Blossom (functional)
# 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 ${\mathcal {B}}[f],$ is completely characterised by the three properties:

       ${\mathcal {B}}[f](u_{1},\dots ,u_{d})={\mathcal {B}}[f]{\big\pi (u_{1},\dots ,u_{d}){\big )},\,$
    (where π is any permutation of its arguments).
       ${\mathcal {B}}[f](\alpha u+\beta v,\dots )=\alpha {\mathcal {B}}[f](u,\dots )+\beta {\mathcal {B}}[f](v,\dots ),{\text{ when }}\alpha +\beta =1.\,$
       ${\mathcal {B}}[f](u,\dots ,u)=f(u).\,$
