Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces

Exact expressions are derived for Hamilton’s angle characteristic for a single reflecting or refracting surface of toric or conic form. The configuration of the base planes and the initial and final coordinates are all ultimately defined with respect to a base ray that is neither assumed to be normally incident nor taken to be incident at an axial or equatorial point. The expressions that emerge form a basis for several aspects of the modeling and design of systems without symmetries. Significant simplifications are realized by appropriately subdividing the derivation, and one of the principal components is a study of the change in the form of the restricted angle characteristic when the coordinates and base planes are reconfigured.