Minimal energy spherical splines on Clough–Tocher triangulations for Hermite interpolation

Abstract

We present a study of the minimal energy method applied to the Hermite interpolation problem over Clough–Tocher partitions on the unit sphere. A subset of spline coefficients is found by satisfying nodal interpolating conditions. The rest of the coefficients are found through energy minimization subject to C1 conditions. We show that the error in approximation of a given sufficiently smooth function by the minimal energy Hermite interpolating spline depends on the mesh size of the underlying triangulation cubically. In addition, we prove that minimizers of energy functionals with different homogeneous extensions are equivalent in the sense that they all converge to the sampled function, and the order of convergence is independent of the extension. We conclude with numerical examples.