On Parameterizations of plane rational curves and their syzygies

Let $C$ be a plane rational curve of degree $d$ and $p:\tilde C \rightarrow C$itsnormalization\mathrm{.}Weareinterestedinthesplittingtype$(a,b)$of$C$, where ${\mathcal{O}}_{{\mathbb{P}}^1}(-a-d)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\le d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in ${\mathbb{P}}^{k+1}$.