4. Properties of Bézier curves of any degree

Generalizing the idea of a cubic Bézier curve, a Bézier curve of degree at most $n$ based on $n+1$ control points $P _ 0,\dots, P _ n$ is defined by

$P(t) = J _ {n,0}(t)P _ 0 + \dots + J _ {n,n}(t) P _ n$,

where the Bernstein polynomial $J _ {n,k}(t)$ is defined by $J _ {n,k}(t) = \binom nk (1-t)^{n-k} t^k$.

The properties of cubic Bézier curves, listed above, generalize as you might expect they would. In addition, here is a nice derivative property that explains some of the derivative properties already mentioned for cubic Bézier curves:

• Derivative at any point: If $P(t)$ is the Bézier curve with control points $P _ 0,\dots, P _ n$, then $P'(t) = n Q(t)$, where $Q(t)$ is the Bézier curve of degree at most $n-1$ based on the control points $\Delta P _ 0 = P _ 1 - P _ 0$, $\Delta P _ 1 = P _ 2 - P _ 1$,..., $\Delta P _ {n-1} = P _ n - P _ {n-1}$.

  Thus we get a factor of $n$, reminiscent of the derivative of $x^n$, and another Bézier curve based on differences of control points of $P(t)$.