Structural Tolerance and Delaunay Triangulation

In this paper we consider the tolerance of a geometric or combinatorial structure associated to a set of points as a measure of how much the set of points can be perturbed while leaving the (topological or combinatorial) structure essentially unchanged. We concentrate on studying the Delaunay triangulation and show that its tolerance can be computed in O(n) time if the triangulation is given as part of the input, while the problem has complexity 2(n logn) if the triangulation is not known. We also study the problem of computing the tolerance of the edges of the triangulation, and show that the tolerance of all the edges can be computed in O(n logn) time. Finally, we extend our study to some subgraphs of the Delaunay triangulation.  1999 Published by Elsevier Science B.V. All rights reserved.