Topological Invariance of Parametric Objects Part Iii:preservation of the Topological Form of Objects Deened by Parametric Patches

Given a nite curvilinear simplicial complex in R3, made up of properlyjoined parametric patches de ned in terms of control points, it is of interest to know under what conditions the complex will retain its original topological form when the control points are perturbed. For example, the patches might be triangular B ezier surface patches, and the complex may represent the boundary of a solid in a solid-modeling application. In this paper we give su cient conditions guaranteeing that topological form is preserved, where this is interpreted to mean that there is a homeomorphism, de ned on the whole ambient space R3, mapping the object onto the perturbed object. The main conditions to be satis ed are that the original object should be continuously perturbed in a way that introduces no self-intersections of patches, no extraneous intersections of adjacent patches, and no intersections of disjoint patches. The results apply directly to most surface modeling schemes, and they are of interest in several areas of application. This paper is Part III in a series of three papers. In Parts I and II we gave conditions precluding self-intersection and extraneous intersection of B ezier curves and triangular B ezier patches, along with corresponding perturbation analyses. The results of Part II, on triangular B ezier patches, provide one means of satisfying the hypotheses of the theorems of the present paper. The second author acknowledges, with appreciation, partial funding for this work received under National Science Foundation Grant Number MII-9308346. The views expressed herein are of this author, not of this funding agency. The research of the third author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.