Blossom (functional)

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 ${\displaystyle {\mathcal {B}}[f],}$ is completely characterised by the three properties:

• It is a symmetric function of its arguments:

${\displaystyle {\mathcal {B}}[f](u_{1},\dots ,u_{d})={\mathcal {B}}[f]{\big (}\pi (u_{1},\dots ,u_{d}){\big )},\,}$

(where π is any permutation of its arguments).

• It is affine in each of its arguments:

${\displaystyle {\mathcal {B}}[f](\alpha u+\beta v,\dots )=\alpha {\mathcal {B}}[f](u,\dots )+\beta {\mathcal {B}}[f](v,\dots ),{\text{ when }}\alpha +\beta =1.\,}$

• It satisfies the diagonal property:

${\displaystyle {\mathcal {B}}[f](u,\dots ,u)=f(u).\,}$

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