Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 77 / 3 / iii

Use unit knot spacing and the centered Cardinal cubic B-spline convention $C(t)={\sum }_i{P}_i{M}_3(t-i)$, where $M_3$ is supported on $[-2,2]$. On the span $[i,i+1]$, with $s=t-i$, the active control points are $P_{i-1},P_i,P_{i+1},P_{i+2}$, and their B-spline weights areFor $t=11/4$, take $i=2$, $s=3/4$. The weights and their derivatives areThe PDF has $P_3=(0,0,0)$ and $P_4=(0,1,0)$, so the point and tangent vector areThus, under this centered convention,The original PDF specifies neither the knot origin nor its indexing relative to $P_i$. If instead the convention associates span $[i,i+1]$ with $P_i,P_{i+1},P_{i+2},P_{i+3}$, the identical local weights multiply $P_2,P_3,P_4,P_5$ at the requested parameter, givingThese are parameter translations of the same uniform construction, rather than contradictory evaluations with the same knots. A specified knot convention is necessary to select the numerical pair uniquely; changing the knot spacing would also rescale the derivative.