A radiance formula shows that the halo arcs produced by ice crystals with two degrees of rotational freedom, the "tangent arcs," are loci on which a mathematical function, the Jacobian, is zero. This enables complete analytic theories of many halo arcs to be given. In other cases, points of contact with the circular halos, and intersections with the solar vertical and parhelic circle, can be found analytically, giving a qualitative description of the forms of the arcs.
© 1976 Optical Society of America
Richard White, "An analytic theory of certain halo arcs," J. Opt. Soc. Am. 66, 768-772 (1976)