Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

TIME REPARAMETRIZATION OF PIECEWISE PYTHAGOREAN-HODOGRAPH $C^1$ HERMITE INTERPOLANTS

  • Kong, Jae-Hoon (Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University) ;
  • Kim, Gwang-Il (Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University)
  • Received : 2011.10.24
  • Accepted : 2012.02.17
  • Published : 2012.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we show two ways of the time reparametrization of piecewise Pythagorean-hodograph $C^1$ Hermite interpolants. One is the time reparametrization with no shape change, and the other is that with shape change. We show that the first reparametrization does not depend on the boundary data and that it is uniquely determined by the size of parameter domain, up to the general cases. We empirically show that the second parametrization can cause the change of the shape of interpolant.

Keywords

References

  1. R.T. Farouki and T. Sakkalis, Pythagorean hodographs. IBM Journal of Research and Development, 34 (1990), 736-752. https://doi.org/10.1147/rd.345.0736
  2. R.T. Farouki, Pythagorean hodograph curves in practical use. R.E. Barnhill (ed.) Geometric Processing for Design and Manufacturing, SIAM (1992), 3-33.
  3. R.T. Farouki and C.A. Neff, Hermite interpolation by Pythagorean-hodograph quintics. Mathematics of Computation, 64 (1995), 1589-1609. https://doi.org/10.1090/S0025-5718-1995-1308452-6
  4. R.T. Farouki and S. Shah, Real-time CNC interpolator for Pythagorean hodograph curves. Computer Aided Geometric Design, 13 (1996), 583-600.
  5. R.T. Farouki, J. Manjunathaiah, D. Nicholas, G.-F. Yuan and S. Jee, Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves. Computer Aided Geometric Design, 3 (1998), 631-640.
  6. R.T. Farouki, J. Manjunathaiah and G.-F. Yuan, G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines. International Journal of Machine Tools and Manufacture, 39 (1999), 123-142 https://doi.org/10.1016/S0890-6955(98)00018-2
  7. H. Pottmann, Curve design with rational Pythagorean-hodograph curves. Advances in Computational Mathematics, 3 (1995), 147-170. https://doi.org/10.1007/BF03028365
  8. D.J. Walton and D.S. Meek, A Pythagorean hodograph quintic spiral. Computer-Aided Design, 28 (1996), 12, 943-950. https://doi.org/10.1016/0010-4485(96)00030-9
  9. D.J. Walton and D.S. Meek, Geometric Hermite interpolation with Tschirnhausen cubics. Journal of Computational and Applied Mathematics, 81 (1997), 299-309. https://doi.org/10.1016/S0377-0427(97)00066-6
  10. R.T. Farouki and J. Peters, Smooth curve design with double-Tschirnhausen cubics. Annals of Numerical Mathematics, 3 (1996), 63-82.
  11. R.T. Farouki and T. Sakkalis, Pythagorean hodograph space curves. Advances in Computational Mathematics, 2 (1994), 41-66. https://doi.org/10.1007/BF02519035
  12. R.T. Farouki, M. al-Kandari and T. Sakkalis, Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves. Advances in Computational Mathematics, 17 (2002), 369-383. https://doi.org/10.1023/A:1016280811626
  13. B. Juttler and C. Maurer, Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling. Computer-Aided Design, 31 (1999), 73-83. https://doi.org/10.1016/S0010-4485(98)00081-5
  14. C. Manni, A. Sestini, R.T. Farouki and C.Y. Han, Characterization and construction of helical polynomial space curves. Journal of Computational and Applied Mathematics, 162 (2004), 365-392. https://doi.org/10.1016/j.cam.2003.08.030
  15. R.T. Farouki, The conformal map $z\rightarrow z^2$ of the hodograph plane. Computer Aided Geometric Design, 11 (1994), 363-390. https://doi.org/10.1016/0167-8396(94)90204-6
  16. H.I. Choi, D.S. Lee and H.P. Moon, Clifford algebra, spin representation, and rational parametrization of curves and surfaces. Advances in Computational Mathematics, 17 (2001), 5-48.
  17. G.I. Kim, J.H. Kong and S. Lee, First order Hermite interpolation with spherical Pythagorean-hodograph curves. Journal of Applied Mathematics and Computing, 23 (2007), 1-2, 73-86. https://doi.org/10.1007/BF02831959
  18. J.H. Kong, S.P. Jeong, S. Lee and G.I. Kim, $C^1$ Hermite interpolation with simple planar PH curves by speed reparametrization. Computer Aided Geometric Design, 25 (2008), 214-229 https://doi.org/10.1016/j.cagd.2007.11.006
  19. G. Albrecht and R.T. Farouki, Construction of $C^2$ Pythagorean-hodograph interpolating splines by the homotopy method. Advances in Computational Mathematics, 5 (1996), 4, 417-442. https://doi.org/10.1007/BF02124754
  20. J.H. Kong, S.P. Jeong and G.I. Kim, C1 Hermite interpolation using PH curves with undetermined junction points. Bull. Korean Math. Soc., 49(2012), 1, 175-195. https://doi.org/10.4134/BKMS.2012.49.1.175
  21. B. Juttler, Hermite interpolation by Pythagorean hodograph curves of degree seven. Mathematics of Computation, 70 (2001), 1089-1111.
  22. R.T. Farouki and C.Y. Han, Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Computer Aided Geometric Design, 20 (2003), 435-454. https://doi.org/10.1016/S0167-8396(03)00095-5