Discrete Beta-splines arise when a Beta-spline curve is subdivided; that is, extra knots are inserted so that the curve is expressed in terms of a larger number of control vertices and Beta-splines. Their properties and an algorithm for their computation are given in “Discrete Beta-Splines” by Joe ( Computer Graphics , vol. 21 , pp. 137-144). We prove a stronger version of one of these properties, from which a new algorithm for computing discrete Beta-splines is obtained. This algorithm can also be used to compute discrete B-splines. We give a comparison of operation counts for this algorithm versus other algorithms, and for two methods to compute the new control vertices of Beta-spline and B-spline curves and surfaces.

Quartic Beta-splines have third-degree arc-length or geometric continuity at simple knots and are determined by three β or shape parameters. We present a general explicit formula for quartic Beta-splines, and determine and illustrate the effects of varying the β parameters on the shape of a quartic Beta-spline curve. We show that quartic (and higher degree) rational Beta-splines with arc-length continuity satisfy the same continuity conditions as (nonrational) Beta-splines. We also show that the torsion continuous spline curves presented by Boehm ("Smooth Curves and Surfaces.” In Geometric Modeling: Algorithms and New Trends , G. E. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp. 175-184.) are equivalent to nonrational quartic Beta-spline curves, and determine the relationship between the shape parameters for the two types of curves. Finally, we present an algorithm for inserting a new knot and determining the refined control polygon.

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.

Goodman (1985) and Joe (1986) have given explicit formulas for (cubic) Beta-splines on uniform knot sequences with varying s1 and s2 values at the knots, and nonuniform knot sequences with varying s2 values at the knots, respectively. The advantage of the latter formula is that it can also be used for knot sequences with multiple knots. Discrete Beta-splines arise when a Beta-spline curve is subdivided, i.e. the knot sequence is refined so that the curve is expressed in terms of a larger number of control vertices and Beta-splines. We prove that discrete Beta-splines satisfy the same properties as discrete B-splines, and present an algorithm for computing discrete Beta-splines and the new control vertices using the explicit formula of Joe (1986).