In numerically controlled(NC) machining simulation, the Z-map is frequently used to represent the workpiece. Since the Z-map is usually represented by a set of z-axis aligned vectors, the machining process can be simulated through updating it with the intersection points between the vectors and the tool swept surface. In our previous work[A fast Z-map update method for linearly moving tools], we achieved a fast Z-map update method for linearly moving tools. On the other hand, typical NC simulation codes contain lots of circular tool movements especially for the free-form surface machining. This paper represents how to directly calculate the intersection points for the circularly moving tools. For fast computation, we express each of intersection points with a single-variable non-linear function and calculate the candidate interval in which the unique intersection point exists. Then, we prove the existence of a solution and its uniqueness in this candidate interval. Based on these properties, we can effectively apply numerical methods to finally calculate the solution of the non-linear function within a given precision. Due to space limitations, we focus on the filleted-end mills, the most general case of the APT(automatically programmed tool) tools. Since other tools such as ball-end mills and flat-end mills can be regarded as the special cases of the filleted-end mills, our result can be applied to the circular motions of any APT tools. Combining it to our previous result on the linear motion, we can cover most of NC simulation codes. Finally, our implementation and its result show that the new numerical approach is more efficient and reliable, in comparison to the previous methods.