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

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

Two methods for the $L^2$ approximation of smooth space curves by spatial Pythagorean-hodograph (PH) curves are considered. The first method employs direct approximation in ${\mathbb R}^3$, which results in a non-linear system of equations that may be solved by a numerical iteration or optimization scheme. The second method performs the optimization in the quaternion preimage space of PH curves, which results in a linear system of equations. The preimage of a curve in ${\mathbb R}^3$ is a surface in the quaternion space, and an appropriate locus on this surface must be identified for the given space curve. This is accomplished by observing that PH curves equipped with a rational rotation-minimizing frame (RMF) have preimages that are geodesic loci on the surface, and computing a discretized approximation to the RMF on the given curve. The methods are also extended to the approximation of spatial B-spline curves. Detailed algorithm descriptions are provided for both methods, and several computed examples illustrate their performance.

• Language: English
• Keywords: $L^2$ approximation, quaternions, Pythagorean-hodograph curves, B-splines, 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: 2027
• Number of pages: 16 str.
• Numbering: Vol. 491, art. 117987
• PID: 20.500.12556/RUL-185330
• UDC: 519.6
• ISSN on article: 0377-0427
• DOI: 10.1016/j.cam.2026.117987
• COBISS.SI-ID: 286529283
• Publication date in RUL: 31.07.2026

Record is a part of a journal

• Title: Journal of computational and applied mathematics
• Shortened title: J. comput. appl. math.
• Publisher: Elsevier
• ISSN: 0377-0427
• COBISS.SI-ID: 27496960

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: $L^2$ aproksimacija, kvaternioni, krivulje s pitagorejskim hodografom, B-zlepki, praslika