C1 Smoothing of Polyhedra with Implicit Surface Patches

Polyhedral "smoothing" is an efficient construction scheme for generating complex boundary models of solid physical objects. This paper presents efficient algorithms for generating families of curved solid objects with boundary topology relaled to the input polyhedron. Individual facets of a polyhedron are replaced by low degree implicit algebraic surface patches with local support. These quintic patches replace the CO contacts of planar facets with C l continuity along all interpatch boundaries. Selection of suitable instances of implicit surfaces as well as local control of the individual surface patches are achieved via simultaneous interpolation and weighted least-squares approximation. Asymptotic degree bounds are also given for the lowest degree implicitly defined, algebraic splines required La CI-smooth the vertices, edges, and facets of an arbitrary polyhedron in three dimensional real space ]R3. ·Supported in part by NSF gran~ GGR 90-00028 and AFOSR contract 91-0276