A Flexible $C^2$ 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 harmonic energy functional, which can in turn be exploited to promote conformality in surface parameterizations. We develop an efficient parallel algorithm for computing the harmonic energy and its gradient with respect to the control vertices. A software implementation (in CUDA and MATLAB) is provided.