Multiple-knot and rational cubic beta-splines

ABSTRACT

Goodman (Properties of Beta-splines. J. Approx. Theory 44, 2 (June 1985), 132-153) gave an explicit formula for cubic Beta-splines on a uniform knot sequence with varying &bgr;1 and &bgr;2 values at the knots. We establish an alternative explicit formula for cubic Beta-splines on a nonuniform knot sequence with constant &bgr;1 = 1 and varying &bgr;2 values at the knots. This alternative formula can also be used if the knot sequence contains multiple knots, and is useful for knot insertion. We show how to efficiently evaluate a cubic Beta-spline curve at many values using this formula. We introduce rational cubic Beta-spline curves and surfaces that have extra weight parameters for shape control, and show that they satisfy the same geometric continuity conditions and properties as nonrational cubic Beta-spline curves and surfaces.

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INDEX TERMS

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

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

General Terms:
Algorithms