Birational models for varieties of Poncelet curves

We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a degree$-c$ plane curve having the Poncelet property with respect to $S$. We prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one and use this to describe a birational model of the variety of Poncelet curves for $c$ odd.