The second fundamental form of a surface and its relation to the dioptric power matrix, sagitta and lens thickness

The dioptric power matrix of a single refracting surface is shown to be directly related to the differential form known in differential geometry as the second fundamental form. Consequently it is a complete description of the local refractive character of the surface. Generalized equations are derived for sagitta and lens thickness for surfaces of any form. The equations hold approximately for points close to the optical axis except in the case of paraboloidal surfaces: for paraboloidal surfaces they hold exactly. Lens thickness approximately obeys what is termed the principle of form independence. The value of the second fundamental form is approximately double the sagitta. Formal support is provided for Keating's concept of a generalized matric vergence of astigmatic wave-fronts.