Béziers and B-splines as Multiaffine Maps

Abstract

It is a classical principle in mathematics that polynomials in a single variable of degree n are essentially equivalent to symmetric polynomials in n variables that are linear in each variable separately. We shall apply this principle to the Bézier and B-spline curves and surfaces that are used in computer aided geometric design. The main result is a method of labeling the Bézier points that control a curve segment or surface patch or the de Boor points that control a B-spline curve with symmetric, multivariate labels. The properties of these labels make it simple to understand or to reconstruct the basic algorithms in this area, such as the de Casteljau Algorithm and the de Boor Algorithm.