The single point machining head includes a probe that is removably mounted on a carriage that moves radially relative to a central axis of the machining head. A system controller moves the carriage in a radial direction while the machining head is fed in a vertical direction so that the probe contacts the surface of a workpiece. Each point of contact on the workpiece becomes a datapoint including the radial position and the vertical position when contact occurred. The surface of the workpiece is scanned with the probe and a profile is generated using the points of contact. A system controller calculates a difference between the measured profile and a desired profile to be machined to determine the amount of material to remove.