An explicit derivation of discretely shaped Beta-spline basis functions of arbitrary order

摘要: Beta-splines are a class of splines with applications in the construction curves and surfaces for computer-aided geometric design. One salient features Beta-spline is that thus constructed geometrically continuous , more general notion continuity than one used ordinary B-splines. The basic building block order k set Beta-polynomials degree – 1, which to form basis functions. coefficients functions certain shape parameters β s ; i . In this paper, we present symbolic derivation as polynomials over field K n real rational indeterminates We show, constructively, existence uniqueness satisfying design objectives continuity, minimum spline order, invariance under translation, linear independence, an explicit procedure their computation. presented here, resulting procedure, valid case discretely-shaped arbitrary uniform knot sequences. By extending z representing lengths parametric intervals, result can be generalized non-uniform sequences, shown Seroussi & Barsky (1991).