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A Symbolic Derivation of Betasplines of Arbitrary Order
Seroussi, Gadiel; Barsky, Brian
HPL9187
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Abstract: Betasplines are a class of splines with applications in the construction of curves and surfaces for computeraided geometric design. One of the salient features of the Betaspline is that the curves and surfaces thus constructed are geometrically continuous, a more general notion of continuity than the one used in ordinary Bsplines. The basic building block for Betaspline of order k is a set of betapolynomials of degree k1, which are used to form the Betaspline basis functions. The coefficients of the Betapolynomials are functions of certain shape parameters Beta sub s; sub i. In this paper, we present a symbolic derivation of the Betapolynomials as polynomials over the field K sub n of real rational functions in the indeterminates Beta sub s; sub i. We prove, constructively, the existence and uniqueness of Betapolynomials 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 discretelyshaped Betasplines of arbitrary order, over uniform knot sequences. By extending the field K sub n with symbolic indeterminates z sub s representing the lengths of the parametric intervals, the result is generalized to discretelyshaped Betasplines over nonuniform knot sequences.
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