Constructive Approximation

, Volume 12, Issue 3, pp 385–408 | Cite as

de Boor-Fix dual functionals and algorithms for Tchebycheffian B-spline curves

  • P. J. Barry
Article

Abstract

The de Boor-Fix dual functionals are a potent tool for deriving results about piecewise polynomial B-spline curves. In this paper we extend these functionals to Tchebycheffian B-spline curves and then use them to derive fundamental algorithms that are natural generalizations of algorithms for piecewise polynomial B-spline algorithms. Then, as a further example of the utility of this approach, we introduce “geometrically continuous Tchebycheffian spline curves,” and show that a further generalization works for them as well.

AMS classification

41A15 65D17 

Key words and phrases

Blossoming de Boor-Fix dual functionals Connection matrix Differentiation Evaluation Geometric continuity Knot insertion Tchebycheffian B-spline Total positivity Zeros 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. J. Barry, R. N. Goldman, C. A. Micchelli (1993):Knot insertion algorithms for geometrically continuous splines determined by connection matrices. Adv. Comput. Math.,1: 139–171.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    P. J. Barry, R. N. Goldman (1993):Knot insertion algorithms. In: Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces (T. Lyche, R. N. Goldman, eds.), Philadelphia: SIAM, pp. 89–133.Google Scholar
  3. 3.
    P. J. Barry (19−):Properties of functions in an auxiliary spline space. To appear in Aequationes Math.Google Scholar
  4. 4.
    W. Boehm (1980):Inserting new knots into B-spline curves. Comput. Aided Design,12: 199–201.CrossRefGoogle Scholar
  5. 5.
    C. de Boor (1972):On calculating with B-splines. J. Approx. Theory,6: 50–62.MATHCrossRefGoogle Scholar
  6. 6.
    C. de Boor, G. Fix (1973):Spline approximation by quasi-interpolants. J. Approx. Theory,8: 19–45.MATHCrossRefGoogle Scholar
  7. 7.
    C. de Boor, K. Höllig (1987):B-splines without divided differences. In: Geometric Modeling: Algorithms and New Trends (G. Farin, ed.). Philadelphia: SIAM, pp. 21–27.Google Scholar
  8. 8.
    N. Dyn, A. Ron (1988):Recurrence relations for Tchebycheffian B-splines. J. Anal. Math.,51: 118–138.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    N. Dyn, C. A. Micchelli (1988):Piecewise polynomial spaces and geometric continuity of curves. Numer. Math.,54: 319–337.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    R. N. Goldman, P. J. Barry (1992):Wonderful triangle: A simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design. In: Mathematical Methods in Computer Aided Geometric Design II (T. Lyche, L. L. Schumaker, eds.), Boston: Academic Press, pp. 297–320.Google Scholar
  11. 11.
    A. Habib, R. N. Goldman (to appear): Theories of contact specified by connection matrices. Comput. Aided Geom. Design.Google Scholar
  12. 12.
    S. Karlin (1968): Total Positivity. Stanford, CA: Stanford University Press.MATHGoogle Scholar
  13. 13.
    T. Lyche (1985):A recurrence relation for Chebyshevian B-splines. Constr. Approx.,1: 155–173.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Pottman (1993):The geometry of Tchebycheffian splines. Comput. Aided Geom. Design,10: 181–210.MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Pottman, M. Wagner (1994):Helix splines as an example of affine Tchebycheffian splines. Adv. Comput. Math.,2: 123–142.MathSciNetGoogle Scholar
  16. 16.
    L. Ramshaw (1987): Blossoming: A Connect-the-Dots Approach to Splines. Digital Research Center Technical Report 19. Palo Alto, CA.Google Scholar
  17. 17.
    L. Ramshaw (1989):Blossoms are polar forms. Comput. Aided Geom. Design,6: 323–358.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    L. L. Schumaker (1981): Spline Functions: Basic Theory. New York: Wiley.MATHGoogle Scholar
  19. 19.
    H.-P. Seidel (1989):A new multiaffine approach to splines. Comput. Aided Geom. Design,6, 23–32.MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    H.-P. Seidel (1993):Polar forms for geometrically continuous spline curves of arbitrary degree. ACM Trans. Graphics,12: 1–34.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • P. J. Barry
    • 1
  1. 1.Computer Science DepartmentUniversity of MinnesotaMinneapolisUSA

Personalised recommendations