Structure of cell decompositions in Extremal Szemer\'edi-Trotter examples

The symmetric case of the Szemer\'edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\'edi Trotter is densely related to a successful instance of the recipe. We obtain this result by getting structural information on cell decompositions for extremal Szemer\'edi-Trotter examples. We obtain analogous results for unit circles.