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

摘要

The investigation of ā-Bernstein polynomials and ā-Bezier curves is continued in this paper. It is shown that convolution of the parameters ā = (ā_1,...,ā_n) is fundamental for (1) the definition of ā-Bernstein polynomials, (2) a simplified derivation of the ā-de Casteljau algorithm, (3) the recurrences that give the blossoming of ā-Bernstein polynomials and ā-Bezier curves, (4) the dual functional property and the ā-dual functional property for an ā-Bezier curve - it is necessary to make this distinction - and (5) the ā-degree elevation.