Tuning Subdivision by Minimising Gaussian Curvature Variation Near Extraordinary Vertices

We present a method for tuning primal stationary subdivision schemes to give the best possible behaviour near extraordinary vertices with respect to curvature variation.
Current schemes lead to a limit surface around extraordinary vertices for which the Gaussian curvature diverges, as demonstrated by Karčiauskas et al. [ KPR04 ]. Even when coefficients are chosen such that the subsubdominant eigenvalues, , equal the square of the subdominant eigenvalue, , of the subdivision matrix [ DS78 ] there is still variation in the curvature of the subdivision surface around the extraordinary vertex as shown in recent work by Peters and Reif [ PR04 ] illustrated by Karčiauskas et al. [ KPR04 ].