A method of machining a workpiece may include continuously rotating the workpiece, continuously rotating a tool having at least one cutting surface, and positioning the tool relative to the workpiece so that the at least one cutting surface engages the workpiece at a first discrete location at a periphery of the workpiece. The method may further include continuing to rotate the workpiece and the tool so that the at least one cutting surface engages a second discrete location at the periphery of the workpiece, and controlling a tool surface velocity VT relative to the workpiece surface velocity VW so that the first and second discrete locations are discontinuous. The tool may make multiple iterative passes over the workpiece to engage subsequent discrete locations, wherein the first discrete location, second discrete location, and multiple subsequent discrete locations may form a machined surface that extends continuously around the workpiece.

A method of machining a workpiece may include continuously rotating the workpiece, continuously rotating a tool having at least one cutting surface, and positioning the tool relative to the workpiece so that the at least one cutting surface engages the workpiece at a first discrete location at a periphery of the workpiece. The method may further include continuing to rotate the workpiece and the tool so that the at least one cutting surface engages a second discrete location at the periphery of the workpiece, and controlling a tool surface velocity VT relative to the workpiece surface velocity VW so that the first and second discrete locations are discontinuous. The tool may make multiple iterative passes over the workpiece to engage subsequent discrete locations, wherein the first discrete location, second discrete location, and multiple subsequent discrete locations may form a machined surface that extends continuously around the workpiece.

Oscillation for a direct distance between the centers of a tool T and a workpiece W to meet l+al(1cos(Mt)) and vertical oscillation to meet alsin(Mt) are applied alone or in combination with time t, where denotes an angular velocity of a workpiece, M denotes the number of sides of a polygon, and a denotes an adjustment parameter. Such oscillation enables adjustment as to how the machining surface is made concave or convex.