Smooth Low Degree Approximations of Polyhedra

We present efficient algorithms to construct both C1 and C 2 smooth meshes of cubic and quintic A-patches to approximate a given polyhedron P in three dimensions. The A~patch is a smooth and single-sheeted zero-contour patch of a trivariate polynomial in Bernstein-Bezier (BB) form defined within a tetrahedron. The smooth mesh constructions rely on a novel scheme to build an inner simplicial hull E consisting of tetrahedra and defined by the faces of the given polyhedron p, A single cubic or quintic A-patch is then constructed within each tetrahedron of the simplicial hull :E with the resulting surface being C l or C 2 smooth, respectively. The free parameters of each individual A-patch can be independently controlled to achieve both local and globa1shape deformations and a family of C 1 or C 2 smooth approximations of the original polyhedron.