Abstract :
Rational Bezier curves are typically defined as projections of polynomial Bezier curves of one higher dimension. Here we introduce an alternative notion of rational Bezier curves defined in terms of rational Bernstein blending functions. These negative degree Bernstein bases share many properties with their polynomial kin: they are linearly independent, form a partition of unity, satisfy Descartesʹ Law of Signs, obey the standard recurrence relation, and have simple two term formulas for differentiation and degree elevation. But whereas the Bernstein bases of positive degree can represent only polynomial functions, Bernstein bases of negative degree can exactly represent arbitrary functions analytic in a neighborhood of zero. Moreover, whereas the Bernstein bases of positive degree can uniformly approximate arbitrary continuous functions on a compact interval, the Bernstein bases of negative degree can uniformly approximate all continuous functions that vanish at minus infinity. Nevertheless, despite these differences, most of the standard algorithms for Bezier curves, such as reparametrization, trimming, and conversion to and from monomial form extend quite naturally to their rational Bezier brethren.
