  The Ince-Gauss beams, separable in elliptic coordinates, are studied through
a ray-optical approach. Their ray structure can be represented over a
ray-Poincar\'e sphere by generalized Viviani curves (intersections of a
cylinder and a sphere). This representation shows two topologically different
regimes, in which the curve is composed of one or two loops. The overall beam
shape is described by the ray caustics that delimit the beams' bright regions.
These caustics are inferred from the generalized Viviani curve through a
geometric procedure that reveals connections with other physical systems and
geometrical constructions. Depending on the regime, the caustics are composed
either of two confocal ellipses or of segments of an ellipse and a hyperbola
that are confocal. The weighting of the rays is shown to follow the two-mode
meanfield Gross-Pitaevskii equations, which can be mapped to the equation of a
simple pendulum. Finally, it is shown that the wave field can be accurately
estimated from the ray description.
