A general setting for halo theory

We describe a general framework for systematically treating halos that are due to refraction in preferentially oriented ice wedges, and we construct an atlas of such halos. Initially we are constrained neither by the interfacial angles nor the orientations of real ice crystals. Instead we consider "all possible" refraction halos. We therefore make no assumption regarding the wedge angle, and only a weak assumption regarding the allowable wedge orientations. The atlas is thus a very general collection of refraction halos that includes known halos as a small fraction. Each halo in the atlas is characterized by three parameters: the wedge angle, the zenith angle of the spin vector, and the spin vector expressed in the wedge frame. Together with the sun elevation, the three parameters for a halo not only permit calculation of the halo shape, they also give much information about the halo without extensive calculation, so that often a crude estimate of the halo's appearance is possible merely from inspection of its parameters. As a result, the theory reveals order in what seems initially to be a staggering variety of halo shapes, and in particular it explains why halos look the way they do. Having constructed and studied the atlas, we then see where real or conceivable refaction halos, arising in specific crystal shapes and crystal orientations, fit into the atlas. Although our main goal is to understand halos arising in pyramidal crystals, the results also clarify and unify the classical halos arising in hexagonal prismatic crystals.