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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $n \\ge 0$ be an integer.\nFor $k = 0, \\dots, n$ let\n$$b_{k,n}^{[a,b]}(x) = \\binom{n}{k} \\left(\\frac{x - a}{b - a}\\right)^{k} \\left(\\frac{b - x}{b - a}\\right)^{n-k}.$$\nThese $n + 1$ polynomials form a [[basis|0KCQ9Y3QKA]] of the space of real [[polynomials|1G0B73WPX8]] of degree at most $n$.\nLet $p$ be a polynomial of degree at most $n$.\nIts *Bernstein–Bézier form* on $[a, b]$ is its expansion\n$$p(x) = \\sum_{k=0}^{n} c_k\\, b_{k,n}^{[a,b]}(x).$$\nThe uniquely determined numbers $c_0, \\dots, c_n$ are its *Bézier coefficients* of degree $n$ on $[a, b]$.\n\n## Remarks\n\nOn $[0, 1]$ the basis consists of the [[Bernstein basis polynomials|0C4VWBFY73]] $\\binom{n}{k} x^k (1-x)^{n-k}$.\nSince the basis functions are nonnegative on $[a, b]$ and sum to $1$, the values of $p$ on $[a, b]$ lie between $\\min_k c_k$ and $\\max_k c_k$, and $p(a) = c_0$, $p(b) = c_n$.\nExample: on $[0, 1]$ with $n = 2$, $x = \\tfrac{1}{2} \\cdot 2x(1-x) + 1 \\cdot x^2$, so $x$ has Bézier coefficients $0, 1/2, 1$.\nNon-example: $p(x) = x^3$ has no Bernstein–Bézier form of degree $2$, since it is not a polynomial of degree at most $2$; one must take $n \\ge 3$.\nIn several variables one uses barycentric coordinates $\\lambda_0, \\dots, \\lambda_d$ of a simplex and the basis $\\frac{n!}{\\alpha_0! \\cdots \\alpha_d!} \\lambda_0^{\\alpha_0} \\cdots \\lambda_d^{\\alpha_d}$ with $\\abs{\\alpha} = n$.\n",
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