Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram.

Question

A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal the number of sites. Euler’s formula: V - E + F = 2 demonstrates the relationship between these variables. Furthermore, the relation between the vertexes and edges is obtained as: = 3 − 6 with some assumptions. Also in three dimensions, any Voronoi diagram has vertexes, edges, regions and faces. I want to know in three dimensions, is there any relation between these variables like two dimensions? I am new to computational geometry and happy with any kind of regard.

Answer 1

Euler's formula is valid for any CW-complex. Similar to 2D case, Voronoi diagram in 3D can be represented is a CW-complex on $S^3$ (~= $R^3$ + one point in infinity). So, the alternating sum of numbers of vertices ($k_0$), edges ($k_1$), faces ($k_2$), and 3D regions ($k_3$) is $$ {k_{0}-k_{1}+k_{2}-k_{3} = \chi(S^3) = 1 + (-1)^3 = 0} $$ (see formula for $\chi(S^n)$ here)