Smoothing cleans up slivers

Herbert Edelsbrunner, Xiang-Yang Li, Gary Miller, Andreas Stathopoulos, Dafna Talmor, Shang-Hua Teng, Alper Üngör and Noel Walkington

Abstract:

A sliver is a tetrahedron whose four vertices lie close to a plane and whose perpendicular projection to that plane is a convex quadrilateral with no short edge. Slivers are both undesirable and ubiquitous in 3-dimensional Delaunay triangulations. Even when the point-set is well-spaced, slivers may result. This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers. It also gives deterministic algorithms that compute the perturbation of $n$ points in time O($n \log n$) with one processor and in time O($\log n$) with O($n$) processors.