Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 70 / 2 / i / Solution

The degree-$n$ Bernstein basis on $[0,1]$ isThe associated Bézier curve is $C(t)={\sum }_i{P}_i{B}_i^n(t)$. The basis is nonnegative and has partition of unity, since ${\sum }_i{B}_i^n(t)=(t+1-t{)}^n=1$. Thus the curve is a convex combination of its control points, giving the convex hull enclosure useful for geometric bounds. The same identity makes the construction invariant under application of affine functions.

It interpolates its endpoint controls, $C(0)=P_0$, $C(1)=P_n$, and its endpoint tangent vectors are $n(P_1-P_0)$ and $n(P_n-P_{n-1})$. These follow fromThe control polygon therefore gives direct geometric editing and tangent control. Repeated interpolation by the De Casteljau algorithm evaluates and subdivides the curve without converting to a monomial representation. Together, positive geometric weights, simple endpoint behavior and stable subdivision make this basis useful in computer aided geometric design.