A Geometric Approach to Bézier Curves

  • Hans-Peter Seidel
Conference paper
Part of the Focus on Computer Graphics book series (FOCUS COMPUTER)


Using techniques from classical geometry we present a purely geometric approach to Bézier curves and B-splines. The approach is based on the intersection of osculating flats: The osculating 1-flat is simply the tangent line, the osculating 2-flat is the osculating plane, etc. The intersection of osculating flats leads to the so-called polar form. We discuss the main properties of the polar form and show how polar forms lead to a simple new labeling scheme for Bézier curves and B-splines.


Geometric Approach Polar Form Spline Curve Bezier Curve Quadratic Case 
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  1. [1]
    W. Boehm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD. Computer-Aided Geom. Design, 1: 1–60, 1984.MATHCrossRefGoogle Scholar
  2. [2]
    W. Dahmen, C.A. Micchelli and H.-P. Seidel. Blossoming Begets B-Splines Built Better by B-Patches. Research Report RC 16261(#72182), IBM Research Division, Yorktown Heights, 1990. To appear in Math. Comp..Google Scholar
  3. [3]
    C. de Boor. B-form basics. In G. Farin, editor, Geometric Modeling, Algorithms and New Trends, 131–148, SIAM, 1987.Google Scholar
  4. [4]
    C. de Boor. On calculating with B-splines. J. Approx. Th., 6: 50–62, 1972.MATHCrossRefGoogle Scholar
  5. [5]
    C. de Boor. A Practical Guide to Splines. Springer, New York, 1978.MATHCrossRefGoogle Scholar
  6. [6]
    P. de Casteljau. Courbes et surfaces à pôles. Technical Report, André Citroen, Paris, 1963.Google Scholar
  7. [7]
    P. de Casteljau. Formes it Pôles. Hermes, Paris, 1985.Google Scholar
  8. [8]
    P. de Casteljau. Outillages méthodes calcul. Technical Report, André Citroen, Paris, 1959.Google Scholar
  9. [9]
    R.N. Goldman. Blossoming and knot insertion algorithms for B-spline curves. Computer-Aided Geom. Design, 7: 69–81, 1990.MATHCrossRefGoogle Scholar
  10. [10]
    L. Ramshaw. Béziers and B-splines as multiaffine maps. In Theoretical Foundations of Computer Graphics and CAD, 757–776, Springer, 1988.Google Scholar
  11. [11]
    L. Ramshaw. Blossoming: A connect-the-dots approach to splines. Technical Report, Digital Systems Research Center, Palo Alto, 1987.Google Scholar
  12. [12]
    L. Ramshaw. Blossoms are polar forms. Computer-Aided Geom. Design, 6: 323–358, 1989.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    L. Ramshaw. The polar forms of polynomial curves. In Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces, SIGGRAPH ‘81 Course Notes #26, 1.1–1.29, ACM SIGGRAPH, 1991.Google Scholar
  14. [14]
    H.-P. Seidel. Geometric Constructions and Knot Insertion for Geometrically Continuous Spline Curves of Arbitrary Degree. Technical Report CS-90–24, Dept. of Computer Science, University of Waterloo, 1990.Google Scholar
  15. [15]
    H.-P. Seidel. Knot insertion from a blossoming point of view. Computer-Aided Geom. Design, 5: 81–86, 1988.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    H.-P. Seidel. A new multiaffine approach to B-splines. Computer-Aided Geom. Design, 6: 23–32, 1989.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    H.-P. Seidel. Polar forms and triangular B-Spline surfaces. In Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces, SIGGRAPH ‘81 Course Notes #26, 8.1–8.52, ACM SIGGRAPH, 1991.Google Scholar
  18. [18]
    H.-P. Seidel. Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles. Constr. Approx., 7: 257–279, 1991.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© EUROGRAPHICS The European Association for Computer Graphics 1992

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  • Hans-Peter Seidel

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