Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 66 / 6 / b / Solution

Use degree elevation of Bernstein coefficients. If $b_j$ are degree-$p$ Bézier curve controls, the identical curve has degree-$(p+1)$ controlsThis follows by multiplying each degree-$p$ Bernstein basis polynomial by $(1-t)+t$ and collecting the degree-$(p+1)$ terms, so the operation changes the representation, not the curve.

For the quadratic controls in part (a), the cubic controls areRaise these cubic Bézier curve controls once more. ThenThus the requested quartic Bézier curve controls, expressed in the original polygon, areEach expression is an affine combination whose coefficients sum to one. Endpoint positions and derivatives are preserved at each degree-raising step, as is every point of the span.