Optimal approximation of spherical squares by tensor product quadratic B\'ezier patches

TL;DR

This paper corrects previous errors in approximating spherical squares with tensor product quadratic Bézier patches, providing a detailed analysis, a numerical algorithm for optimal approximation, and exploring smooth spline approximations of spheres.

Contribution

It offers a corrected and detailed approach to optimal approximation of spherical squares, including a new numerical algorithm and analysis of smooth spline approximations.

Findings

The previous results on approximation are incorrect.

A new numerical algorithm for optimal approximation is proposed.

Smooth G^1 splines of six patches are not good approximations.

Abstract

In [1], the author considered the problem of the optimal approximation of symmetric surfaces by biquadratic B\'ezier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal approximation of spherical squares. A detailed analysis and a numerical algorithm are given, providing the best approximant according to the (simplified) radial error, which differs from the one obtained in [1]. The sphere is then approximated by the continuous spline of two and six tensor product quadratic B\'ezier patches. It is further shown that the $G^1$ smooth spline of six patches approximating the sphere exists, but it is not a good approximation. The problem of an approximation of spherical rectangles is also addressed and numerical examples indicate that several optimal approximants might exist in some cases, making the problem extremely difficult to handle. Finally, numerical examples are provided that confirm theoretical results.