3D linear and circular interpolations are a basic part for the machining of complex shapes. Until now, because of the absence of appropriate algorithms for the generation of 3D lines and circles, a full accomplishment for available machine tool resolution is difficult. this paper presents new algorithms for 3D linear and circular interpolation in the reference pulse technique. In 3D space, the line or circle is not expressed as an implicit function, it is only defined as the intersection of two surfaces. A 3D line is defined as the intersection of two planes, and a 3D circle is defined as the intersection of a plane and the surface of a sphere. Based on these concepts, interpolation algorithms are designed to follow intersection curves in 3D space, and a real-time 3D linear and circular interpolator was developed in software using a PC. The algorithm implemented in a PC showed promising results in interpolation error and speed performance. It is expected that it can be applied to the next generation computerized numerical control systems for the machining of 3D lines, circles and some other complex shapes.