11.5 General offsets

In 3-axis NC machining, not only ball-end cutters but also cylindrical and toroidal (fillet-end) cutters are used as shown in Fig. 11.31. While the center of a ball-end cutter moves along an offset surface, the reference point on cylindrical and toroidal cutters moves along the so-called general offset. General offset surfaces were first introduced by Brechner [44] and have been extended further, from the differential geometric as well as algebraic points of view, by Pottmann [325]. If we denote by the parametric representation of the cutter in the initial position, where the reference point on the axis of the cutter is chosen to be at the origin of the Cartesian coordinate system, then represents a so-called reflected cutter. Then the general offset is given by

and , , , are chosen such that there are parallel tangent plane at and [325]. As a consequence, the tangent planes at corresponding points and of the progenitor surface and its general offset are parallel. Thus, the general offset is the sum of the progenitor surface and the reflected cutter. If both surfaces are convex, the general offset is the Minkowski sum of the progenitor surface and the reflected cutter. The general offset surface for a cylindrical cutter is given by [325]

where is the radius of the cutter, is a unit vector along the tool axis and is a unit vector parallel to the bottom silhouette line of the cutter. The general offset surface for a toroidal cutter it is given by [439,355]

where is the radius of the toroidal cutter, is the corner radius of the cutter. The first two terms construct the classical offset with offset distance at the cutter contact point and the third term is a vector parallel to the bottom silhouette line of the cutter with magnitude , which is the radius of the spine circle of the torus.

Pottmann et al. [331] and Glaeser et al. [127] investigate collision-free 3-axis milling of free-form surfaces based on general offsets. They show that if some conditions on the curvature of the surface are fulfilled locally, and in certain cases also globally, there will be no unwanted collision of the cutting tool with the surface.

Figure 11.31: General offsets (adapted from [250]): (a) cylindrical cutter, (b) toroidal cutter