Extending Catmull-Clark Subdivision and PCCM with Polar Structures

We complete and bring together two pairs of surface constructions that use polynomial pieces of degree (3,3) to associate a smooth surface with a mesh. The two pairs complement each other in that one extends the subdivisionmodeling paradigm, the other the NURBS patch approach to free-form modeling. Both Catmull-Clark [3] and polar subdivision [7] generalize bi-cubic spline subdivision. Together, they form a powerful combination for smooth object design: while Catmull-Clark subdivision is more suitable where few facets join, polar subdivision nicely models regions where many facets join, as when capping extruded features. We show how to easily combine the meshes of these two generalizations of bi-cubic spline subdivision. A related but different generalization of bi-cubic splines is to model non-tensor-product configurations by a finite set of smoothly connected bi-cubic patches. PCCM [12] does so for layouts where Catmull-Clark would apply. We show that a single NURBS patch can be used where polar subdivision would be applied. This spline is singularly parametrized, but, using a novel technique, we show that the surface is C1 and has bounded curvatures.