G1 – Smooth biquintic approximation of Catmull-Clark subdivision surfaces

Abstract

In this paper a construction of a globally G1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are biquintic G1 around extraordinary vertices and bicubic elsewhere; the smoothness around regular vertices neighboring EV with even valence will be C1, while it results to be C2 in the other occurrences of regular vertices. Each Bézier point is computed using a locally defined mask around the neighboring mesh vertices. To define G1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C0 scheme of Loop and Schaefer (2008), while for others, explicit masks are deduced by providing closed-form solutions of the G1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate numerically that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided.