Application of Ambient Isotopy to Surface Approximation and Interval Solids

Given a nonsingular compact 2-manifold without boundary, we present methods for establishing a family of surfaces which can approximate so that each approximant is ambient isotopic to . 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 so that the offsets of at distances are nonsingular. In doing so, a normal tubular neighborhood, , of is constructed. Then, each approximant of lies inside . Comparisons between these global and local constraints are given.