φφ-optics and generalized Bianchi–Auerbach equation

We consider the family of regular plane closed strictly convex curves and the family of differential equations of special type associated to positive-valued, periodic and continuous functions. Each curve and the solutions of a fixed differential equation determine a family of disjoint closed curves called φφ-optics. In particular cases we obtain the family of αα-isoptics and the family of arc-optic curves. The family of arc-optic curves is closely related to the solutions of the Ulam problem. We generalize the Bianchi–Auerbach equation derived for the problem of floating body to general case of φφ-optics and we give some of its applications. We present the necessary and sufficient conditions for some geometric quantities connected in a natural way to φφ-optics to be constant.