A practical method for computing with piecewise Chebyshevian splines

TL;DR

This paper introduces a practical method for efficiently computing with piecewise Chebyshevian splines, enabling better design and refinement of spline-based curves with higher accuracy and ease of implementation.

Contribution

It develops transition functions for B-spline basis computation, generalizes knot insertion and degree raising, and enhances numerical evaluation accuracy for Chebyshevian splines.

Findings

Transition functions enable efficient basis computation.

The method generalizes classical spline algorithms.

Numerical evaluation is more accurate and easier to implement.

Abstract

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces 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 a good-for-design space, 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. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms.