Abstract: A method and apparatus for forming tapered surfaces on male and female parts to be interfitted one within the other along the complementary tapered surfaces thereof wherein the male part is mounted on the chuck of a lathe wherein the chuck axis extends along the centerline of the lathe, the tool on the lathe is mounted for movement at an angle to the longitudinal centerline of the lathe, and the external surface of the male part is cut while rotating the part in one direction to provide a tapered surface by moving the cutting tool along a line extending at the angle. The male part is removed from the chuck and the female part is mounted on the chuck. The cutting tool is moved transversely of the lathe without changing the angle or vertical position of the cutting edge, and the internal surface of the female part is cut by moving the same cutting tool along a path parallel to the first-mentioned path of the tool while rotating the female part in the opposite direction.

Abstract: A method and apparatus for producing an aspherical surface, for example, an aspherical optical surface, on a workpiece, on which the workpiece is rotated about a first axis, a cutting tool is rotated about a second axis, and at a radial distance from the second axis, and the second and first axes meet at a point and at an angle with respect to one another, the second axis being moved with respect to the first axis so that the point moves along the first axis and the angle changes.

Abstract: A lathe for machining interior surfaces of revolution in hollow parts supported on a rotational axis by a head-stock and including a pair of transversely and longitudinally movable carriages arranged symmetrically on the work piece rotatable axis, the carriages each supporting at least one tool holder bar projecting along the work piece rotational axis and having a cooperating cross-sectional configuration by which the tool holder bars are supported along their length by engagement with each other. Alternatively the carriages may support turrets each carrying a plurality of cooperable tool holding bars in a manner such that the operative pair of tool holding bars derives similar support one from the other. In both embodiments, the cutting tools supported by the bars are disposed on a diametric plane inclined with respect to the holder bar in engagement plane to resolve all cutting forces into exclusively torsion on the tool holder bars.