Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T05:33:46.325Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albrecht, Gudrun and T.|Farouki, Rida, Construction of C2 Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math. 5 (1996), no.4, 417–442.CrossRefGoogle Scholar
[2]Allgower, Eugene L. and Georg, Kurt, Numerical continuation methods, Springer Series in Computational Mathematics, vol. 13, Springer-Verlag, Berlin, 1990, An introduction.Google Scholar
[3]Byrtus, Marek and Bastl, Bohumír, G1 Hermite interpolation by PH cubics revisited, Comput. Aided Geom. Design 27 (2010), no. 8, 622–630.CrossRefGoogle Scholar
[4]Choi, Hyeong In, Farouki, Rida T., Kwon, Song-Hwa, and Moon, Hwan Pyo, Topological criterion for selection of quintic Pythagorean-hodograph Hermite interpolants, Comput. Aided Geom. Design 25 (2008), no. 6, 411–433.CrossRefGoogle Scholar
[5]Choi, Hyeong In, Lee, Doo Seok, and Moon, Hwan Pyo, Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comput. Math. 17 (2002), no.1-2, 5–48, Advances in geometrical algorithms and representations.CrossRefGoogle Scholar
[6]Farin, Gerald, Hoschek, Josef, and Kim, Myung-Soo, Handbook of Computer Aided Geometric Design,first ed., Elsevier, Amsterdam, 2002.Google Scholar
[7]Farouki, Rida T., The conformal map z » z 2 of the hodograph plane, Comput. Aided Geom. Design 11 (1994), no. 4, 363–390.CrossRefGoogle Scholar
[8]Farouki, Rida T., Pythagorean-hodograph curves: algebra and geometry inseparable, Geometry and Computing, vol. 1, Springer, Berlin, 2008.CrossRefGoogle Scholar
[9]Farouki, Rida T., Giannelli, Carlotta, Manni, Carla, and Sestini, Alessandra, Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures, Comput. Aided Geom. Design 25 (2008), no. 4-5, 274–297.CrossRefGoogle Scholar
[10]Farouki, Rida T. and Sakkalis, Takis, Pythagorean hodographs, IBM J. Res. Develop. 34 (1990), no. 5, 736–752.CrossRefGoogle Scholar
[11]Han, Chang Yong, Geometric Hermite interpolation by monotone helical quintics, Comput. Aided Geom. Design 27 (2010), no. 9, 713–719.CrossRefGoogle Scholar
[12]Jaklič, Gašper, Kozak, Jernej, Krajnc, Marjeta, Vitrih, Vito, and Zagar, Emil, Geometric La-grange interpolation by planar cubic Pythagorean-hodograph curves, Comput. Aided Geom. Design 25 (2008), no. 9, 720–728.CrossRefGoogle Scholar
[13]Jaklič, Gašper, Kozak, Jernej, Krajnc, Marjeta, Vitrih, Vito, and Zagar, EmilOn interpolation by planar cubic G2 Pythagorean-hodograph spline curves, Math. Comp. 79 (2010), no. 269, 305–326.CrossRefGoogle Scholar
[14]Jaklič, Gašper, Kozak, Jernej, Krajnc, Marjeta, Vitrih, Vito, and Zagar, EmilAn approach to geometric interpolation by Pythagorean-hodograph curves, Adv. Comput. Math. (2012), no. 37, 123–150.CrossRefGoogle Scholar
[15]Juüttler, B., Hermite interpolation by Pythagorean hodograph curves of degree seven, Math. Comp. 70 (2001), no. 235, 1089–1111 (electronic).CrossRefGoogle Scholar
[16]Jüttler, Bert and Maurer, C, Cubic Pythagorean-hodograph spline curves and applications to sweep surface modeling, Comput. Aided Geom. Design 31 (1999), 73–83.CrossRefGoogle Scholar
[17]Kwon, Song-Hwa, Solvability of G1 Hermite interpolation by spatial Pythagorean-hodograph cubics and its selection scheme, Comput. Aided Geom. Design 27 (2010), no. 2, 138–149.CrossRefGoogle Scholar
[18]Lyche, Tom and Mørken, Knut, A metric for parametric approximation, Curves and surfaces in geometric design (Chamonix-Mont-Blanc, 1993), Peters, A K, Wellesley, MA, 1994, pp. 311–318.Google Scholar
[19]S.|Meek, Dereck and Walton, D.J., Hermite interpolation with Tschirnhausen cubic spirals, Comput. Aided Geom. Design 14 (1997), no. 7, 619–635.Google Scholar
[20]S.|Meek, Dereck and Walton, D.J., G2 curve design with a pair of Pythagorean hodograph quintic spiral segments, Comput. Aided Geom. Design 24 (2007), no. 5, 267–285.Google Scholar
[21]Pelosi, Francesca, T.|Farouki, Rida, Manni, Carla, and Sestini, Alessandra, Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics, Adv. Comput. Math. 22 (2005), no. 4, 325–352.CrossRefGoogle Scholar
[22]Sakkalis, Takis and Farouki, Rida T., Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3, J.Comput. Appi. Math. 236 (2012), no. 17, 4375–4382.CrossRefGoogle Scholar
[23]Šír, Zbyněk and Jüttler, Bert, C2 Hermite interpolation by Pythagorean hodograph space curves, Math. comp. 76 (2007), no. 259, 1373–1391.CrossRefGoogle Scholar