CODEWITHSUNDEEP
graphics3dmesh
Kafros
Kafros

Reputation: 176

What is the maximum number of faces in a triangular mesh of n vertices in 3 dimensions?

In 2D the maximum number of faces for n vertices in a 'perfect'(non-overlapping) mesh is f = 2n - 4. Is there an equivalent result for 3 dimensions?

Upvotes: 4

Views: 1837

Answers (1)

Nico Schertler
Nico Schertler

Reputation: 32617

The Euler characteristic chi is defined as:

chi = V - E + F

where V, E, and F are the numbers of vertices, edges and faces, respectively.

For closed triangular meshes, we know that each edge has two incident faces and each face has three incident edges. Therefore:

3 * F = 2 * E
E = 3/2 * F

Hence,

chi = V - 3/2 * F + F
    = V - 1/2 F
  F = 2 * (V - chi)

In the 2D case for planar graphs, chi is 2, resulting in your definition F = 2 * V - 4.

For any 3D surface, the Euler characteristic can be calculated from its genus. In general, the more handles the surface has, the smaller its Euler characteristic. Hence, chi (and by that F) is not limited. However, for a fixed surface topology, the number of faces (relative to the number of vertices) is fixed.

Upvotes: 2

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