Equivalence of Topological Form for Curvilinear Geometric Objects

Given a curvilinear geometric object in R 3 , made up of properly-joined para-metric patches deened in terms of control points, it is of interest to know under what conditions the object 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 geometric object may represent the boundary of a solid in a solid-modeling application. In this paper we give suucient conditions guaranteeing that topological form is preserved by an ambient isotopy. The main conditions to be satissed are that the original object should be continuously perturbed in a way that introduces no self-intersections of any patch, and such that the patches remain properly joined. The patches need only have C 0 continuity along the boundaries joining adjacent patches. The results apply directly to most surface modeling schemes, and they are of interest in several areas of application.