## 3 Results  
### Verification with Datasets in the Literature  
Due to the limited availability of publicly accessible datasets in the field of sphericity error research, current research papers often refer to the same set of publicly available spherical datasets for comparative analyses. Notably, the four datasets [15; 16; 31; 33] used in the experiments of this section have been widely cited in numerous high-quality publications in the field of sphericity error research. As a result, these datasets hold considerable authority. By conducting experiments using these four datasets, we can effectively compare our approach with the leading algorithms in the field of sphericity error evaluation under identical experimental conditions. This reasoning highlights one of the main reasons for selecting these four datasets for experimentation in our study. The experiments were conducted under identical conditions. The number of populations (\(N\)) was set to 30, the local enhancement parameter (\(n\)) was set to 12, the initial temperature (\(T\)) was set to 100, the annealing rate (\(q\)) was set to 0.9, and the temperature threshold was set to 0.001. Using the aforementioned settings, each dataset was independently calculated ten times. The mean and standard deviation of the results were calculated and compared with the results obtained using state-of-the-art algorithms. Dataset 1, obtained from reference [34], comprises 100 randomly generated coordinate points within two concentric spheres with radii of 10.0 and 9.0. The data distribution is illustrated in Figure 7, where the sphericity error is determined as 1.0 based on five data points from the inner and outer spheres. The sphericity error of dataset 1 has been validated in several studies [21; 34], and the algorithm's accuracy in this paper was initially assessed using dataset 1. Ten sets of replication experiments were performed for dataset 1. The mean experimental result is 1.0000000000003 mm, with a standard deviation of 2.0 \(\times\) 10\({}^{-14}\) mm. Figure 8 displays one convergence curve from the ten experiments. Table 1 presents a comparison with the results reported in the literature. Figure 7: The distribution of dataset 1. Dataset 2, consisting of 384 coordinate points, is provided in reference [16]. The 384 coordinate points were obtained through real spherical sampling using the birdcage method and were evenly distributed along 12 lines on the sphere. Figure 9 illustrates the distribution of dataset 2. The reference provides the optimal solution as 0.015385 mm. Ten sets of replication experiments were conducted for dataset 2. Table 2 presents the results of the ten experiments, with a mean experimental result of 0.01538487054930 mm and a standard deviation of 3.0 \(\times\) 10\({}^{-14}\) mm. Figure 10 displays one convergence curve from the ten experiments. The least square solution obtained by our algorithm is 0.01639862 mm, and the final convergence solution is 0.01538487054931 mm. Table 3 presents a comparison with the results reported in the literature. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Spherical Center Coordinate (X, Y, Z)** & **Sphericity Error** \\ \hline Least squares & (\(-\)0.01913, 0.08935, 0.01762) & 1.15224 \\ [34] & (0.00000, 0.00000, 0.00000) & 1.00000 \\ [21] & (0.0000000, 0.000000, 0.0000000) & 1.000000000 \\ Ours & (\(-5.3\times 10^{-14}\), 0.1 \(\times 10^{-14}\), 1.6 \(\times 10^{-14}\)) & 1.0000000000003 \\ \hline \hline \end{tabular} The number of zeros after the decimal points represents the precision provided in the original data from the cited references.  
\end{table} 
Table 1: Comparison with results based on dataset 1 in the literature (rounding is as given in the respective papers; units: mm). Figure 8: Algorithm optimization process (corresponding experiments based on dataset 1). Figure 9: The distribution of dataset 2. Figure 14 displays a convergence curve from one of the ten experiments.