Optimal approximation of spherical squares by tensor product quadratic Bézier patches

Aleš Vavpetič and Emil Žagar

Applied Mathematics and Computation, 2023, vol. 457, issue C

Abstract: In [1], the author considered the problem of the optimal approximation of symmetric surfaces by biquadratic Bézier 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 Bézier patches. It is further shown that the G1 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 the theoretical results.

Keywords: Bézier patch; Spherical square; Optimal approximation; Sphere approximation (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:457:y:2023:i:c:s009630032300365x

DOI: 10.1016/j.amc.2023.128196

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