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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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Characteristics of an Object

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Characteristics of an object l.jpg

Characteristics of an Object


Object characteristics l.jpg

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 l.jpg

(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 l.jpg

(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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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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