Blossom (functional)

Blossom (functional)

In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bezier and spline curves and surfaces.

The blossom of a polynomial "ƒ", often denoted mathcal{B} [f] , is completely characterised by the three properties:
* It is a symmetric function of its arguments::: mathcal{B} [f] (u_1,dots,u_d) = mathcal{B} [f] ig(pi(u_1,dots,u_d)ig),,: (where "π" is any permutation of its arguments).
* It is affine in each of its arguments::: mathcal{B} [f] (salpha + teta,dots) = smathcal{B} [f] (alpha,dots) + tmathcal{B} [f] (eta,dots), ext{ when }s + t = 1.,
* It satisfies the diagonal property::: mathcal{B} [f] (u,dots,u) = f(u).,

References

*cite paper | author=Ramshaw, Lyle | title = Blossoming: A Connect-the-Dots Approach to Splines | publisher=Digital Systems Research Center | date=1987 | url=ftp://ftp.digital.com/pub/compaq/SRC/research-reports/abstracts/src-rr-019.html | accessdate=2006-06-28

*cite paper | author=Casteljau, Paul de Faget de | authorlink = Paul de Casteljau | title = POLynomials, POLar Forms, and InterPOLation | date = 1992 | editor= Schumaker et al. | book = Mathematical methods in computer aided geometric design II | publisher = Academic Press Professional, Inc.

*cite book | author=Farin, Gerald | title = Curves and Surfaces for CAGD: A Practical Guide | year = 2001 | publisher = Morgan Kaufmann | edition = fifth edition | id = ISBN 1-55860-737-4


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