Shape-preserving interpolation of irregular data by bivariate curvature-based cubic L1 splines in spherical coordinates

We investigate C1-smooth bivariate curvature-based cubic L1 interpolating splines in spherical coordinates. The coefficients of these splines are calculated by minimizing an integral involving the L1 norm of univariate curvature in four directions at each point on the unit sphere. We compare these curvature-based cubic L1 splines with analogous cubic L2 interpolating splines calculated by minimizing an integral involving the square of the L2 norm of univariate curvature in the same four directions at each point. For two sets of irregular data on an equilateral tetrahedron with protuberances on the faces, we compare these two types of curvature-based splines with each other and with cubic L1 and L2 splines calculated by minimizing the L1 norm and the square of the L2 norm, respectively, of second derivatives. Curvature-based cubic L1 splines preserve the shape of irregular data well, better than curvature-based cubic L2 splines and than second-derivative-based cubic L1 and L2 splines. Second-derivative-based cubic L2 splines preserve shape poorly. Variants of curvature-based L1 and L2 splines in spherical and general curvilinear coordinate systems are outlined.