On parameterizations of plane rational curves and their syzygies

Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ 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 $(a,b)=(k,d-k)$ ($2kleq d)$, 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 $PP^{k+1}$.