More pointsets with many rich lines

We present some new sharp constructions for the Szemer\'{e}di-Trotter theorem. These constructions generalize previous work of Erd\H{o}s, Elekes, Sheffer and Silier, Guth and Silier, and the author. In the past, arguments showing the optimality of many of these constructions have required some elementary number theory and have been rather technical, thus limiting the scope of the results. We replace these number-theoretic arguments with purely incidence-geometric ones, allowing for simpler proofs and more general results.