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Title:An approach to geometric interpolation by Pythagorean-hodograph curves
Authors:ID Jaklič, Gašper (Author)
ID Kozak, Jernej (Author)
ID Knez, Marjetka (Author)
ID Vitrih, Vito (Author)
ID Žagar, Emil (Author)
Work type:Not categorized
Typology:1.01 - Original Scientific Article
Organization:IAM - Andrej Marušič Institute
Abstract:The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree ▫$n$▫ is studied independently of the dimension ▫$d \ge 2$▫. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.
Keywords:mathematics, parametric curve, PH curve, geometric interpolation, Lagrange interpolation, Hermite interpolation, cubic curves, homotopy
Year of publishing:2012
Number of pages:str. 123-150
Numbering:Vol. 37, no. 1
PID:20.500.12556/RUP-7733 This link opens in a new window
COBISS.SI-ID:16051289 This link opens in a new window
Publication date in RUP:03.04.2017
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Secondary language

Title:Pristop k geometrijski interpolaciji s krivuljami, katerih hodograf je pitagorejski
Abstract:V članku obravnavamo problem geometrijske interpolacije s polinomskimi krivuljami stopnje ▫$n$▫, katerih hodograf je pitagorejski (PH krivulje), neodvisno od dimenzije ▫$d \ge 2$▫. V nasprotju s klasičnimi pristopi, kjer uporabljajo posebne strukture, ki so odvisne od dimenzije (kompleksna številka, kvaternioni,...), uporabimo osnovno definicijo PH lastnosti skupaj s pogoji geometrijske interpolacije. Analiza dobljenega sistema nelinearnih enačb sledi tehniki, podobni cilindrični algebraični dekompoziciji in močno temelji na računalniških algebraičnih sistemih. Nelinearne enačbe so v celoti zapisane z geometrijskimi količinami in so neodvisne od dimezije. Za utemeljitev eksistence (in v nekaterih primerih števila) sprejemljivih rešitev, uporabimo analizo robnih območij, konstrukcijo rešitev za posebne podatke in homotopijo. Splošni pristop uporabimo za analizo Hermiteove in Lagrangeove interpolacije s kubičnimi krivuljami. S tem razširimo nekatere znane rezultate in jih podkrepimo z numeričnimi primeri.
Keywords:matematika, parametrična krivulja, PH krivulja, geometrijska interpolacija, Lagrangeova interpolacija, Hermiteova interpolacija, kubične krivulje, homotopija


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