A bevel head apparatus includes tilt, rotation, and vertical positioning with a cutting torch of a numerically controlled plate cutting machine. The apparatus has a back panel that attaches to the cutting machine and provides vertical movement of the cutting torch, a drive box coupled with the back panel houses electronics and mechanical components, an arm coupled to the drive box provides rotational movement of the cutting torch, and a pantograph coupled with the arm provides tilt movement to the cutting torch attached to the pantograph. Tilt, rotation, and vertical positioning of the cutting torch is controlled by the electronics and mechanical components. The bevel head apparatus also include a laser for measuring distance to determine and control a height of the cutting torch above the flat plate.