Fixing a graph $\tau$ on a surface $\Sigma_g$ of genus $g \geq 1$ which lifts to a triangulation on the universal cover, we set up the problem of which surfaces with projective structures admit a circle packing on it with nerve isotopic to $\tau$, and if some surface does, then whether the packing is rigid, and the relation with the uniformization map.
In this first part, we formulate the problem in terms of what we call a cross ratio parameter space $\mathcal{C}_{\tau}$, which turns out to be identified with the space of all pairs of a projective structure on $\Sigma_g$ and a packing with nerve $\tau$ on it, and propose the following conjecture: The composition of the forgetting map $f : \mathcal{C}_{\tau} \to \mathcal{P}_g$ to the space $\mathcal{P}_g$ of all projective structures on $\Sigma_g$ with the uniformization map@ $u : \mathcal{P}_g \to \mathcal{T}_g$ to the Teichm\"uller space $\mathcal{T}_g$ is a homeomorphism.