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

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