Generalized Bernstein Polynomials and Bézier Curves: an Application of Umbral Calculus to Computer Aided Geometric Design

Bernstein polynomials and Bézier curves are of fundamental importance for Computer Aided Geometric Design (CAGD). They are used for the design of curves and they are the starting point for several generalizations: in particular to higher dimensions and to B-splines. Powerful algorithms are available for both their algebraic construction and their visualization, and their basic theory (explained beautifully in Farin’s book [5]) has been examined repeatedly from different new angles: see, e.g., [6,7,8] for the “barycentric” point of view, [15,17,18] for “blossoming”, and [3] for the “natural generalization of Bézier curves”. In the present paper we introduce an approach to the generalization of Bernstein polynomials and Bézier curves which seems to be entirely new. It is based on the Umbral Calculus which was first described in its classical form by John Blissard in the 1850’s. After a short phase of early success the Umbral Calculus was largely rejected by the mathematics community due to “lack of rigor”; professional jealousy and disdain for its inventor – Blissard was a Land vicar – played an important role, too (cf. [2]). But in the late 1960’s the Umbral Calculus was revived, rehabilitated, and put on firm foundations by Gian-Carlo Rota and his co-workers; the basic work is [16], the book [11] gives an extensive and lucid presentation, and a shorter introduction can be found in [12]. The Umbral Calculus allows a unified and algebraically simple treatment of classical polynomials and classical (combinatorial) numbers with respect to generating functions, recursion formulas, characterising differential equations, creation operators, addition theorems, formula for derivatives, reciprocity formulas, expansion theorems, etc. on the basis of viewing formal power series as the “umbra” (latin for: shadow) of linear functionals on polynomials. The Umbral Calculus is thus a mix of linear algebra, the theory of formal power series and classical analysis. An overview with basic definitions and formulas for both Bernstein polynomials and Umbral Calculus will be given in Section 2 and 3.