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:
![{\displaystyle {\mathcal {B}}[f],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/695e0432d24ca9d2b5956c67cc532614db86760d)
- 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 )},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bbd48cc9236fd324ec99d98a876d659a42f9c53)
- 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.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0600097aeca068f9070452322049297b08b513a)
- It satisfies the diagonal property:
=f(u).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b87b8b0650dd79e72885f20a183aa2cd9c263bdf)
References
- Ramshaw, Lyle (November 1989). "Blossoms are polar forms". Computer Aided Geometric Design. 6 (4): 323–358. doi:10.1016/0167-8396(89)90032-0.
- Casteljau, Paul de Faget de (1992). "POLynomials, POLar Forms, and InterPOLation". In Larry L. Schumaker; Tom Lyche (eds.). 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.