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

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

: (where π is any permutation of its arguments).

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

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

References

| last = Ramshaw | first = Lyle

| date = November 1989

| doi = 10.1016/0167-8396(89)90032-0

| issue = 4

| journal = Computer Aided Geometric Design

| pages = 323–358

| title = Blossoms are polar forms

| url = https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-34.html

| volume = 6}}

{{cite book | author=Casteljau, Paul de Faget de | author-link = Paul de Casteljau | chapter= POLynomials, POLar Forms, and InterPOLation | year = 1992 | editor= Larry L. Schumaker |editor2= Tom Lyche | title = Mathematical methods in computer aided geometric design II | publisher = Academic Press Professional, Inc. | isbn = 978-0-12-460510-7}}
{{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 }}