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Characteristics of an Object - PowerPoint PPT Presentation

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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Characteristics of an Object

• 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

• 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.

• 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.

• 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.

• 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

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

Mobius Strip

• 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.