Abstract: A programmable machine for automatically cutting and/or resurfacing a selected arcuate pattern upon either a convex or concave lap, particularly such laps as may then be used for forming and polishing opthalmic lenses. A lap positioning unit and a cutter positioning unit are located respectively on adjacent upright and horizontal supports so as to bring the lap and cutter into mutual engagement, each unit initially being longitudinally adjustable from a respective pivot point by a selected amount so as to determine the particular arcuate curvature obtained, which curvatures jointly form the two-dimentional arcuate pattern chosen for the lap face. In operation, the lap is oscillated vertically and a rotary milling cutter is driven back and forth transverse thereto.