Curves and Surfaces for CAGD, 5th Edition

5

The Bernstein Form of a Bézier Curve

Bézier curves can be defined by a recursive algorithm, which is how de Casteljau first developed them. It is also necessary, however, to have an explicit representation for them; this will facilitate further theoretical development considerably.

5.1 Bernstein Polynomials

We will express Bézier curves in terms of Bernstein polynomials, defined explicitly by

(5.1)

where the binomial coefficients are given by

There is a fair amount of literature on these polynomials. We cite just a few: Bernstein [53], Lorentz ...