Characteristics of an Object

1
Characteristics of an Object
2
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

3
(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.

4
(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.

5
Genus (g) of a manifold

• Applicable only for manifolds
• (Naïve) Number of handles.
• Sphere has g0 cube has g0 torus has g1
  coffee cup has g1.

6
Euler Characteristic (e) of a manifold

• e V-EF (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 e2-2g
• Sphere or Cube e2-2(0)2
• Torus e2-2(1)0
• Verify Cube has 8 vertices, 12 edges, 6 faces
• e V-EF 8-126 2

7
Orientability of an object

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

Mobius Strip
8
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.