Polygonizing Implicit Surfaces With Guaranteed Topology

An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuses on blended ellipsoids, but could be extended to include any smooth, bounded implicit surface. A polygonization algorithm and an incremental repolygonization algorithm are provided. Treating an implicit surface as a gradient system allows theorems from Morse theory to describe implicit surface topology. An implicit surface changes topology only when a critical value of its defining function changes sign. These critical points may be found using interval analysis. Techniques for modifying the polygonization to accommodate such changes in topology are given