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:
 * It is a symmetric function of its arguments:
 * $$\mathcal{B}[f](u_1,\dots,u_d) = \mathcal{B}[f]\big(\pi(u_1,\dots,u_d)\big),\,$$
 * (where &pi; is any permutation of its arguments).


 * It is affine in each of its arguments:
 * $$\mathcal{B}[f](\alpha u + \beta v,\dots) = \alpha\mathcal{B}[f](u,\dots) + \beta\mathcal{B}[f](v,\dots),\text{ when }\alpha + \beta = 1.\,$$


 * It satisfies the diagonal property:
 * $$\mathcal{B}[f](u,\dots,u) = f(u).\,$$