Blossom (functional) explained

In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces.

The blossom of a polynomial ƒ, often denoted

l{B}[f], is completely characterised by the three properties:

• It is a symmetric function of its arguments:

l{B}[f](u _{1,...,u} _{d)} = l{B}[f] ( \pi(u _{1,...,u} _{d)),}

(where π is any permutation of its arguments).

• It is affine in each of its arguments:

l{B}[f](\alpha u + \beta v, ... ) = \alpha l{B}[f](u,...) + \beta l{B}[f](v,...), when \alpha + \beta = 1 .

• It satisfies the diagonal property:

l{B}[f](u,...,u) = f(u).

References

• Ramshaw . Lyle . November 1989 . 10.1016/0167-8396(89)90032-0 . 4 . Computer Aided Geometric Design . 323–358 . Blossoms are polar forms . 6.
• Book: Casteljau, Paul de Faget de . Paul de Casteljau . POLynomials, POLar Forms, and InterPOLation . 1992 . Larry L. Schumaker . Tom Lyche . Mathematical methods in computer aided geometric design II . Academic Press Professional, Inc. . 978-0-12-460510-7.
• Book: Farin, Gerald . Curves and Surfaces for CAGD: A Practical Guide . 2001 . Morgan Kaufmann . fifth . 1-55860-737-4 .