Geometric approximation of the sphere by triangular polynomial spline patches

A sphere is a fundamental geometric object widely used in (computer aided) design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by patches if measure quality (simplified) radial error. As consequence, approximations unit spline underlying on particular regular triangulations arising from tetrahedron, octahedron and icosahedron inscribed are provided. Some low total degree with corresponding smoothness analyzed detail several numerical examples shown confirming approximants.