It is frequently necessary to machine a part-spherical surface on a workpiece, in particular on the end of a rod. This can be done most accurately by rotating the rod about its longitudinal axis, which crosses the center of curvature of the surface to be machined, and pressing it axially against a concave grinding or machining tool rotated about a tool axis that forms an obtuse angle of less than 180.degree. to the workpiece axis.
Typically as described in "Maschinen fur die Bearbeitung yon Endoprothesen" by G. Scherber (Industriediamantrundschau 17, 1983) the tool is tubular and has a diameter substantially smaller than that of the rod whose end surface is being ground. Thus the workpiece is engaged over an annular region that extends from its outer periphery to the tool axis so that as the workpiece and tool are rotated every portion of the tool surface is contacted by the workpiece surface. The result is therefore a perfectly part-spherical surface centered on the point where the tool and workpiece axes intersect. Under ideal operating conditions the tool surface will wear uniformly so that the tubular rod forming the tool gets shorter but otherwise remains usable.
Such a procedure is extremely effective when dealing with a large-diameter workpiece more than a few centimeters in diameter, for instance a ball of a hip prosthesis, because the tool can be made fairly rigid and stiff. Nonetheless when trying to hone or burnish an end of a small-diameter rod, this method does not work because the tool, which must be of similarly small diameter, inherently deforms. Thus putting an accurately finished part-spherical surface on the end of a shaft having a diameter in the millimeter range is a very time-consuming and slow process, in part because one must work with very low pressure to avoid deforming the tool.