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 is completely characterised by the three properties:
![\mathcal{B}[f],](../I/m/695e0432d24ca9d2b5956c67cc532614db86760d.svg)
- It is a symmetric function of its arguments:
- (where π is any permutation of its arguments).
 = \mathcal{B}[f]\big(\pi(u_1,\dots,u_d)\big),\,](../I/m/7bbd48cc9236fd324ec99d98a876d659a42f9c53.svg)
- It is affine in each of its arguments:
 = \alpha\mathcal{B}[f](u,\dots) + \beta\mathcal{B}[f](v,\dots),\text{ when }\alpha + \beta = 1.\,](../I/m/d0600097aeca068f9070452322049297b08b513a.svg)
- It satisfies the diagonal property:
 = f(u).\,](../I/m/b87b8b0650dd79e72885f20a183aa2cd9c263bdf.svg)
References
- Ramshaw, Lyle (1987). "Blossoming: A Connect-the-Dots Approach to Splines". Digital Systems Research Center. Retrieved 2006-06-28.
- Casteljau, Paul de Faget de (1992). "POLynomials, POLar Forms, and InterPOLation". In Larry L. Schumaker; Tom Lyche. 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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