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

• It is a symmetric function of its arguments:

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

(where π is any permutation of its arguments).

• It is affine in each of its arguments:

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

• It satisfies the diagonal property:

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

• 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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