Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Abstract

It is well known that the complexity of the Delaunay triangulation of n points in ℝ, i.e., the number of its simplices, can be Ω(n). In particular, in R, the number of tetrahedra can be quadratic. Put another way, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the three-dimensional Delaunay triangulation of the points when the sampling density increases. More precisely, we show that the complexity is O(n) for general polyhedral surfaces and O(nn√n) for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points.