Geometric approximation of the sphere by triangular polynomial spline patches

TL;DR

This paper presents a method for optimally approximating spherical surfaces using parametric polynomial spline patches, focusing on minimizing radial error and providing practical approximations for common spherical triangulations.

Contribution

It introduces a novel approach for approximating spheres with polynomial patches, including optimal solutions and analysis of smoothness and approximation quality.

Findings

Optimal polynomial approximations for spherical triangles are achieved.

Numerical examples demonstrate high-quality approximations.

Approximations for spherical triangulations from tetrahedron, octahedron, and icosahedron are provided.

Abstract

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.