Bernstein-Bézier form

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)