A round trip to B-splines via de Casteljau

ABSTRACT

B-splines are constructed from Bézier curves solely using de Casteljau's construction. Divided differences are not being used, nor is Mansfield's recurrence formula presupposed. Yet, it is shown how to differentiate, subdivide, and evaluate a B-spline. These results are derived from the corresponding techniques of Bézier curves.

REFERENCES

• 1: BOEHM, W. Ober die Konstruktion von B-Spline-Kurven. Computing 18, 2 (1977), 161-166.
• 2: BOEHM, W. Inserting new knots into B-spline curves. Comput.-Aided Des. 12, 4 {July 1980), 199-201.
• 3: DE BOOR, C. On calculating with B-splines. J. Approx. Theory 6, 1 {1972), 50-62.
• 4: DE BOOR, C. Splines as Linear Combinations of B-sptines, A Survey Approx. Theory II. Academic Press, New York, 1976.
• 5: DE BOOR, C., HOLLIG, K. B-splines without divided differences. In Geometric Modeling, Algorithms and New Trends, G. Farin, Ed. SIAM, Philadelphia, Pa., pp. 21-27.
• 6: PRAUTZSCH, H. Unterteihzngsalgorithmen fi~r Bdzier-und B-Spline-Fliichen. Diplom Arbeit, TU Braunschweig, 1983.
• 7: Hartmut Prautzsch, Some remarks on three B-spline constructions, Mathematical methods in computer aided geometric design, Academic Press Professional, Inc., San Diego, CA, 1989
• 8: SABIN, M.A. The use of p}ecewise forms for the numerical representation of shape. Dissertation, MTA Budapest, 1977.
• 9: SABLONNIERE, P. Spline and B6zier polygons associated with a polynomial spline curve. Comput.-Aided Des. 10, 4 (July 1978), 257-261.
• 10: Hans-Peter Seidel, A new multiaffine approach to B-splines, Computer Aided Geometric Design, v.6 n.1, p.23-32, February 1989 [doi>10.1016/0167-8396(89)90004-6]
• 11: STARK, E. Mehrfach differenzierbare B~zier-Kurven und BAzier-Fl~ichen. Dissertation, TU Braunschweig, 1976.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

General Terms:
Algorithms, Theory