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 Karˇciauskas et al. [KPR04]. Even when coefficients are chosen such that the subsubdominant eigenvalues, μ, equal the square of the subdominant eigenvalue, λ, 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 Karˇciauskas et al. [KPR04]. In our tuning method we optimise within the space of subdivision schemes with bounded curvature to minimise this variation in curvature around the extraordinary vertex. To demonstrate our method we present results for the Catmull-Clark [CC78], 4-8 [Vel01, VZ01] and 4-3 [PS03] subdivision schemes. We compare our results to previous work on the tuning of these schemes and show that the coefficients derived with this method give a significantly smaller curvature variation around extraordinary vertices.