View source for Bernstein-Bézier form

''Bernstein form, Bézier polynomial''

The Bernstein polynomial of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301101.png" /> for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301102.png" />, defined on the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301103.png" />, is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301104.png" /></td> </tr></table>

with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301105.png" /></td> </tr></table>

The polynomial was introduced in 1912 (see, e.g., [[#References|[a3]]]) by S.N. Bernstein (S.N. Bernshtein) and shown to converge, uniformly on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301106.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301107.png" />, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301108.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b1301109.png" /> is continuous, thus providing a wonderfully short, probability-theory based, constructive proof of the Weierstrass approximation theorem (cf. [[Weierstrass theorem|Weierstrass theorem]]).

The Bernstein polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011010.png" /> is of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011011.png" /> and agrees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011012.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011013.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011014.png" />. It depends linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011015.png" /> and is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011016.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011017.png" /> 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., [[#References|[a1]]] and [[Approximation of functions, linear methods|Approximation of functions, linear methods]]), with the Bernstein operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011018.png" />, the prime example. See also [[Bernstein polynomials|Bernstein polynomials]].

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011019.png" />-sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011020.png" /> is evidently linearly independent, hence a basis for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011021.png" />-dimensional linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011022.png" /> of all polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011023.png" /> which contains it. It is called the Bernstein–Bézier basis, or just the Bernstein basis, and the corresponding representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011024.png" /></td> </tr></table>

is called the Bernstein–Bézier form, or just the Bernstein form, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011025.png" />. 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., [[#References|[a2]]]) for representing a polynomial curve, that is, the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011026.png" /> of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011027.png" /> under a vector-valued polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011028.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011029.png" /> in that form readily provide information about the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011030.png" /> and its derivatives at both endpoints of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011031.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130110/b13011032.png" />, 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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.A. DeVore,   "The approximation of continuous functions by positive linear operators" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Farin,   "Curves and surfaces for computer aided geometric design" , Acad. Press  (1993)  (Edition: Third)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.G. Lorentz,   "Bernstein polynomials" , Univ. Toronto Press  (1953)</TD></TR></table>