Numerical Checking of C1 for Arbitrary Degree Quadrilateral Subdivision Schemes

We derive a numerical method to confirm that a subdivision scheme based on quadrilateral meshes is C 1 at the extraordinary points. We base our work on Theorem 5.25 in Peters and Reif's book "Subdivision Surfaces", which expresses it as a condition on the derivatives within the characteristic ring around the EV. This note identifies instead a sufficient condition on the control points in the natural configuration from which the conditions of Theorem 5.25 can be established.

[1]  Denis Zorin,et al.  A Method for Analysis of C1 -Continuity of Subdivision Surfaces , 2000, SIAM J. Numer. Anal..

[2]  Ulrich Reif,et al.  A unified approach to subdivision algorithms near extraordinary vertices , 1995, Comput. Aided Geom. Des..

[3]  Georg Umlauf,et al.  Analyzing a generalized Loop subdivision scheme , 2007, Computing.

[4]  Ron Goldman,et al.  Recursive subdivision without the convex hull property , 1986, Comput. Aided Geom. Des..

[5]  Jörg Peters,et al.  Subdivision Surfaces , 2002, Handbook of Computer Aided Geometric Design.

[6]  M. Sabin,et al.  NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes , 2009, SIGGRAPH 2009.

[7]  Nira Dyn.  Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials , 2002, Tutorials on Multiresolution in Geometric Modelling.

[8]  Nira Dyn,et al.  Convergence and C1 analysis of subdivision schemes on manifolds by proximity , 2005, Comput. Aided Geom. Des..

[9]  G. Umlauf.  A Technique for Verifying the Smoothness of Subdivision Schemes , 2022 .

[10]  Ewald Quak,et al.  Tutorials on Multiresolution in Geometric Modelling, Summer School Lecture Notes , 2002 .

[11]  Neil A. Dodgson,et al.  Tuning Subdivision by Minimising Gaussian Curvature Variation Near Extraordinary Vertices , 2006, Comput. Graph. Forum.

[12]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[13]  A. A. Ball,et al.  Conditions for tangent plane continuity over recursively generated B-spline surfaces , 1988, TOGS.