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Bézier form : ウィキペディア英語版
Bernstein polynomial

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 (), became important in the form of Bézier curves.
==Definition==
The ''n'' + 1 Bernstein basis polynomials of degree ''n'' are defined as
: b_(x) = x^ \left( 1 - x \right)^, \quad \nu = 0, \ldots, n.
where 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
:B_n(x) = \sum_^ \beta_ b_(x)
is called a Bernstein polynomial or polynomial in Bernstein form of degree ''n''. The coefficients \beta_\nu are called Bernstein coefficients or Bézier coefficients.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Bernstein polynomial」の詳細全文を読む



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