A Flexible C2 Subdivision Scheme on the Sphere: With Application to Biomembrane Modelling

We construct a $C^2$ multiscale approximation scheme for functions defined on the (Riemann) sphere. Based on a three-directional box-spline, a flexible $C^2$ scheme over a valence 3 extraordinary vertex can be constructed. Such a flexible $C^2$ subdivision scheme is known to be impossible for arbitrary valences. The subdivision scheme can be used to model spherical surfaces based on a recursively subdivided tetrahedron, with only valence 3 and 6 vertices in the resulted triangulations. This adds to the toolbox of subdivision methods a high order, high regularity scheme which can be beneficial to scientific computing applications. For instance, the scheme can be used in the numerical solution of the Canham--Helfrich--Evans models for spherical and toroidal biomembranes. Moreover, the characteristic maps of the subdivision scheme endow the underlying simplicial complex with a conformal structure. This in particular means that the special subdivision surfaces constructed here comes with a well-defined harmon...

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