Beta-splines are a class of splines with applications in the construction of curves and surfaces for computer-aided geometric design. One of the salient features of the Beta-spline is that the curves and surfaces thus constructed are geometrically continuous, a more general notion of continuity than the one used in ordinary B-splines. The basic building block for Beta-splines of order k is a set of Beta-polynomials of degree k-1, which are used to form the Beta-spline basis functions. The coefficients of the Beta-polynomials are functions of certain shape parameters Betas;i. In this paper, we present a symbolic derivation of the Beta-polynomials as polynomials over the field Kn of real rational functions in the indeterminates Betas;i. We prove, constructively, the existence and uniqueness of Beta-polynomials satisfying the design objectives of geometric continuity, minimum spline order, invariance under translation, and linear independence, and we present an explicit symbolic procedure for their computation. The initial derivation, and the resulting procedure, are valid for the general case of discretely-shaped Beta-splines of arbitrary order, over uniform knot sequences. By extending the field Kn with symbolic indeterminates zs representing the lengths of the parametric intervals, the result is generalized to discretely-shaped Beta-splines over non-uniform knot sequences.




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