  The Poincar\'e sphere is a graphical representation in a three-dimensional
space for the polarization of light. Similarly, an optical element with
spatially varying birefringence can be represented by a surface on a
four-dimensional "Poincar\'e hypersphere". A projection of this surface onto
the traditional Poincar\'e sphere provides an intuitive geometric description
of the polarization transformation performed by the element, as well as the
induced geometric phase. We apply this formalism to quantify the effects of
birefringence on the image quality of an optical system.
