Bernsteinpolynomials

Bernstein polynomials are a set of functions used in approximation theory and computer-aided design. They form a basis for the vector space of polynomials. For a given non-negative integer n, the Bernstein basis polynomials of degree n are defined as B_{i,n}(x) = \binom{n}{i} x^i (1-x)^{n-i} for i = 0, 1, ..., n. Here, \binom{n}{i} represents the binomial coefficient.

Any polynomial of degree up to n can be expressed as a unique linear combination of these basis polynomials. This representation is known as the Bernstein form of the polynomial. Specifically, a polynomial p(x) of degree at most n can be written as p(x) = \sum_{i=0}^n c_i B_{i,n}(x), where c_i are the coefficients.

A key application of Bernstein polynomials is in the definition of Bézier curves. A Bézier curve of degree n is defined by a set of n+1 control points, and its points are parameterized using Bernstein polynomials. As the parameter t varies from 0 to 1, the curve traces a path influenced by the control points.

Bernstein polynomials possess several useful properties. They are non-negative on the interval [0, 1]. Their sum over i from 0 to n is always equal to 1, meaning \sum_{i=0}^n B_{i,n}(x) = 1 for all x. This property ensures that the Bernstein form of a polynomial is a weighted average of its coefficients, which contributes to their use in interpolation and approximation tasks.