Computation and modeling in piecewise Chebyshevian spline spaces

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under arbitrary knot insertion. The interest in piecewise Chebyshevian spline spaces that are good for design is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For all good-for-design spaces, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. To illustrate the benefits of the proposed computational approaches, we provide several examples dealing with different types of piecewise Chebyshevian spline spaces that are good for design.

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