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

• It is a symmetric function of its arguments:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathcal{B}[f](u,\dots,u) = f(u).\, }[/math]

References

• Ramshaw, Lyle (1987). Blossoming: A Connect-the-Dots Approach to Splines. Digital Systems Research Center. https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-19.html. Retrieved 2019-04-19. 
• Ramshaw, Lyle (1989). Blossoms are polar forms. Digital Systems Research Center. https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-34.html. Retrieved 2019-04-19. 
• Casteljau, Paul de Faget de (1992). "POLynomials, POLar Forms, and InterPOLation". in Larry L. Schumaker. 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.