  A sphere is a fundamental geometric object widely used in (computer aided)
geometric 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 parametric
polynomial patches if the measure of quality is the (simplified) radial error.
As a consequence, optimal approximations of the unit sphere by parametric
polynomial spline patches underlying on particular regular spherical
triangulations arising from a tetrahedron, an octahedron and an icosahedron
inscribed in the unit sphere are provided. Some low total degree spline patches
with corresponding geometric smoothness are analyzed in detail and several
numerical examples are shown confirming the quality of approximants.
