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Interpolation by G2 Quintic Pythagorean-Hodograph Curves

Published online by Cambridge University Press:  28 May 2015

Gašper Jaklič ,
Jernej Kozak ,
Marjeta Krajnc ,
Vito Vitrih  and
Emil Žagar
Gašper Jaklič*
Affiliation:
FMF, University of Ljubljana, and IAM, University of Primorska, Jadranska 19, 1000 Ljubljana, Slovenia
Jernej Kozak*
Affiliation:
FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Marjeta Krajnc*
Affiliation:
FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Vito Vitrih*
Affiliation:
FAMNIT and IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
Emil Žagar*
Affiliation:
FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
*
Corresponding author.Email address:gasper.jaklic@fmf.uni-lj.si
Corresponding author.Email address:jernej.kozak@fmf.uni-lj.si
Corresponding author.Email address:marjetka.krajnc@fmf.uni-lj.si
Corresponding author.Email address:vito.vitrih@upr.si
Corresponding author.Email address:emil.zagar@fmf.uni-lj.si
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Abstract

In this paper, the G2 interpolation by Pythagorean-hodograph (PH) quintic curves in ℝd, d ≥ 2, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension d, they supply a G2 quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.

Keywords

65D0565D17Pythagorean-hodograph curveHermite interpolationgeometric continuitynonlinear analysishomotopy
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 3 , August 2014 , pp. 374 - 398
Copyright
Copyright © Global Science Press Limited 2014

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