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 β1 and β2 values at the knots. We establish an alternative explicit formula for cubic Beta-splines on a nonuniform knot sequence with constant β1 = 1 and varying β2 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.