Computing B-spline control points

A B-spline curve over a given knotvector is completely determined by its control points. Using the equivalence between polynomials and symmetric multiaffine maps, de Casteljau, Ramshaw, and Seidel have recently shown how to compute B-spline control points as values of symmetric multiaffine maps at a sequence of consecutive knots. After a short self-contained proof of this fact, in this paper we will use this result to compute the new B-spline control points for knot insertion, degree elevation and osculatory B-spline interpolation. Examples will demonstrate how multiaffine maps may be used both for practical computations and also as a theoretical tool for the derivation of efficient algorithms.