Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes?
First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, $F$ faces number and $g$ genus (number of handles in the mesh). Now my book says
Since for most practical applications the genus is small compared to the number of elements, the right hand side of the equation can be assume to be negligible. Given this and the fact that each triangle is bounded by three edges and that each interior manifold edge is incident to two triangles, one can derive the following
- The number of triangles is twice the number of vertices $F \approx 2V$
- The number of edges is three times the number of vertices $E \approx 3V$