Conventional CNC circular interpolation algorithms are based on interpolating the next value according to the present value, regardless of the particular pulse increment method or data sampling method. In this paper, a new theory and algorithm for any space circular interpolation in a CNC system, called three-point recursion interpolation (TPRI) is proposed, which makes a breakthrough from the conventional interpolation algorithms and which is based on a two-step recursion scheme. The proposed algorithm improves not only the interpolation accuracy, but also the interpolation process, which is carried out by the recursion method based on initial values. The interpolation is easily done by the operations of adding and bit shift without multiplication or division. Thus, compared with the existing space circular interpolation methods, by means of the new algorithm, the computing size is remarkably decreased and the calculation speed is increased greatly, SO that the interpolation process may be enhanced strongly. The interpola-tion step is selected according to the tolerance of a segment of a circle. Also, the interpolation stability of this recursion algorithm is proven and the interpolation error and theoretical machining error are estimated in this paper.