Bézier derivative control polygon
= Bézier derivative control polygon
{c}
{title2=$C^{(r)}=\frac{n!}{(n-r)!}\sum_{j=0}^{n-r}\Delta^rP_j B_j^{n-r}$}
For a degree-$n$ <Bézier curve>, the $r$th <derivative> has degree at most $n-r$ and <control points> $n!\Delta^rP_j/(n-r)!$. The first <derivative> follows by differentiating the <Bernstein basis> and shifting one summation index; iteration proves the general formula. A parameter interval of length $h$ multiplies these <derivative> controls by $h^{-r}$.