On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus (III): Blossoming

Abstract

The investigation of a¯-Bernstein polynomials and a¯-Bézier curves is continued in this paper. It is shown that convolution of the parameters a¯=(a¯1,…,a¯n) is fundamental for (1) the definition of a¯-Bernstein polynomials, (2) a simplified derivation of the a¯-de Casteljau algorithm, (3) the recurrences that give the blossoming of a¯-Bernstein polynomials and a¯-Bézier curves, (4) the dual functional property and the a¯-dual functional property for an a¯-Bézier curve – it is necessary to make this distinction – and (5) the a¯-degree elevation.