Abstract: In order to get rid of the local interference of the tool positions calculated using multi-point machining algorithm, the discriminant of local interference was derived after the analysis on the geometric match between the disc cutter with a concave end and the desired surface. The ratio of curvature radius of intersection curve of the disc cutter and the desired surface at the first contact point to the radius of the cutter, which was belonged to tool position calculated using multi-point machining method, was derived with differential geometry analysis. Combined with the discriminant of local interference, the solution areas of multi-point machining constraint immune to local interference were deduced. Results show that no matter what kind of surface was machined, the tool positions calculated using multi-point machining method would not be interferential with the desired surface locally if the second tangent contact point was selected in the solution areas of multi-point machining constraint, where the corresponding principle direction had larger surface curvature.