A practical method for computing with piecewise Chebyshevian splines

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 p...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 406; p. 114051
Main Authors: Beccari, Carolina Vittoria, Casciola, Giulio, Romani, Lucia
Format: Journal Article
Language:English
Published: Elsevier B.V 01.05.2022
Subjects:
(Piecewise) Chebyshevian splines
B-spline basis
Computational algorithms
Knot insertion
Order elevation
Transition functions
Transition functions
B-spline basis
Order elevation
Knot insertion
Computational algorithms
(Piecewise) Chebyshevian splines
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary: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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2021.114051