## 4 Discussion  
Evaluating the sphericity error based on the minimum zone criterion is an indispensable and challenging task in metrology. Thus, this study aims to achieve a high-precision evaluation of spherical contour sampling data. Consequently, we conducted an in-depth \begin{table}
\begin{tabular}{c c c c c} \hline
**Experiment** & **1** & **2** & **3** & **4** \\ \hline Sphericity Error & 0.00747188442066 & 0.00747188442072 & 0.00747188442067 & 0.00747188442064 \\ \hline
**Experiment** & **6** & **7** & **8** & **9** \\ \hline Sphericity Error & 0.00747188442067 & 0.00747188442066 & 0.00747188442064 & 0.00747188442065 \\ \hline
**Experiment** & **5** & **10** & — & — \\ \hline Sphericity Error & 0.00747188442065 & 0.00747188442064 & — & — \\ \hline \end{tabular}
\end{table} 
Table 11: Results of ten experiments based on measurement data (units: mm). Figure 17: The dataset distribution. Figure 18: Algorithm optimization process. investigation into the spatial distribution characteristics of the sphericity error. We propose treating the local space of the sphericity error as a unimodal function and utilizing computational geometry methods to obtain high-precision solutions for the local space. Subsequently, combined with the advantages of heuristic algorithms in searching the global space, we successfully achieved exact solutions for sphericity errors. To validate the efficacy of the proposed approach, we conducted experiments using datasets that are widely cited in the realm of spherical error research, which are featured in four high-level scholarly papers. The experimental results demonstrate a significant improvement in precision, surpassing the existing achievements in the field of spherical error research by several orders of magnitude. Meanwhile, the uncertainty of the algorithm's solution reaches 10\({}^{-14}\) orders of magnitude, which far exceeds that of the traditional heuristic algorithm. Furthermore, when processing the distribution of data with specific sphericity or dealing with a large number of data points, the global optimal solution can be searched out accurately using the proposed algorithm, which shows stronger robustness. Subsequently, we conducted ablation experiments on the four datasets, as mentioned earlier. The experimental results show that, after removing the local enhancement method, the algorithm's standard deviation decreases from the order of 10\({}^{-14}\) mm to the order of 10\({}^{-2}\)\(\sim\)10\({}^{-7}\) mm. The ablation experiments demonstrate the effectiveness of the proposed local enhancement method in this paper. Finally, we apply the algorithm to evaluate the sphericity errors of the hemispherical shell resonator and obtain its sphericity error of 0.00747188442066 mm with a standard deviation of 2.3 \(\times\) 10\({}^{-14}\) mm for ten experiments. Compared to the results obtained from the CMM, this algorithm can improve the accuracy of sphericity error evaluation by approximately 10%. In this study, each member of the population in the DE algorithm is considered the center point for partitioning local solution spaces. During each iteration of the algorithm, computational geometric techniques are employed to solve for these individual local solution spaces. This approach has yielded outstanding precision in evaluating sphericity errors. However, it has also resulted in an algorithmic time complexity of O(n\({}^{4}\)), leading to elevated time costs for each evaluation. Furthermore, we conducted assessments of the average time required for each evaluation. Under the specified parameter settings, completing a single evaluation takes approximately 300 s. Compared to the time costs of 0.1 to 1 s in existing studies, the average time cost of our algorithm has increased several-fold. It must be acknowledged that this algorithm is not suitable for components with spherical geometry that require large-scale production and have relatively low precision requirements, such as ball bearings.