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Concept
Bernstein polynomial
Summary
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein.
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 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 is a binomial coefficient.
So, for example,
The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:
The Bernstein basis polynomials of degree n form a basis for the vector space of polynomials of degree at most n with real coefficients. A linear combination of Bernstein basis polynomials
is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients are called Bernstein coefficients or Bézier coefficients.
The first few Bernstein basis polynomials from above in monomial form are:
The Bernstein basis polynomials have the following properties:
if or
for
and where is the Kronecker delta function:
has a root with multiplicity at point (note: if , there is no root at 0).
has a root with multiplicity at point (note: if , there is no root at 1).
The derivative can be written as a combination of two polynomials of lower degree:
The k-th derivative at 0:
The k-th derivative at 1:
The transformation of the Bernstein polynomial to monomials is and by the inverse binomial transformation, the reverse transformation is
The indefinite integral is given by
The definite integral is constant for a given n:
If , then has a unique local maximum on the interval at .
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