Bernstein form

In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials.

A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.

Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone–Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves.

The n + 1 Bernstein basis polynomials of degree n are defined as

where ${\displaystyle {n \choose \nu }}$ is a binomial coefficient.

The Bernstein basis polynomials of degree n form a basis for the vector space Π_n of polynomials of degree at most n.

A linear combination of Bernstein basis polynomials

is called a Bernstein polynomial or polynomial in Bernstein form of degree n.

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