## 2 Methods
### Sphericity Error Spatial Distribution Characteristics  
From Equation (1) and Figure 1, it can be observed that the problem of evaluating the sphericity error based on the minimum zone criterion is non-differentiable and unconstrained. Finding an ideal sphere center \(\mathbf{p}_{0}^{*}\) within three-dimensional space, such that the difference in the radii is minimized when the concentric spheres encompass the spherical contour, poses a highly challenging problem. Meanwhile, existing studies suggest that the ideal sphere center \(\mathbf{p}_{0}^{*}\) is not unique [10; 21]. There are several local optimal solutions near the optimal solution of the sphericity error, and the sphericity error decreases rapidly with a large gradient at the location far from the optimal solution [21].  
Therefore, we have undertaken a study on the spatial distribution characteristics of spherical error. We conducted an analysis of the data sourced from reference [15]. Subsequently, we performed 27,000 samplings from the global solution space, calculating the spherical errors for each sampled point. Figure 2a illustrates the distribution of these sampled points, with the color gradient representing the magnitudes of the associated spherical error values. Following this, within this array of 27,000 points, we scrutinized the variations in spherical error values along the directions parallel to the XYZ axes. This scrutiny was carried out with the center \(O_{\text{LS}}\) (X, Y, Z) of the least squares sphere as the central point, independently assessing the variations along each axis.  
Meanwhile, Figure 2b-d show the trend of the sphericity error near the center \(O_{\text{LS}}\) (X, Y, Z) of the least squares sphere, where the three lines passing through \(O_{\text{LS}}\) and running parallel to the X-axis, Y-axis, and Z-axis, respectively, are constructed. It is clearly discernible from Figure 2b-d that, at positions considerably distant from the ideal solution, there is a significant gradient in the variation of the spherical error. Conversely, as the position approaches the ideal solution, the evolution of the spherical error gradually becomes more intricate.  
Figure 1: Geometric structure of sphericity error evaluation based on the minimum zone criterion.  
Subsequently, we arbitrarily elected several localized solution spaces from the global solution space, analyzing the spherical error variation of the sampling points within these specific solution spaces. Figure 2e-g illustrate the distribution of sampling points within these localized solution spaces. It can be observed from Figure 2e-g that within smaller regions, the variation of the spherical error manifests a relatively uncomplicated nature, approximating the characteristics of a unimodal function. If the global solution space is divided into different size subregions, then each subregion can be approximated as a single-peaked function for processing. In local spatial domains, unimodal functions can be effectively solved using computational geometry methods, yielding high-precision results.  
The algorithm presented in this paper primarily leverages the characteristics of spherical error gradient variations and the approximate unimodal nature of spherical error changes within local spaces for its design. This is the foundation of the local enhancement principle in this paper.  
Figure 2: Schematic representation of spherical error distribution characteristics. (**a**) Spherical error distribution of sampling points in the global space; (**b**) change in sphericity along the X-axis; (**c**) change in sphericity along the Y-axis; (**d**) change in sphericity along the Z-axis; (**e**) schematic distribution of sampling points in Local Space A; (**f**) schematic distribution of sampling points in Local Space B; (**g**) schematic distribution of sampling points in Local Space C.