Computer numerical control (CNC) machines play an important role in computer-aided manufacturing; they allow real-time tracking of geometrical objects for processing and production purposes. The requirement that this tracking should be done in real time puts strict limitations on the kinds of geometrical objects that can currently be treated by CNC machines. Developing new software controllers (interpolators) for larger classes of geometrical objects is of vital importance in CNC machining.

While linear and circular path segments, as well as Pythagorean hodographs, can be handled by current interpolators, general conic sections are still a problem. In this paper, the authors develop such interpolators, and analyze their complexity and convergence behavior. This is achieved by use of the rational Bézier forms of conic sections. Whereas the arc length function can be integrated in closed form for the case of a parabola, the elliptic and hyperbolic segments require the evaluation of elliptic integrals. Several methods are proposed for the numerical evaluation of the resulting elliptic integrals, of which the arithmetic-geometric mean recursion turns out to be best in real-time applications.