Equidistant Smoothing of Polyhedra with Arbitrary Topologies

Smoothing of polyhedron with arbitrary topology is an important issue in CAGD and CAD/CAM, but so far it is deemed to be difficult to smooth the complex corners of a polyhedron. In this paper, the concept of distance surfaces of a surface and a solid is introduced, and the incisive properties of such surfaces are addressed which provide a theoretical foundation for modifying a general corner. The method is based on making constricted volume and the maximum distance the volume can be constricted is given too. It is shown that by the proposed method in this paper any polyhedron can be G¹ smoothed with quadraic and, sometimes toroidal surfaces. The new approach is suitable for engineering design and NC machining. The associated algorithm based on the classification theorem of corners is simple, fast and robust.