Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form

Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in CP2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.