A polygon machining method whereby first polygon machining using a polygon cutter is carried out on a workpiece, followed by machining using tools other than the polygon cutter, then second polygon machining using the polygon cutter again. The secondary portions may be under separate commitment control that is asynchronous to the commitment control for the mandatory portions, or may be under no commitment control at all. The support portion has a cylindrical body which is bendable and can maintain a curved shape in a state where the sheath tube is inserted. A gas valve is disposed in the capped end of the cylindrical can for pressurizing the interior portion of the can with air.