Understanding recurrence relations for Chebyshevian B-splines via blossoms

The purpose of this article is to show how naturally recurrence relations for most general Chebyshevian B-splines emerge from blossoms. In particular, this work gives a new insight into previous results by Lyche [A recurrence relation for Chebyshevian B-splines, Constr. Approx. 1 (1985) 155–178], the importance of which it underlines.     

Towards existence of piecewise Chebyshevian B-spline bases

The paper addresses the problem of how to ensure existence of blossoms in the context of piecewise spaces built from joining different extended Chebyshev spaces by means of connection matrices. The interest of this issue lies in the fact that existence of blossoms is equivalent to existence of B-spline bases in all associated spline spaces. As is now classical, blossoms are defined in a geometric way by means of intersections of osculating flats. In such a piecewise context, intersecting a number of osculating flats is a tough proposition. In the present paper, we show that blossoms exist if an only if Bézier points exist, which significantly simplifies the problem. Existence of blossoms also proves to be equivalent to existence of Bernstein bases. In order to establish the latter results, we start by extending to the piecewise context some results which are classical for extended Chebyshev spaces. AMS subject classification 65D17, 65D07     

Blossoming stories

We review the main properties of blossoms along with their important repercussions in all aspects of geometric design. Not only are they an elegant and efficient tool to express all classical algorithms, but they are also a fundamental concept, as proven by the fact that their existence is equivalent to the existence of B-spline bases. AMS subject classification 65D17