject((GLdouble) poly->tlist[i]->center.entry[0], (GLdouble) poly->tlist[i]->center.entry[1], (GLdouble)poly->tlist[i]->center.entry[2], modelview_matrix1, projection_matrix1, viewport1, &face_norm_start.entry[0], &face_norm_start.entry[1], &face_norm_start.entry[2]);
Topics Today
• Platonic solids • Corner structure
Topics Today
• Platonic solids• Corner structure
Platonic Solids
shiftingsands.com.au/platonicsolids.html
Platonic Solids
shiftingsands.com.au/platonicsolids.html
Platonic Solids
davidf.faricy.net/polyhedra/Platonic_Solids.html
Platonic Solids
Are We Missing Anything?
Are We Missing Anything?
• All regular polyhedron must be convex.
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3?
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3?
– m=3: tetrahedron
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3?
– m=3: tetrahedron– m=4: octahedron
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3?
– m=3: tetrahedron– m=4: octahedron– m=5: icosahedron
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3? • When n=4?
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3? • When n=4?
– m=3, Hexahedron (cube)
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3? • When n=4?• When n=5?
Are We Missing Anything?
• All regular polyhedron must be convex.• When n=3? • When n=4?• When n=5?
– m=3: dodecahedron
Are We Missing Anything?
• For example, is it possible to have– m=3 and n=3 but f<>4?
Euler Characteristics
• L=V-E+F=2• Why?
Elementary Collapse on Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
Elementary Collapse for Edges
• V=E for closed a simple planar curve.
Elementary Collapse for Edges
• V=E for closed a simple planar curve.• What about 3D surfaces?
Elementary Collapse for Edges
• V=E for closed a simple planar curve.• What about 3D surfaces?
– Need to consider merging faces.
Elementary Collapse for Faces
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C CD
Proof of Euler’s Theorem on a Cube
A B A B
E E
G GH H
F
C C
Proof of Euler’s Theorem on a Cube
A B A B
E E
G G
F
C C
Proof of Euler’s Theorem on a Cube
A B A B
G G
F
C C
Proof of Euler’s Theorem on a Cube
A B A B
F
C
Proof of Euler’s Theorem on a Cube
B B
F
Proof of Euler’s Theorem on a Cube
F
Proof of Euler’s Theorem on a Cube
FF
Another Look
A B A B
E E
G GH H
F
C CD
Another Look
A B A
E E
G GH H
F
C CD D
B
F
Dual of a Hexahedron
A
E
GH
CD
B
F
Dual of a Hexahedron
A
E
GH
CD
F
Dual of a Hexahedron
A
E
GH
CD
F
Dual of a Hexahedron
A
E
GH
CD
F
Dual of a Hexahedron
Dual Shape
• What is the dual of– Octahedron– Icosahedron– Dodecahedron– Tetrahedron
• Does the dual operation change the Euler characteristic?
• What operations will change it?
Are We Missing Anything?
• For example, is it possible to have– m=3 and n=3 but f<>4?
Are We Missing Anything?
• For example, is it possible to have– m=3 and n=3 but f<>4?
• No, we are not.
Proof
v-e+f=2n=number of edges in the polyongm=number of faces (edges) meeting at a
vertex
Proof
v-e+f=2n=number of edges in the polygonm=number of faces (edges) meeting at a
vertexWe have
2e=nf
Proof
v-e+f=2n=number of edges in the polygonm=number of faces (edges) meeting at a
vertexWe have
2e=nfmv=nf
Proof
v-e+f=22e=nfmv=nfWhen m=3, n=3 what is f?nf/m-nf/2+f=2
Proof
v-e+f=22e=nfmv=nfWhen m=3, n=3 what is f?nf/m-nf/2+f=23f/3-3f/2+f=2
Proof
v-e+f=22e=nfmv=nfWhen m=3, n=3 what is f?nf/m-nf/2+f=23f/3-3f/2+f=2f/2=2
Proof
v-e+f=22e=nfmv=nfWhen m=3, n=3 what is f?nf/m-nf/2+f=23f/3-3f/2+f=2f/2=2f=4
Any questions?
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