Abstract: A machine for generating surfaces of various characteristics on workpieces. It is similar in overall construction to the usual lathe which has a headstock for rotating the workpiece about its axis and a tool carriage which is mounted on a pivot for swinging movement relative to the headstock so that the tool, when adjusted into engagement with the face of the workpiece, will produce a spherical surface thereon. However, the present invention provides a special eccentric control unit in cooperation with the tool carriage for advancing or retracting it relative to the face of the workpiece, as the tool carriage swings, so that aspherical surfaces will be produced thereon by the tool. This eccentric unit is such that it can be readily set to obtain, accurately, surfaces of various preselected characteristics.