Discretization and interpolation of curves are two frequently adopted practices when machining complex curves using computer numerically controlled (CNC) machines. Both practices stem from the need to sample curves at discrete time intervals corresponding to the sampling period of the CNC machine. This paper proposes new techniques for discretization and interpolation that account for the change of tool orientations in five-axis machining. First, the method for discretization proposed in this paper is based on sampling the curves such that specified contour, feedrate, and orientation errors are not exceeded. Second, the interpolator proposed in this paper will be able to avoid excessive angular speeds arising from sampling the curves based on the feedrate alone. [S1087-1357(00)01401-5]