Capillary floating and the billiard ball problem

We establish a connection between capillary floating in neutral equilibrium and the billiard ball problem. This allows us to reduce the question of floating in neutral equilibrium at any orientation with a prescribed contact angle for infinite homogeneous cylinders to a question about billiard caustics for their orthogonal cross-sections. We solve the billiard problem. As an application, we characterize the possible contact angles and exhibit an infinite family of real analytic non-round cylinders that float in neutral equilibrium at any orientation with constant contact angles.