Bernstein-Bézier form

Bernstein form, Bézier polynomial

The Bernstein polynomial of order for a function , defined on the closed interval , is given by the formula

with

The polynomial was introduced in 1912 (see, e.g., [a3]) by S.N. Bernstein (S.N. Bernshtein) and shown to converge, uniformly on the interval as , to in case is continuous, thus providing a wonderfully short, probability-theory based, constructive proof of the Weierstrass approximation theorem (cf. Weierstrass theorem).

The Bernstein polynomial is of degree and agrees with in case is a polynomial of degree . It depends linearly on and is positive on in case is positive there, and so has served as the starting point of the theory concerned with the approximation of continuous functions by positive linear operators (see, e.g., [a1] and Approximation of functions, linear methods), with the Bernstein operator, , the prime example. See also Bernstein polynomials.

The -sequence is evidently linearly independent, hence a basis for the -dimensional linear space of all polynomials of degree which contains it. It is called the Bernstein–Bézier basis, or just the Bernstein basis, and the corresponding representation

is called the Bernstein–Bézier form, or just the Bernstein form, for . Thanks to the fundamental work of P. Bézier and P. de Casteljau, this form has become the standard way in computer-aided geometric design (see, e.g., [a2]) for representing a polynomial curve, that is, the image of the interval under a vector-valued polynomial . The coefficients in that form readily provide information about the value of and its derivatives at both endpoints of the interval , hence facilitate the concatenation of polynomial curve pieces into a more or less smooth curve.

Somewhat confusingly, the term "Bernstein polynomial" is at times applied to the polynomial , the term "Bézier polynomial" is often used to refer to the Bernstein–Bézier form of a polynomial, and, in the same vein, the term "Bézier curve" is often used for a curve that is representable by a polynomial, as well as for the Bernstein–Bézier form of such a representation.

References

[a1]	R.A. DeVore, "The approximation of continuous functions by positive linear operators" , Springer (1972)
[a2]	G. Farin, "Curves and surfaces for computer aided geometric design" , Acad. Press (1993) (Edition: Third)
[a3]	G.G. Lorentz, "Bernstein polynomials" , Univ. Toronto Press (1953)