$C^2$ Hermite interpolation by Pythagorean Hodograph space curves
HTML articles powered by AMS MathViewer
- by Zbyněk Šír and Bert Jüttler;
- Math. Comp. 76 (2007), 1373-1391
- DOI: https://doi.org/10.1090/S0025-5718-07-01925-4
- Published electronically: February 1, 2007
- PDF | Request permission
Abstract:
We solve the problem of $C^2$ Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of $C^2$ space boundary data (two points with associated first and second derivatives) we construct a four–dimensional family of PH interpolants of degree $9$ and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree $9$ which is globally $C^2$, and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.References
- Min-Ho Ahn, Gwang-Il Kim, and Chung-Nim Lee, Geometry of root-related parameters of PH curves, Appl. Math. Lett. 16 (2003), no. 1, 49–57. MR 1938190, DOI 10.1016/S0893-9659(02)00143-X
- Hyeong In Choi, Doo Seok Lee, and Hwan Pyo Moon, 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. MR 1902534, DOI 10.1023/A:1015294029079
- Hyeong In Choi and Chang Yong Han, Euler-Rodrigues frames on spatial Pythagorean-hodograph curves, Comput. Aided Geom. Design 19 (2002), no. 8, 603–620. MR 1937124, DOI 10.1016/S0167-8396(02)00165-6
- Roland Dietz, Josef Hoschek, and Bert Jüttler, An algebraic approach to curves and surfaces on the sphere and on other quadrics, Comput. Aided Geom. Design 10 (1993), no. 3-4, 211–229. Free-form curves and free-form surfaces (Oberwolfach, 1992). MR 1235153, DOI 10.1016/0167-8396(93)90037-4
- 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, DOI 10.1016/0167-8396(94)90204-6
- Rida T. Farouki and Takis Sakkalis, Pythagorean-hodograph space curves, Adv. Comput. Math. 2 (1994), no. 1, 41–66. MR 1266023, DOI 10.1007/BF02519035
- R. T. Farouki and C. A. Neff, Hermite interpolation by Pythagorean hodograph quintics, Math. Comp. 64 (1995), no. 212, 1589–1609. MR 1308452, DOI 10.1090/S0025-5718-1995-1308452-6
- R. T. Farouki, J. Manjunathaiah and S. Jee (1998), Design of rational cam profiles with Pythagorean-hodograph curves. Mech. Mach. Theory 33, 669-682.
- R. T. Farouki, K. Saitou, and Y.-F. Tsai, Least-squares tool path approximation with Pythagorean-hodograph curves for high-speed CNC machining, The mathematics of surfaces, VIII (Birmingham, 1998) Info. Geom., Winchester, 1998, pp. 245–264. MR 1732987
- Rida T. Farouki, Mohammad al-Kandari, and Takis Sakkalis, Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves, Adv. Comput. Math. 17 (2002), no. 4, 369–383. MR 1916985, DOI 10.1023/A:1016280811626
- Rida T. Farouki, Mohammad al-Kandari, and Takis Sakkalis, Structural invariance of spatial Pythagorean hodographs, Comput. Aided Geom. Design 19 (2002), no. 6, 395–407. MR 1917337, DOI 10.1016/S0167-8396(02)00123-1
- Rida T. Farouki, Pythagorean-hodograph curves, Handbook of computer aided geometric design, North-Holland, Amsterdam, 2002, pp. 405–427. MR 1928550, DOI 10.1016/B978-044451104-1/50018-6
- Rida T. Farouki, Carla Manni, and Alessandra Sestini, Spatial $C^2$ PH quintic splines, Curve and surface design (Saint-Malo, 2002) Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003, pp. 147–156. MR 2042481
- Rida T. Farouki and Chang Yong Han, Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves, Comput. Aided Geom. Design 20 (2003), no. 7, 435–454. MR 2011551, DOI 10.1016/S0167-8396(03)00095-5
- Rida T. Farouki, Chang Yong Han, Carla Manni, and Alessandra Sestini, Characterization and construction of helical polynomial space curves, J. Comput. Appl. Math. 162 (2004), no. 2, 365–392. MR 2028035, DOI 10.1016/j.cam.2003.08.030
- Josef Hoschek and Dieter Lasser, Fundamentals of computer aided geometric design, A K Peters, Ltd., Wellesley, MA, 1993. Translated from the 1992 German edition by Larry L. Schumaker. MR 1258308
- B. Jüttler and C. Mäurer (1999), Cubic Pythagorean Hodograph Spline Curves and Applications to Sweep Surface Modeling, Comp. Aided Design 31, 73–83.
- B. Jüttler, Hermite interpolation by Pythagorean hodograph curves of degree seven, Math. Comp. 70 (2001), no. 235, 1089–1111. MR 1826577, DOI 10.1090/S0025-5718-00-01288-6
- Jack B. Kuipers, Quaternions and rotation sequences, Princeton University Press, Princeton, NJ, 1999. A primer with applications to orbits, aerospace, and virtual reality. MR 1670862
- Hwan Pyo Moon, Rida T. Farouki, and Hyeong In Choi, Construction and shape analysis of PH quintic Hermite interpolants, Comput. Aided Geom. Design 18 (2001), no. 2, 93–115. MR 1822574, DOI 10.1016/S0167-8396(01)00016-4
- Z. Šír (2003), Hermite interpolation by space PH curves. Proceedings of the 23rd Conference on Geometry and Computer Graphics, University Plzeň, 193-198.
- Z. Šír and B. Jüttler (2005), Constructing acceleration continuous tool paths using pythagorean hodograph curves. Mech. Mach. Theory. 40(11), 1258-1272.
- Z. Šír and B. Jüttler (2005), Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces, in R. Martin, H. Bez and M. Sabin, editors, The Mathematics of Surfaces XI, Springer, 364-380.
- D. S. Meek and D. J. Walton, Geometric Hermite interpolation with Tschirnhausen cubics, J. Comput. Appl. Math. 81 (1997), no. 2, 299–309. MR 1459031, DOI 10.1016/S0377-0427(97)00066-6
- D. J. Walton and D. S. Meek, A generalisation of the Pythagorean hodograph quintic spiral, J. Comput. Appl. Math. 172 (2004), no. 2, 271–287. MR 2095321, DOI 10.1016/j.cam.2004.02.008
Bibliographic Information
- Zbyněk Šír
- Affiliation: Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, Austria
- Address at time of publication: Charles University, Sokolovská 83, 18675 Prague, Czech Republic
- Email: zbynek.sir@jku.at
- Bert Jüttler
- Affiliation: Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, Austria
- Email: bert.juettler@jku.at
- Received by editor(s): May 24, 2005
- Received by editor(s) in revised form: October 27, 2005
- Published electronically: February 1, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1373-1391
- MSC (2000): Primary 68U07; Secondary 53A04, 65D17
- DOI: https://doi.org/10.1090/S0025-5718-07-01925-4
- MathSciNet review: 2299779