Isotopic approximations and interval solids

Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number r so that the offsets Foð^rÞ of F at distances ^r are nonsingular. In doing so, a normal tubular neighborhood, FðrÞ; of F is constructed. Then, each approximant of F lies inside FðrÞ: Comparisons between these global and local constraints are given. q 2004 Elsevier Ltd. All rights reserved.