# Difference between revisions of "Bernstein-Bézier form"

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''Bernstein form, Bézier polynomial'' | ''Bernstein form, Bézier polynomial'' | ||

− | The Bernstein polynomial of order | + | 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 | 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., [[#References|[a3]]]) by S.N. Bernstein (S.N. Bernshtein) and shown to converge, uniformly on the interval | + | 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 | + | 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 | + | 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 | + | 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 | + | 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>< | + | <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=50068