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			In [[numerical analysis]], a '''blossom''' is a [[functional (mathematics)\|functional]] that can be applied to any [[polynomial]], but is mostly used for [[Bézier curve\|Bézier]] and [[Spline (mathematics)\|spline]] curves and surfaces.

			The blossom of a polynomial ''ƒ'', often denoted <math>\mathcal{B}[f],</math> is completely characterised by the three properties:
			* It is a symmetric function of its arguments:
			:: <math>\mathcal{B}[f](u_1,\dots,u_d) = \mathcal{B}[f]\big(\pi(u_1,\dots,u_d)\big),\,</math>
			: (where ''π'' is any [[permutation]] of its arguments).
			* It is affine in each of its arguments:
			:: <math>\mathcal{B}[f](\alpha u + \beta v,\dots) = \alpha\mathcal{B}[f](u,\dots) + \beta\mathcal{B}[f](v,\dots),\text{ when }\alpha + \beta = 1.\,</math>
			* It satisfies the diagonal property:
			:: <math>\mathcal{B}[f](u,\dots,u) = f(u).\,</math>

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			*{{cite paper \| author=Ramshaw, Lyle \| title = Blossoming: A Connect-the-Dots Approach to Splines \| publisher=Digital Systems Research Center \| year=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 \| year = 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 \| isbn = 1-55860-737-4 }}

			[[Category:Numerical analysis]]

130.183.92.142 at 10:42, 8 November 2011

2011-11-08T10:42:46Z

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130.183.92.142 at 10:42, 8 November 2011