Using the extended Jones calculus for oblique incidence, we have succeeded in representing light propagation through birefringent media on the Poincaré sphere for arbitrary propagation directions. The situation is far more complicated than for the case of perpendicular transmission. To achieve a simple representation method we had to define appropriate conventions. One can use the method to evaluate the efficiency of existing viewing-angle compensation foils. The Poincaré sphere turns out to be a valid geometrical method for the design of new compensation foils that optimize the contrast ratio. The different parameters of a homogeneous compensation foil that is applied to a liquid-crystal display to improve the viewing-angle properties can be systematically determined.