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Characteristics of an ObjectPowerPoint Presentation

Characteristics of an Object

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**darin** - Follow User

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Characteristics of an Object. Object Characteristics. Topological properties Manifolds (w/ boundaries) /Non-manifold Euler characteristic/Genus/Betti numbers/ Orientability Homeomorphism/ Homology groups etc. Geometric properties Curvature, continuity Surface parameterization

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Object Characteristics

- Topological properties
- Manifolds (w/ boundaries) /Non-manifold
- Euler characteristic/Genus/Betti numbers/
- Orientability
- Homeomorphism/ Homology groups etc.

- Geometric properties
- Curvature, continuity
- Surface parameterization
- Convexity and concavity
- Silhouette visibility

(Layman) Manifold Definitions

- Manifolds
- 2D: Every edge has exactly two incident triangles.
- 3D: Every triangle has exactly two incident tetrahedrons.

- Manifolds with boundaries
- 2D: Every edge has either one or two incident triangles.
- 3D: Every triangle has either one or two incident tetrahedrons.

- Non-manifolds
- That does not have the above restrictions.

(Expert) Manifold Definitions

- Manifolds
- 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 boundaries
- 2D: …(as above) or a half-disk.
- 3D:…(as above) or a half-ball

- Non-manifolds
- That does not have the above restrictions.

Genus (g) of a manifold

- Applicable only for manifolds
- (Naïve) Number of “handles”.
- Sphere has g=0; cube has g=0; torus has g=1; coffee cup has g=1.

Euler Characteristic (e) of a manifold

- e = V-E+F (V: Vertices, E: Edges, F: Faces).
- Applicable only for manifolds
- In general
- e=(0 dim)-(1 dim)+(2 dim)-(3 dim)+(4 dim)…

- Relationship 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 faces
- e = V-E+F = 8-12+6 = 2

Orientability of an object

- If you have consistent normal direction for a point then the object is orientable. Otherwise, non-orientable.

Mobius Strip

In this course..

- 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.

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