Optimal approximation of spherical squares by tensor product quadratic Bézier patches

Vavpetič, Aleš (author), Žagar, Emil (author)

In cited article E. F. Eisele 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 mentioned article. The sphere is then approximated by the continuous spline of two and six tensor product quadratic Bézier 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 the theoretical results.

• Language: English
• Keywords: Bézier patches, spherical squares, optimal approximation, sphere approximation
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: 12 str.
• Numbering: Vol. 457, art. 128196
• PID: 20.500.12556/RUL-152828
• UDC: 519.6
• ISSN on article: 0096-3003
• DOI: 10.1016/j.amc.2023.128196
• COBISS.SI-ID: 161773059
• Publication date in RUL: 08.12.2023

Record is a part of a journal

• Title: Applied mathematics and computation
• Shortened title: Appl. math. comput.
• Publisher: Elsevier
• ISSN: 0096-3003
• COBISS.SI-ID: 24983808

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: Bézierjeve krpe, sferični kvadrati, optimalna aproksimacija, aproksimacija sfere