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](u1,…,ud)=ℬ[f](π(u1,…,ud)),
(where π is any permutation of its arguments).

* It is affine in each of its arguments:

ℬ[f](αu+βv,…)=αℬ[f](u,…)+βℬ[f](v,…), when α+β=1.

* It satisfies the diagonal property:

ℬ[f](u,…,u)=f(u).

## References

* Ramshaw, Lyle (1987). Blossoming: A Connect-the-Dots Approach to Splines. Digital Systems Research Center. https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-19.html. Retrieved 2019-04-19.
* Ramshaw, Lyle (1989). Blossoms are polar forms. Digital Systems Research Center. https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-34.html. Retrieved 2019-04-19.
* Casteljau, Paul de Faget de (1992). "POLynomials, POLar Forms, and InterPOLation". in Larry L. Schumaker. 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.

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