1. Gopi -ICS280F02 - Slide 1 Characteristics of an Object

2. Gopi -ICS280F02 - Slide 2 Object Characteristics Topological propertiesTopological properties –Manifolds (w/ boundaries) /Non-manifold –Euler characteristic/Genus/Betti numbers/ –Orientability –Homeomorphism/ Homology groups etc. Geometric propertiesGeometric properties –Curvature, continuity –Surface parameterization –Convexity and concavity –Silhouette visibility

3. Gopi -ICS280F02 - Slide 3 (Layman) Manifold Definitions ManifoldsManifolds –2D: Every edge has exactly two incident triangles. –3D: Every triangle has exactly two incident tetrahedrons. Manifolds with boundariesManifolds with boundaries –2D: Every edge has either one or two incident triangles. –3D: Every triangle has either one or two incident tetrahedrons. Non-manifoldsNon-manifolds –That does not have the above restrictions.

4. Gopi -ICS280F02 - Slide 4 (Expert) Manifold Definitions ManifoldsManifolds –2D: Neighborhood of every point belonging to the object is homeomorphic to an open disc. –3D: Neighborhood of every point belonging to the object is homeomorphic to an open ball. Manifolds with boundariesManifolds with boundaries –2D: …(as above) or a half-disk. –3D:…(as above) or a half-ball Non-manifoldsNon-manifolds –That does not have the above restrictions.

5. Gopi -ICS280F02 - Slide 5 Genus (g) of a manifold Applicable only for manifoldsApplicable only for manifolds (Naïve) Number of “handles”.(Naïve) Number of “handles”. Sphere has g=0; cube has g=0; torus has g=1; coffee cup has g=1.Sphere has g=0; cube has g=0; torus has g=1; coffee cup has g=1.

6. Gopi -ICS280F02 - Slide 6 Euler Characteristic (e) of a manifold e = V-E+F (V: Vertices, E: Edges, F: Faces).e = V-E+F (V: Vertices, E: Edges, F: Faces). Applicable only for manifoldsApplicable only for manifolds In generalIn general –e=(0 dim)-(1 dim)+(2 dim)-(3 dim)+(4 dim)… Relationship between e and g: e=2-2gRelationship between e and g: e=2-2g –Sphere or Cube: e=2-2(0)=2 –Torus: e=2-2(1)=0 Verify: Cube has 8 vertices, 12 edges, 6 facesVerify: Cube has 8 vertices, 12 edges, 6 faces –e = V-E+F = 8-12+6 = 2

7. Gopi -ICS280F02 - Slide 7 Orientability of an object If you have consistent normal direction for a point then the object is orientable. Otherwise, non- orientable.If you have consistent normal direction for a point then the object is orientable. Otherwise, non- orientable. Mobius Strip

8. Gopi -ICS280F02 - Slide 8 In this course.. 2D orientable manifolds with boundaries.2D orientable manifolds with boundaries. Remember: manifolds with boundaries is a superset of manifolds, and non-manifold is a superset of manifolds with boundaries. Non-manifold actually means that “need not” be a manifold; not “is not” a manifold.