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''Bernstein form, Bézier polynomial''
 
''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
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The Bernstein polynomial of order $n$ for a function $f$, defined on the closed interval $[0,1]$, 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>
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\begin{equation*} B_n f ( x ) : = B _ { n } ( f , x ) : = \sum _ { j = 0 } ^ { n } f \left( \frac { j } { n } \right) b _ { j } ^ { n } ( x ), \end{equation*}
  
 
with
 
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>
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\begin{equation*} b _ { j } ^ { n } ( x ) : = \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { j } ( 1 - x ) ^ { n - j } , j = 0 , \ldots , n. \end{equation*}
  
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]]).
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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 $[0,1]$ as $n \rightarrow \infty$, to $f$ in case $f$ 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]].
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The Bernstein polynomial $B _ { n } f$ is of degree $\leq n$ and agrees with $f$ in case $f$ is a polynomial of degree $\leq 1$. It depends linearly on $f$ and is positive on $[0,1]$ in case $f$ 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, $B _ { n }$, 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
+
The $( n + 1 )$-sequence $\{ b _ { j } ^ { n } : j = 0 , \dots , n \}$ is evidently linearly independent, hence a basis for the $( n + 1 )$-dimensional linear space $\Pi _ { n }$ of all polynomials of degree $\leq n$ 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>
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\begin{equation*} p = \sum _ { j = 0 } ^ { n } a _ { j } b _ { j } ^ { n } \end{equation*}
  
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.
+
is called the Bernstein–Bézier form, or just the Bernstein form, for $p \in \Pi _ { n }$. 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 $\{ p ( t ) : 0 \leq t \leq 1 \}$ of the interval $[0,1]$ under a vector-valued polynomial $p$. The coefficients $a _ { j }$ in that form readily provide information about the value of $p$ and its derivatives at both endpoints of the interval $[0,1]$, 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.
+
Somewhat confusingly, the term  "Bernstein polynomial"  is at times applied to the polynomial $b _ { j } ^ { n }$, 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====
 
====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>
+
<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>

Latest revision as of 16:55, 1 July 2020

Bernstein form, Bézier polynomial

The Bernstein polynomial of order $n$ for a function $f$, defined on the closed interval $[0,1]$, is given by the formula

\begin{equation*} B_n f ( x ) : = B _ { n } ( f , x ) : = \sum _ { j = 0 } ^ { n } f \left( \frac { j } { n } \right) b _ { j } ^ { n } ( x ), \end{equation*}

with

\begin{equation*} b _ { j } ^ { n } ( x ) : = \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { j } ( 1 - x ) ^ { n - j } , j = 0 , \ldots , n. \end{equation*}

The polynomial was introduced in 1912 (see, e.g., [a3]) by S.N. Bernstein (S.N. Bernshtein) and shown to converge, uniformly on the interval $[0,1]$ as $n \rightarrow \infty$, to $f$ in case $f$ is continuous, thus providing a wonderfully short, probability-theory based, constructive proof of the Weierstrass approximation theorem (cf. Weierstrass theorem).

The Bernstein polynomial $B _ { n } f$ is of degree $\leq n$ and agrees with $f$ in case $f$ is a polynomial of degree $\leq 1$. It depends linearly on $f$ and is positive on $[0,1]$ in case $f$ 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., [a1] and Approximation of functions, linear methods), with the Bernstein operator, $B _ { n }$, the prime example. See also Bernstein polynomials.

The $( n + 1 )$-sequence $\{ b _ { j } ^ { n } : j = 0 , \dots , n \}$ is evidently linearly independent, hence a basis for the $( n + 1 )$-dimensional linear space $\Pi _ { n }$ of all polynomials of degree $\leq n$ which contains it. It is called the Bernstein–Bézier basis, or just the Bernstein basis, and the corresponding representation

\begin{equation*} p = \sum _ { j = 0 } ^ { n } a _ { j } b _ { j } ^ { n } \end{equation*}

is called the Bernstein–Bézier form, or just the Bernstein form, for $p \in \Pi _ { n }$. 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., [a2]) for representing a polynomial curve, that is, the image $\{ p ( t ) : 0 \leq t \leq 1 \}$ of the interval $[0,1]$ under a vector-valued polynomial $p$. The coefficients $a _ { j }$ in that form readily provide information about the value of $p$ and its derivatives at both endpoints of the interval $[0,1]$, 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 $b _ { j } ^ { n }$, 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

[a1] R.A. DeVore, "The approximation of continuous functions by positive linear operators" , Springer (1972)
[a2] G. Farin, "Curves and surfaces for computer aided geometric design" , Acad. Press (1993) (Edition: Third)
[a3] G.G. Lorentz, "Bernstein polynomials" , Univ. Toronto Press (1953)
How to Cite This Entry:
Bernstein-Bézier form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein-B%C3%A9zier_form&oldid=22101
This article was adapted from an original article by C. de Boor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article