The Euler-Poincare Formula
The Euler-Poincaré formula describes the relationship of the number of vertices, the number of edges and the number of faces of a manifold. It has been generalized to include potholes and holes that penetrate the solid. To state the Euler-Poincaré formula, we need the following definitions :
- V : the number of vertices
- E : the number of edges
- F : the number of faces
- G : the number of holes that penetrate the solid, usually referred to as genus in topology
- S : the number of shells. A shell is an internal void of a solid. A shell is bounded by a 2-manifold surface, which can have its own genus value. Note that the solid itself is counted as a shell. Therefore, the value for S is at least 1.
- L : the number of loops, all outer and inner loops of faces are counted.
Then, the Euler-Poincaré formula is the following :
V - E + F - (L - F) - 2 (S - G) = 0