B-form basics

0. Introduction; notation This paper lists the essential facts about the representation of polynomials in m variables as Bernstein polynomials. An expanded version may appear elsewhere. While univariate Bernstein polynomials are well studied see, e.g., Lorentz’ classical book Lorentz (1953), the multivariate version has only attracted attention sporadically. Lorentz’ book devotes just one page to the two most direct generalizations: the tensor product or coordinate degree generalization, and the total degree generalization which is the topic of the present paper. Motivation for the paper comes from computer-aided geometric design where, through the initiative of de Casteljau and Bézier, the Bernstein polynomials of mostly one variable have become the main tool for the representation and computational use of pp (:= piecewise polynomial) functions. Farin’s work Farin (1979), Farin (1980) brought popularity and understanding to the use of bivariate Bernstein polynomials, and my own understanding starts from that work. My own interest has been started and repeatedly reinforced by work with smooth pp functions in two or more variables (de Boor and Höllig (1983), (1986)), in which their representation in terms of Bernstein polynomials, i.e., their Bnet, for short, plays an essential role, since it reflects so nicely, and far better than other standard representations, the interplay between the geometry of the underlying triangular partition and the smoothness requirements. For the sake of brevity, and since there are several people and ideas responsible, I am proposing here the term B-form (and correspondingly, B-net) for what would, more properly, be called the barycentric-Bernstein-de Casteljau-Bézier-Farin· · · -form. I apologize to de Casteljau and Farin and · · · for the slight they might feel. While the bivariate and trivariate situation is of most practical interest, I have chosen here to record the facts in the general m-dimensional context. This forces careful consideration of notation and brings out the essential mathematical aspects and surprising beauty of the B-form. I will adhere to the following notational conventions: I won’t bother with boldface, arrows, or underlines to distinguish points in IR from other objects. The j-th component of a point x ∈ IR I will denote by x(j) (rather than xj). I will use standard multi-index notation throughout. Thus