Rates of convergence of control polygons

It is well known (cf. [Farin '77] for a proof) that the sequence of control polygons associated with the Bernstein-Bezier representation of a curve using polynomials of degree n converges to the curve as n goes to infinity. Similarly, it was shown in [Lane & Riesenfeld '80] that if a uniform floating B-spline curve is uniformly refined, then the resulting sequence of control polygons also converges to the curve. In this paper we present a simple general method for treating such convergence questions which actually provides precise rates of convergence. We illustrate the method by applying it to B-spline curves which are refined by increasing the degrees and/or refining the knot sequences.