G2 tensor product splines over extraordinary vertices

Abstract

We present a second order smooth filling of an n-valent Catmull-Clark spline ring with n biseptic patches. While an underdetermined biseptic solution to this problem has appeared previously, we make several advances in this paper. Most notably, we cast the problem as a constrained minimization and introduce a novel quadratic energy functional whose absolute minimum of zero is achieved for bicubic polynomials. This means that for the regular 4-valent case, we reproduce the bicubic B-splines. In other cases, the resulting surfaces are aesthetically well behaved. We extend our constrained minimization framework to handle the case of input mesh with boundary. © 2008 The Author(s) Journal compilation © 2008 The Eurographics Association and Blackwell Publishing Ltd.