On $L^2$ approximation by planar Pythagorean-hodograph curves

Farouki, Rida T. (author), Knez, Marjetka (author), Vitrih, Vito (author), Žagar, Emil (author)

The $L^2$ approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex valued functions of a real parameter. Because of the nonlinear nature of polynomial PH curves, constructing $L^2$ approximants involves solving a nonlinear optimization problem. However, a simplified method that requires only the solution of a linear system may be developed by formulating the $L^2$ approximation in the preimage space. The extension of the methodology to approximation by PH B-spline curves is also addressed, and several examples are provided to illustrate its implementation and potential.

• Language: English
• Keywords: $L^2$ approximation, complex polynomial, Pythagorean-hodograph curve, Pythagorean-hodograph spline, preimage
• 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: 2025
• Number of pages: Str. 296-310
• Numbering: Vol. 233
• PID: 20.500.12556/RUL-167348
• UDC: 519.6
• ISSN on article: 1872-7166
• DOI: 10.1016/j.matcom.2025.02.001
• COBISS.SI-ID: 226119171
• Publication date in RUL: 17.02.2025

Record is a part of a journal

• Title: Mathematics and computers in simulation
• Shortened title: Math. comput. simul.
• Publisher: Elsevier, International Association for Mathematics and Computers in Simulation
• ISSN: 1872-7166
• COBISS.SI-ID: 23159813

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: $L^2$ aproksimacija, PH krivulja, PH zlepek, praslika