  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.
