## 2 Methods
### Adaptive Local Enhancement Method
#### 2.4.2 Computational Geometric Method for Local Solution Spaces  
Once the local solution spaces are determined based on the positions of population individuals, a precise solution need to be conducted for each local solution space. Since we regard the problem of solving the spherical deviation error under the constraint of local solution space as a unimodal function problem, we employ the computational geometry approach shown below.  
In the three-dimensional space, the local enhancement process of individuals is shown in Figure 5. According to the predetermined search interval, the initial \(Z\)-directional search interval is determined as \(\big{[}Z_{L}^{i}=z_{i}-L_{i},Z_{R}^{i}=z_{i}+L_{i}\big{]}\), and then the search is performed on two planes \(\ Z_{1}^{i},Z_{2}^{i}\), where \(Z_{1}^{i}=(Z_{R}^{i}+2Z_{L}^{i})/3\),\(Z_{2}^{i}=(2Z_{R}^{i}+Z_{L}^{i})/3\). The initial search interval in the \(Y\) direction is \(\big{[}Y_{L}^{i}=y_{i}-L_{i},Y_{R}^{i}=y_{i}+L_{i}\big{]}\) on two planes \(\ Z_{1}^{i},Z_{2}^{i}\), and a search for the local optimal solution in the one-dimensional direction, \(Y_{1}^{i}=(Y_{R}^{i}+2Y_{L}^{i})/3\) and \(Y_{1}^{i}=(Y_{R}^{i}+2Y_{L}^{i})/3\), is performed on both planes. Shrinking the search interval according to strategy 1, four positions, \(\gamma_{1}Z_{1}X_{\text{best}}\), \(\gamma_{2}Z_{1}X_{\text{best}}\), \(\gamma_{2}Z_{1}X_{\text{best}}\), and \(\gamma_{2}Z_{2}X_{\text{best}}\), of the local optimal solution can be obtained when satisfying the following condition: \(X_{\text{R}}-X_{\text{L}}<1\times 10^{-n}\). The local optimal solutions on the \(Z_{1}^{i},Z_{2}^{i}\) planes are compared, respectively, and the search interval in the \(Y\)-direction on the two planes, \(Z_{1}^{i},Z_{2}^{i}\), is shrunken according to strategy 2. After satisfying the condition \(Y_{\text{R}}-Y_{\text{L}}<1\times 10^{-n}\), the optimal solutions \(f\big{(}X,Y,Z_{1}^{i}\big{)}\) and \(f\big{(}X,Y,Z_{2}^{i}\big{)}\) on the \(Z_{1}^{i},Z_{2}^{i}\) planes are obtained. Afterward, the \(Z\)-directional search space is shrunken according to strategy 3. While satisfying \(Z_{R}-Z_{L}<1\times 10^{-n}\), the local optimal solution is obtained.  
\[\text{Strategy 1}:\ \left\{\begin{array}{l}X_{L}^{i}=X_{1}^{i-1},if:f(X_{1} ^{i-1},Y,Z)>f(X_{2}^{i-1},Y,Z)\\ X_{R}^{i}=X_{1}^{i-1},else\\ \end{array}\right.\] \[\text{Strategy 2}:\ \left\{\begin{array}{l}Y_{L}^{i}=Y_{1}^{i-1},if:f( \gamma_{1}Z_{X\text{best}},Y_{1}^{i-1},Z)>f(\gamma_{2}X_{\text{best}^{i}},Y_{2 }^{i-1},Z)\\ Y_{R}^{i}=Y_{2}^{i-1},else\\ \end{array}\right.\] \[\text{Strategy 3}:\ \left\{\begin{array}{l}Z_{L}^{i}=Z_{1}^{i-1},if:f(Z_{best},Y_{ best},Z_{1}^{i-1})>f(X_{\text{best}},Y_{best},Z_{2}^{i-1})\\ Z_{R}^{i}=Z_{1}^{i-1},else\\ \end{array}\right.\]  
Figure 4: Local solution spaces’ sizes and distribution.  
Furthermore, for the one-dimensional search area corresponding to \(Y_{1}^{i}\) on the \(Z_{1}^{i}\) plane in Figure 5, the detailed search process is illustrated in Figure 6. The middle points, \(X_{1}^{i-1}=(X_{R}^{l-1}+2X_{L}^{i-1})/3\) and \(X_{2}^{i-1}=(2X_{R}^{i-1}+X_{R}^{i-1})/3\), are determined based on the neighborhood searching range of the individuals, \(\left[X_{L}^{i-1}=x_{i-1}-L_{i-1},X_{R}^{i-1}=x_{i-1}+L_{i-1}\right]\), after which the magnitudes of \(f(X_{1}^{i-1})\) and \(f(X_{2}^{i-1})\) are compared. If \(f(X_{1}^{i-1})>f(X_{2}^{i-1})\), then set \(X_{L}^{i}=X_{1}^{i-1}\); otherwise, set \(X_{R}^{i}=X_{2}^{i-1}\). Through the strategy above, used to shrink the search interval, the optimal solution is considered as the sphericity error at the \(X_{R}^{i}\) position when \(X_{R}^{i}-X_{L}^{i}<1\times 10^{-n}\) is satisfied.