Two separate geometric design procedures are presented for calculating aspheric surfaces. The first calculates individual reflecting surfaces for correcting the optical path length (OPL) of a system. The second jointly calculates pairs of surfaces for the simultaneous correction of OPL and offense against the sine condition (OCS). The procedures remain valid for extreme focal ratios. Applications of these procedures are made to the design of Arecibo-style (stationary spherical primary) telescopes, coma-correctors for such telescopes, three-mirror aplanats with deep spherical primaries, and two-mirror aplanats. Unusual new forms of aplanatic telescope and microscope objectives have emerged from these applications.