Using a new formalism involving projection from the sphere of directions to the stereographic plane, and associated complex variables, explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic (absorbing) and chiral (optically active). This enables three types of polarization singularity to be classified and explored: singular axes, which are degeneracies where the two refractive indices are equal, and which for a transparent non–chiral crystal condense pairwise onto the optic axes; C points, where the polarization is purely circular (right– or left–handed), with topological index +1, +½ or +¼ and whose positions are independent of the chirality; and L lines, where the polarization is purely linear, dividing direction space into regions with right– and left–handedness. A local model captures essential features of the general theory. Interference figures generated by slabs of crystal viewed directly or through a polarizer and/or analyser enable the singularities to be displayed directly.
† Dedicated to Professor J. F. Nye, FRS, and Professor S. Ramaseshan on their 80th birthdays.