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

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

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.

• Jezik: Angleški jezik
• Ključne besede: $L^2$ approximation, quaternions, Pythagorean-hodograph curves, B-splines, preimage
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2027
• Št. strani: 16 str.
• Številčenje: Vol. 491, art. 117987
• PID: 20.500.12556/RUL-185330
• UDK: 519.6
• ISSN pri članku: 0377-0427
• DOI: 10.1016/j.cam.2026.117987
• COBISS.SI-ID: 286529283
• Datum objave v RUL: 31.07.2026

Gradivo je del revije

• Naslov: Journal of computational and applied mathematics
• Skrajšan naslov: J. comput. appl. math.
• Založnik: Elsevier
• ISSN: 0377-0427
• COBISS.SI-ID: 27496960

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: $L^2$ aproksimacija, kvaternioni, krivulje s pitagorejskim hodografom, B-zlepki, praslika