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Available in print format 		Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(e) ISSN 0025-5718(p)

     

On interpolation by Planar cubic $ G^2$ pythagorean-hodograph spline curves

Author(s): Gasper Jaklic; Jernej Kozak; Marjeta Krajnc; Vito Vitrih; Emil Zagar.
Journal: Math. Comp. 79 (2010), 305-326.
MSC (2000): Primary 41A05, 41A15, 41A25, 41A30, 65D05, 65D07, 65D17; Secondary 65D10
Posted: July 29, 2009
MathSciNet review: 2552228
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Abstract | References | Similar articles | Additional information

Abstract: In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic $ G^2$ Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.

References:

1.
Gudrun Albrecht and Rida T. Farouki, Construction of $ C^2$ Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math. 5 (1996), no. 4, 417-442. MR 1414289 (97k:65033)

2.
Carl de Boor, A practical guide to splines, revised ed., Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York, 2001. MR 1900298 (2003f:41001)

3.
Rida T. Farouki, The conformal map $ z\to z^2$ of the hodograph plane, Comput. Aided Geom. Design 11 (1994), no. 4, 363-390. MR 1287495 (95f:65034)

4.
-, Pythagorean-hodograph curves: Algebra and geometry inseparable, Geometry and Computing, vol. 1, Springer, Berlin, 2008. MR 2365013 (2008k:65027)

5.
Rida T. Farouki, Bethany K. Kuspa, Carla Manni, and Alessandra Sestini, Efficient solution of the complex quadratic tridiagonal system for $ C\sp 2$ PH quintic splines, Numer. Algorithms 27 (2001), no. 1, 35-60. MR 1847983 (2002e:65074)

6.
Rida T. Farouki, Carla Manni, and Alessandra Sestini, Shape-preserving interpolation by $ G^1$ and $ G^2$ PH quintic splines, IMA J. Numer. Anal. 23 (2003), no. 2, 175-195. MR 1974222 (2004c:65011)

7.
Rida T. Farouki and C. Andrew Neff, Hermite interpolation by Pythagorean hodograph quintics, Math. Comp. 64 (1995), no. 212, 1589-1609. MR 1308452 (95m:65025)

8.
Rida T. Farouki and Takis Sakkalis, Pythagorean hodographs, IBM J. Res. Develop. 34 (1990), no. 5, 736-752. MR 1084084 (92a:65063)

9.
Yu Yu Feng and Jernej Kozak, On $ G^2$ continuous cubic spline interpolation, BIT 37 (1997), no. 2, 312-332. MR 1450963 (98c:65014)

10.
Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, Vito Vitrih, and Emil Žagar, Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curves, Comput. Aided Geom. Design 25 (2008), no. 9, 720-728. MR 2468201

11.
Bert Jüttler, Hermite interpolation by Pythagorean hodograph curves of degree seven, Math. Comp. 70 (2001), no. 235, 1089-1111 (electronic). MR 1826577 (2002c:65027)

12.
Dereck S. Meek and D. J. Walton, Geometric Hermite interpolation with Tschirnhausen cubics, J. Comput. Appl. Math. 81 (1997), no. 2, 299-309. MR 1459031 (98b:65011)

13.
-, Hermite interpolation with Tschirnhausen cubic spirals, Comput. Aided Geom. Design 14 (1997), no. 7, 619-635. MR 1467315 (98d:65020)

14.
Francesca Pelosi, Maria Lucia Sampoli, Rida T. Farouki, and Carla Manni, A control polygon scheme for design of planar $ C^2$ PH quintic spline curves, Comput. Aided Geom. Design 24 (2007), no. 1, 28-52. MR 2286365

15.
Zbyněk Šır and Bert Jüttler, Euclidean and Minkowski Pythagorean hodograph curves over planar cubics, Comput. Aided Geom. Design 22 (2005), no. 8, 753-770. MR 2173576 (2006i:65021)

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Additional Information:

Gasper Jaklic
Affiliation: FMF, University of Ljubljana, Slovenia and PINT, University of Primorska, Koper, Slovenia
Address at time of publication: Jadranska 19, 1000 Ljubljana, Slovenia
Email: gasper.jaklic@fmf.uni-lj.si

Jernej Kozak
Affiliation: FMF and IMFM, University of Ljubljana, Slovenia
Address at time of publication: Jadranska 19, 1000 Ljubljana, Slovenia
Email: jernej.kozak@fmf.uni-lj.si

Marjeta Krajnc
Affiliation: IMFM, University of Ljubljana, Slovenia
Address at time of publication: Jadranska 19, 1000 Ljubljana, Slovenia
Email: marjetka.krajnc@fmf.uni-lj.si

Vito Vitrih
Affiliation: PINT, University of Primorska, Koper, Slovenia
Address at time of publication: Muzejski trg 2, 6000 Koper, Slovenia
Email: vito.vitrih@upr.si

Emil Zagar
Affiliation: FMF and IMFM, University of Ljubljana, Slovenia
Address at time of publication: Jadranska 19, 1000 Ljubljana, Slovenia
Email: emil.zagar@fmf.uni-lj.si

DOI: 10.1090/S0025-5718-09-02298-4
PII: S 0025-5718(09)02298-4
Received by editor(s): June 6, 2008
Received by editor(s) in revised form: March 25, 2009
Posted: July 29, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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