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 $ℬ[f],$ is completely characterised by the three properties:

• It is a symmetric function of its arguments:

$ℬ[f](u_1,\dots ,u_d)=ℬ[f](\pi (u_1,\dots ,u_d)),$

(where π is any permutation of its arguments).

• It is affine in each of its arguments:

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

• It satisfies the diagonal property:

$ℬ[f](u,\dots ,u)=f(u).$

References

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