“Discrete Beta-splines” by Joe

Goodman (1985) and Joe (1986) have given explicit formulas for (cubic) Beta-splines on uniform knot sequences with varying ß1 and ß2 values at the knots, and nonuniform knot sequences with varying ß2 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).

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