ST - T C SC - 2 00 1 - 11 The Probabilistic Complexity of the Voronoi Diagram of Points ona Polyhedron

It is well known that the complexity, i.e., the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as (n 2): Interest has recently arisen as to what happens, both in deterministic and probabilistic situations, when the three dimensional points are restricted to lie on the surface of a 2-dimensional object. In this paper we consider the situation when the points are drawn from a 2-dimensional Poisson distribution with rate n over a xed union of triangles in R 3 : We show that with high probability the complexity of their Voronoi diagram is ~ O (n) : This implies, for example, that the complexity of the Voronoi diagram of points chosen from the surface of a general xed polyhedron in R 3 will also be ~ O (n) with high probability.