The Cartesian machining method has been used extensively for cutting path generation of compound surfaces on NC machine tools. Most of the available algorithms, however, are based on the so called CC-Cartesian method, in which the cutter contact points are generated first. Each array of points is then offset along the surface normal to obtain a sequence of cutter location points on which a cutting path is generated. The problem of such a method, however, is that the detection and elimination of gouging is difficult as the sequences of cutter location points may not locate on a plane. An alternative approach is proposed in this work by offsetting the part surface first and then generating the cutting paths on the offset surface. A prototype system capable of modeling compound surfaces composed of analytical CSG primitives and parametric NURB surfaces is developed. Each of the surface elements is offset individually. The cutter center points are then determined directly from the offset surfaces rather than the cutter contact points. All cutting paths are definitely parallel to each other and, hence, the gouging problem can be dealt with easily. To enhance the computational efficiency, the NURB surface element is approximated by triangular meshes so that the offsetting procedures can be carried out. Several examples are presented to verify the feasibility of the proposed strategy.