C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

Bastl, Bohumír; Bizzarri, Michal; Knez, Marjetka; Lávička, Miroslav; Michálkova, Kristýna; Šír, Zbiněk; Vitrih, Vito; Žagar, Emil

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Title:	C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs
Authors:	ID Bastl, Bohumír (Author) ID Bizzarri, Michal (Author) ID Knez, Marjetka (Author) ID Lávička, Miroslav (Author) ID Michálkova, Kristýna (Author) ID Šír, Zbiněk (Author) ID Vitrih, Vito (Author) ID Žagar, Emil (Author)
Files:	http://dx.doi.org/10.1016/j.cam.2013.08.007
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	In this paper the $C^1$ Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates $C^1$ data at one point and they are then joined together with a $C^1$ continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally $C^1$ continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.
Keywords:	mathematics, parametric curve, PH curve, Pythagorean-hodograph, Hermite interpolation, biarc, cubic curve
Year of publishing:	2014
Number of pages:	str. 65-78
Numbering:	Vol. 257
PID:	20.500.12556/RUP-7739
ISSN:	0377-0427
UDC:	519.65
COBISS.SI-ID:	16726361
Publication date in RUP:	03.04.2017
Views:	2681
Downloads:	43
Metadata:
:	BASTL, Bohumír, BIZZARRI, Michal, KNEZ, Marjetka, LÁVIČKA, Miroslav, MICHÁLKOVA, Kristýna, ŠÍR, Zbiněk, VITRIH, Vito and ŽAGAR, Emil, 2014, C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs. [online]. 2014. Vol. 257, p. 65–78. [Accessed 5 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.cam.2013.08.007 Copy citation

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Secondary language

Language:	English
Abstract:	V članku je obravnavana $C^1$ Hermiteova interpolacija s prostorskimi loki, katerih hodograf je pitagorejski. Opisan je splošen algoritem za njihovo konstrukcijo. Vsak segment loka, ki je sestavljen iz dveh delov, interpolira pozicijo in tangento v robni točki. Segmeta sta v sredini spojena $C^1$ gladko. Dobljen lok je podan v zaključeni obliki s tremi prostimi parametri. Dva sta izbrana na podlagi asimptotične analize, tretjega pa izberemo tako, da minimiziramo dolžino loka ali elastično napetostno energijo. Končni rezultat je interpolacijski zlepek, ki ga je mogoče konstruirati lokalno in obstaja za poljubno konfiguracijo podatkov Hermiteovega tipa.
Keywords:	matematika, parametrična krivulja, PH krivulja, pitagorejski hodograh, Hermiteova interpolacija, lok, kubična prostorska krivulja

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