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

Choose the knot origin so the span most influenced by $P_i$ is $[i,i+1]$, and use its local parameter $t\in [0,1]$. The three active uniform quadratic B-spline basis functions giveThe central coefficient reaches its maximum at $t=1/2$, so this is indeed the span centered on the strongest influence of $P_i$. Express it in the quadratic Bernstein basis:Matching constant, linear and quadratic coefficients gives the quadratic Bézier curve controlsFor example, $C_i^{\prime }(0)=2(B_1-B_0)=P_i-P_{i-1}$ and $C_i^{\prime }(1)=2(B_2-B_1)=P_{i+1}-P_i$, checking the endpoint derivatives and the $C^1$ joins of neighboring spans.