What is the mathematical relationship between the number of faces in a mesh and its vertices?

Question

An "open and planar quad mesh" (more description below) with 4 mesh faces has 9 vertices, the same mesh with 8 faces has 15 vertices (2 faces at every X-axis row, 4 faces at every Y-axis column)...etc.. What is the mathematical relationship between the number of mesh faces and its number of vertices?

The planar quad mesh in this case is a continuous collection of n rectangular faces sharing 4 edges (the 4 outer boundary edges of the mesh are open). The number of faces in the x direction can be different to the ones in the y direction.

I have looked at the Euler characteristic for a mesh that is homotopic to a sphere but do not know the relationship for the specific type of mesh described above.

See euler characteristic: http://en.wikipedia.org/wiki/Euler_characteristic — Lieven, Feb 17 '13 at 17:36
I would have expected an odd number of vertices for $8$ faces. What is your exact definition of mesh with $n$ faces? — Hagen von Eitzen, Feb 17 '13 at 17:39
@HagenvonEitzen sorry I forgot to add the 0 from domain 0to14 so mesh with 8 faces has 15 vertices. — Arthur Mamou-Mani, Feb 17 '13 at 17:41
@Arthur Mamou-Mani
Only if it's homotopic to a sphere (ie, the shape can be obtained by deforming a sphere, without punching through the shape to get new holes.

http://en.wikipedia.org/wiki/Homotopy — Lieven, Feb 17 '13 at 17:41
@Lieven, thanks, I am currently dealing with a plane. I will add this constraint to the question. — Arthur Mamou-Mani, Feb 17 '13 at 17:44
Is a mesh simply a collection of unit squares in the plane that are connected by sharing a few edges? — Hagen von Eitzen, Feb 17 '13 at 17:47
@HagenvonEitzen yes rectangles sharing 4 edges. I added a def in the question thank you. — Arthur Mamou-Mani, Feb 17 '13 at 17:48
I can build a mesh with $8$ faces and $2\times 9$ vertices, can't I? — Hagen von Eitzen, Feb 17 '13 at 17:49
But there must be some boundary edges, right? So not all quads share all 4 edges with another quad. — Lieven, Feb 17 '13 at 17:50
@HagenvonEitzen 8 rectangular faces in my case creates 15 vertices. This is because it is a single continuous "welded" mesh. This planar quad mesh is a continuous collection of n rectangular faces sharing 4 edges (the 4 outer edges of the mesh are open). — Arthur Mamou-Mani, Feb 17 '13 at 17:51
That's still the case for Hagen's example. You're thinking of a 2x4 rectangle. He's describing a 1x8 rectangle. — , Feb 17 '13 at 18:06
@EmanuelePaolini 2 faces at every X-axis row, 4 faces at every Y-axis column — Arthur Mamou-Mani, Feb 17 '13 at 18:08
@ℝⁿ that's a very good point, is there no general rules for this relationship in this specific case then? — Arthur Mamou-Mani, Feb 17 '13 at 18:09

score 3 · Accepted Answer · answered Feb 17 '13 at 18:02

3

A $n\times n$ square has $n^2$ quads and $(n+1)^2$ vertices.

A rectangular $m\times n$ mesh has $mn$ quads and $(m+1)(n+1)$ vertices.

answered Feb 17 '13 at 18:02

Emanuele Paolini

21,447

Thank you. The number of faces in the x direction can be different to the ones in the y direction. – Arthur Mamou-Mani Feb 17 '13 at 18:04

What is the mathematical relationship between the number of faces in a mesh and its vertices?

1 Answers1