Advances in Computational Mathematics

, Volume 3, Issue 1–2, pp 147–170

Curve design with rational Pythagorean-hodograph curves

  • Helmut Pottmann
Article

Abstract

The dual Bézier representation offers a simple and efficient constructive approach to rational curves with rational offsets (rational PH curves). Based on the dual form, we develop geometric algorithms for approximating a given curve with aG2 piecewise rational PH curve. The basic components of the algorithms are an appropriate geometric segmentation andG2 Hermite interpolation. The solution involves rational PH curves of algebraic class 4; these curves and important special cases are studied in detail.

Keywords

Rational curve hodograph offset curve dual Bézier representation geometric Hermite interpolation curve approximation 

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References

  1. [1]
    W. Blaschke, Untersuchungen über die Geometrie der Speere in der Euklidischen Ebene, Monatshefte für Mathematik und Physik 21 (1910) 3–60.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    O. Bottema and B. Roth,Theoretical Kinematics (North-Holland, Amsterdam/New York/ Oxford, 1979).MATHGoogle Scholar
  3. [3]
    C. de Boor, K. Höllig and M. Sabin, High accuracy geometric Hermite interpolation, Comp. Aided Geom. Design 4 (1987) 267–278.Google Scholar
  4. [4]
    T.D. DeRose, Rational Bézier curves and surfaces on projective domains, in:NURBS for Curve and Surface Design, ed. G. Farin (SIAM, Philadelphia, 1991) pp. 35–45.Google Scholar
  5. [5]
    G. Farin, Rational curves and surfaces, in:Mathematical Methods in CAGD, eds. T. Lyche and L.L. Schumaker (Academic press, 1989) pp. 215–238.Google Scholar
  6. [6]
    G. Farin,Curves and Surfaces for Computer Aided Geometric Design, 3rd ed. (Academic Press, 1992).Google Scholar
  7. [7]
    R.T. Farouki, Hierarchical segmentations of algebraic curves and some applications, in:Mathematical Methods in CAGD, eds. T. Lyche and L.L. Schumaker (Academic Press, 1989) pp. 239–248.Google Scholar
  8. [8]
    R.T. Farouki, Pythagorean-hodograph curves in practical use, in:Geometry Processing for Design and Manufacturing, ed. R.E. Barnhill (SIAM, Philadelphia, 1992) pp. 3–33.Google Scholar
  9. [9]
    R.T. Farouki, The conformal mapzz 2 of the hodograph plane, Comp. Aided Geom. Design 11 (1994) 363–390.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    R.T. Farouki, and C.A. Neff, Hermite interpolation by Pythagorean-hodograph quintics, IBM Research Report RC19234 (1993).Google Scholar
  11. [11]
    R.T. Farouki and H. Pottmann, Polynomial and rational Pythagorean-hodograph curves reconciled, IBM Research Report RC19571 (1994).Google Scholar
  12. [12]
    R.T. Farouki and T. Sakkalis, Pythagorean hodographs, IBM J. Res. Develop. 34 (1990) 736–752.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R.T. Farouki and T. Sakkalis, Pythagorean-hodograph space curves, Adv. Comp. Math. 2 (1994) 41–66.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    R.T. Farouki and T.W. Sederberg, Genus of the offset to a parabola, IBM Research Report RC18980 (1993).Google Scholar
  15. [15]
    R.T. Farouki, K. Tarabanis, J.U. Korein, J.S. Batchelder and S.R. Abrams, Offset curves in layered manufacturing, IBM Research Report RC19408 (1993).Google Scholar
  16. [16]
    J.C. Fiorot and T. Gensane, Characterizations of the set of rational parametric curves with rational offsets, in:Curves and Surfaces in Geometric Design, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (AK Peters, Wellesley, MA, 1994) pp. 153–160.Google Scholar
  17. [17]
    J. Hoschek, Dual Bézier curves and surfaces, in:Surfaces in Computer Aided Geometric Design, eds. R.E. Barnhill and W. Boehm (North-Holland, 1983) pp. 147–156.Google Scholar
  18. [18]
    J. Hoschek, Detecting regions with undesirable curvature, Comp. Aided Geom. Design 1 (1984) 183–192.MATHCrossRefGoogle Scholar
  19. [19]
    J. Hoschek and D. Lasser,Grundlagen der geometrischen Datenverarbeitung, 2nd ed. (Teubner, Stuttgart, 1992).MATHGoogle Scholar
  20. [20]
    W. Lü, Rational offsets by reparametrizations, preprint (1992).Google Scholar
  21. [21]
    W. Lü, Rationality of the offsets to algebraic curves and surfaces, preprint (1993).Google Scholar
  22. [22]
    B. Pham, Offset curves and surfaces: a brief survey, Comp. Aided Design 24 (1992) 223–229.CrossRefGoogle Scholar
  23. [23]
    H. Pottmann, Rational curves and surfaces with rational offsets, Comp. Aided Geom. Design (1995), to appear.Google Scholar
  24. [24]
    H. Pottmann, Applications of the dual Bézier representation of rational curves and surfaces, in:Curves and Surfaces in Geometric Design, eds. P.J. Laurent, A. L. Méhauté and L.L. Schumaker (AK Peters, Wellesley, MA, 1994) pp. 377–384.Google Scholar
  25. [25]
    W. Wunderlich, Algebraische Böschungslinien dritter und vierter Ordnung, Sitzungsberichte der Österreichischen Akademie der Wissenschaften 181 (1973) 353–376.MATHMathSciNetGoogle Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Helmut Pottmann
    • 1
  1. 1.Institut für GeometrieTechnische Universität WienWienAustria

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