DOI QR코드

DOI QR Code

HERMITE INTERPOLATION USING PH CURVES WITH UNDETERMINED JUNCTION POINTS

  • Kong, Jae-Hoon (Department of Mathematics GyeongSang National University) ;
  • Jeong, Seung-Pil (Department of Mathematics GyeongSang National University) ;
  • Kim, Gwang-Il (Department of Mathematics and RINS College of Natural Science GyeongSang National University)
  • Received : 2010.09.30
  • Published : 2012.01.31

Abstract

Representing planar Pythagorean hodograph (PH) curves by the complex roots of their hodographs, we standardize Farouki's double cubic method to become the undetermined junction point (UJP) method, and then prove the generic existence of solutions for general $C^1$ Hermite interpolation problems. We also extend the UJP method to solve $C^2$ Hermite interpolation problems with multiple PH cubics, and also prove the generic existence of solutions which consist of triple PH cubics with $C^1$ junction points. Further generalizing the UJP method, we go on to solve $C^2$ Hermite interpolation problems using two PH quintics with a $C^1$ junction point, and we also show the possibility of applying the modi e UJP method to $G^2[C^1]$ Hermite interpolation.

References

  1. G. Albrecht and R. T. Farouki, Construction of $C^{2}$Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math. 5 (1996), no. 4, 417-442. https://doi.org/10.1007/BF02124754
  2. H. I. Choi, C. Y. Han, H. P. Moon, K. H. Roh, and N. S.Wee, Medial axis transformation and offset curves by Minkowski Pythagorean hodograph curves, Computer-Aided Design 31 (1999), 59-72 https://doi.org/10.1016/S0010-4485(98)00080-3
  3. 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. https://doi.org/10.1023/A:1015294029079
  4. R. T. Farouki, Pythagorean hodograph curves in practical use, Geometry processing for design and manufacturing, 3-33, SIAM, Philadelphia, PA, 1992.
  5. R. T. Farouki, The conformal map $z{\rightarrow}z^{2}$ of the hodograph plane, Comput. Aided Geom. Design 11 (1994), no. 4, 363-390. https://doi.org/10.1016/0167-8396(94)90204-6
  6. R. T. Farouki, The elastic bending energy of Pythagorean-hodograph curves, Comput. Aided Geom. Design 13 (1996), no. 3, 227-241. https://doi.org/10.1016/0167-8396(95)00024-0
  7. R. T. Farouki, M. Al-Kandari, and T. Sakkalis, Hermite interpolation by rotation- invariant spatial Pythagorean-hodograph curves, Adv. Comput. Math. 17 (2002), no. 4, 369-383. https://doi.org/10.1023/A:1016280811626
  8. R. T. Farouki and C. Y. Han, Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves, Comput. Aided Geom. Design 20 (2003), no. 7, 435-454. https://doi.org/10.1016/S0167-8396(03)00095-5
  9. 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.
  10. 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
  11. R. T. Farouki and C. A. Neff, Hermite interpolation by Pythagorean-hodograph quintics, Math. Comp. 64 (1995), no. 212, 1589-1609. https://doi.org/10.1090/S0025-5718-1995-1308452-6
  12. R. T. Farouki and J. Peters, Smooth curve design with double-Tschirnhausen cubics, Ann. Numer. Math. 3 (1996), no. 1-4, 63-82.
  13. R. T. Farouki and T. Sakkalis, Pythagorean hodographs, IBM J. Res. Develop. 34 (1990), no. 5, 736-752. https://doi.org/10.1147/rd.345.0736
  14. R. T. Farouki and T. Sakkalis, Pythagorean hodograph space curves, Adv. Comput. Math. 2 (1994), no. 1, 41-66. https://doi.org/10.1007/BF02519035
  15. R. T. Farouki and S. Shah, Real-time CNC interpolator for Pythagorean hodograph curves, Comput. Aided Geom. Design 13 (1996), 583-600. https://doi.org/10.1016/0167-8396(95)00047-X
  16. Z. Habib and M. Sakai, $G^{2}$ Pythagorean hodograph quintic transition between two circles with shape control, Comput. Aided Geom. Design 24 (2007), no. 5, 252-266. https://doi.org/10.1016/j.cagd.2007.03.004
  17. Z. Habib and M. Sakai, Transition between concentric or tangent circles with a single segment of $G^{2}$ PH quintic curve, Comput. Aided Geom. Design 25 (2008), no. 4-5, 247-257. https://doi.org/10.1016/j.cagd.2007.10.006
  18. B. Juttler, Hermite interpolation by Pythagorean hodograph curves of degree seven, Math. Comp. 70 (2001), no. 235, 1089-1111. https://doi.org/10.1090/S0025-5718-00-01288-6
  19. 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
  20. G. I. Kim and M. H. Ahn, $C^{1}$ Hermite interpolation using MPH quartic, Comput. Aided Geom. Design 20 (2003), no. 7, 469-492. https://doi.org/10.1016/j.cagd.2003.06.001
  21. G. I. Kim, J. H. Kong, and S. Lee, First order Hermite interpolation with spherical Pythagorean-hodograph curves, J. Appl. Math. Comput. 23 (2007), no. 1-2, 73-86. https://doi.org/10.1007/BF02831959
  22. G. I. Kim and S. Lee, Pythagorean-hodograph preserving mappings, J. Comput. Appl. Math. 216 (2008), no. 1, 217-226. https://doi.org/10.1016/j.cam.2007.04.026
  23. J. H. Kong, S. P. Jeong, S. Lee, and G. I. Kim, $C^{1}$ Hermite interpolation with simple planar PH curves by speed reparametrization, Comput. Aided Geom. Design 25 (2008), no. 4-5, 214-229. https://doi.org/10.1016/j.cagd.2007.11.006
  24. C. Manni, A. Sestini, R. T. Farouki, and C. Y. Han, Characterization and construction of helical polynomial space curves, J. Comput. Appl. Math. 162 (2004), no. 2, 365-392. https://doi.org/10.1016/j.cam.2003.08.030
  25. H. P. Moon, Minkowsi Pythagorean hodographs, Comput. Aided Geom. Design 25 (2008), no. 4-5, 739-753.
  26. F. Pelosi, R. T. Farouki, C. Manni, and A. Sestini, Geometric Hermite interpolation by spatial Pythagorean hodograph cubics, Adv. Comput. Math. 22 (2005), no. 4, 325-352. https://doi.org/10.1007/s10444-003-2599-x
  27. F. Pelosi, M. L. Sampoli, R. T. Farouki, and C. 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. https://doi.org/10.1016/j.cagd.2006.09.005
  28. 3H. Pottmann, Curve design with rational Pythagorean-hodograph curves, Adv. Comput. Math. 3 (1995), no. 1-2, 147-170. https://doi.org/10.1007/BF03028365
  29. Z. Sir and B. Juttler, Euclidean and Minkowski Pythagorean hodograph curves over planar cubics, Comput. Aided Geom. Design 22 (2005), no. 8, 753-770. https://doi.org/10.1016/j.cagd.2005.03.002
  30. Z. Sir and B. Juttler, $C^{2}$ Hermite interpolation by Pythagorean hodograph space curves, Mathematics of Computation 76 (2007), no. 259, 1373-1391. https://doi.org/10.1090/S0025-5718-07-01925-4
  31. D. J. Walton and D. S. Meek, A Pythagorean hodograph quintic spiral, Computer-Aided Design 28 (1996), no. 12, 943-950. https://doi.org/10.1016/0010-4485(96)00030-9
  32. D. J. Walton and D. S. Meek, Geometric Hermite interpolation with Tschirnhausen cubics, J. Comput. Appl. Math. 81 (1997), no. 2, 299-309. https://doi.org/10.1016/S0377-0427(97)00066-6
  33. D. J. Walton and D. S. Meek, $C^{2}$ curve design with a pair of Pythagorean Hodograph quintic spiral segments, Comput. Aided Geom. Design 24 (2007), no. 5, 267-285. https://doi.org/10.1016/j.cagd.2007.03.003

Cited by

  1. Planar C1 Hermite interpolation with PH cuts of degree (1,3) of Laurent series vol.31, pp.9, 2014, https://doi.org/10.1016/j.cagd.2014.08.005
  2. Minkowski Pythagorean-hodograph preserving mappings vol.308, 2016, https://doi.org/10.1016/j.cam.2016.05.032
  3. C 1 Hermite interpolation with PH curves by boundary data modification vol.248, 2013, https://doi.org/10.1016/j.cam.2013.01.016