With the advantage of in situ measurement, the three-probe method is commonly used to measure either the error motion of high-precision spindles or the roundness error of artifacts. The roundness error of artifacts or spindle errors can be obtained through solving error-separation equations. Both the time- and frequency-domain solutions of the three-probe method are presented. In addition, the key points of solutions, i.e., the rounding error induced by inconsistency of sampling points, harmonic suppression, and averaging schemes of multiple revolutions into one circle, are described in detail. Experiments were conducted to compare the two solutions and quantify the influence of setup parameters, including rotational speed probe arrangement, consistency of sampling points, and number of revolutions. The results showed that the roundness error of the time-domain solution was inaccurate due to large rounding errors, while that of the frequency-domain solution with the previous average scheme was accurate. In contrast, the spindle error of the frequency-domain solution with the latter average scheme was more reliable. The findings provided a reference to recommend setup parameters depending on the aim of the three-probe method.

The minimum circumscribed circle (MCC), maximum inscribed circle (MIC), and minimum zone circle (MZC) methods for circularity evaluation are difficult to execute due to the lack of specific rules in mathematics, especially the MZC. New accurate algorithms have been proposed to realize the MIC, MCC and MZC evaluation methods. First, the diameter criterion and acute triangle criterion of control points defining the MIC or MCC are presented. The definition of the crossing sector structure is introduced in the minimum zone criterion and transformed into an angular relationship of control points, making it easy to identify the MZC. Then the key points of the algorithms are presented in details including the initial circle or the initial region, centre moving direction, minimum step size and updating rules of control points. Flow charts are provided to make the algorithm producible. Finally, four cases are selected to demonstrate the advantages of the algorithms. The results show that the proposed MZC algorithm is more general than that in the published reference. Compared to the traditional ergodic searching algorithms, the efficiency of the new algorithm for the MIC, the MCC and the MZC evaluations of circularity can be improved by 100% when the point set increases to 100 and above.