Simple mathematical representation of toroidal surfaces

Toroidal surfaces are surfaces of revolution generated by revolving a plane curve about an axis in the plane of the curve. To avoid the generation of self-intersecting surfaces, the extent of the curve is restricted, so that the curve does not intersect the axis. Unlike aspherical surfaces, there is no standard mathematical representation for toroidal surfaces. In the present work we propose equations for the sagitta, the unit normal vector and the principal curvatures of a toroidal surface which are appropriate for ray tracing purposes, and also for the propagation of thin pencils of rays. In addition, we derive the equations corresponding to a revolving curve with the shape of a conic section, from which one can readily deduce the equations corresponding to a toric surface. All these equations are of considerable importance in ophthalmic lens design, where the refraction of thin pencils of rays is required, and it is commonly performed with the aid of Coddington equations.