20.500.12556/RUP-23511 On ▫$L^2$▫ approximation by spatial Pythagorean-hodograph curves 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. Obravnavani sta dve metodi za ▫$L^2$▫ aproksimacijo gladkih prostorskih krivulj s prostorskimi PH krivuljami. Prva metoda uporablja neposredno aproksimacijo v ▫${\mathbb R}^3$▫, kar vodi do nelinearnega sistema enačb, ki ga je mogoče rešiti z numerično iteracijsko ali optimizacijsko metodo. Druga metoda izvaja optimizacijo v kvaternionskem prostoru praslik PH-krivulj, pri čemer dobimo linearni sistem enačb. Praslika krivulje v ▫${\mathbb R}^3$▫ je ploskev v kvaternionskem prostoru, zato je treba za dano prostorsko krivuljo na tej ploskvi določiti ustrezno množico točk. To dosežemo z ugotovitvijo, da imajo PH krivulje, opremljene z racionalnim ogrodjem z minimalno rotacijo (RMF), praslike, ki so geodetke na tej ploskvi, ter z izračunom diskretiziranega približka RMF na dani krivulji. Metodi sta razširjeni tudi na aproksimacijo prostorskih krivulj B-zlepkov. Za obe metodi so podani podrobni opisi algoritmov, njuno učinkovitost pa ponazarja več izračunanih primerov. ▫$L^2$▫ approximation quaternions Pythagorean-hodograph curves B-splines preimage ▫$L^2$▫ aproksimacija kvaternioni krivulje s pitagorejskim hodografom B-zlepki praslika true false true Angleški jezik Slovenski jezik Članek v reviji 2026-08-17 15:34:45 2026-08-17 15:34:45 2026-08-18 03:07:46 0000-00-00 00:00:00 2027 0 0 16 str. [article no.] 117987 Vol. 491 Feb. 2027 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2026-07-22 519.6 0377-0427 10.1016/j.cam.2026.117987 286529283 RAZ_Farouki_Rida_T._2027.pdf RAZ_Farouki_Rida_T._2027.pdf 1 AEEF449DF42C0B38650FD970D4141216 552f9136aad739513c174582adf2db027e4171588ca68f31ae592c6ac54a3ba9 634a47df-9a40-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34599 https://www.sciencedirect.com/science/article/pii/S0377042726006291 1 2d0d14a8-9a40-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34598 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0