Spherical Triangular B-splines with Application to Data Fitting

Abstract

Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties across patch boundaries. Recently, Alfeld et al. [1] introduced the concept of spherical barycentric coordinates which allowed them to formulate Bernstein-Bezier polynomials over the sphere.In this paper we use the concept of spherical barycentric coordinates to develop a similar formulation for triangular B-splines, which we call spherical triangular B-splines. These splines defined over spherical triangulations share the same continuity properties and similar evaluation algorithms with their planar counterparts, but possess none of the annoying degeneracies found when trying to represent closed surfaces using planar parametric surfaces. We also present an example showing the use of these splines for approximating spherical scattered data.