discrete beta-splines
Goodman (1985) and Joe (1986) have given explicit formulas for (cubic) Beta-splines on uniform knot sequences with varying 1 and 2 values at the knots, and nonuniform knot sequences with varying 2 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).
sequence values control algorithm computation