## 3 Results of Micro-Scale Spherical and Cylindrical Surface Modeling  
The micro-scale spherical and cylindrical surface modeling is carried out by the microscope vision system shown in Figure 4a. In this way, the micro-scale spherical surface modeling is performed for the plastic surface shown in Figure 5a. The scale of this figure is represented in millimeters in the \(x\)-axis. The micro laser line projected on the plastic surface is shown in Figure 5b. Thus, the plastic surface is scanned in the \(x\)-axis via the micro laser line to compute the coordinates (\(x_{i,j}\), \(y_{i,j}\)) by means of Equations (26) and (27). Then, the coordinates (\(x_{i,j}\), \(y_{i,j}\)) are substituted in Equations (23) and (23) to compute the micro-scale surface coordinates (\(z_{i,j}\), \(y_{i,j}\)). The surface coordinate \(x_{i,j}\) is provided by the slider device. Thus, one hundred and ninety four images are processed to retrieve the micro-scale spherical surface shown in Figure 6a. In this figure, the \(x\)-axis, \(y\)-axis, and \(z\)-axis are indicated in microns. The accuracy of the surface shown in Figure 6a is determined via relative error [48], which is computed by employing a reference contact method. Thus, the relative error is computed by the next expression  
\[Error\%=\frac{100}{n\cdot m}\sum_{i=0}^{n}\sum_{j=0}^{m}\frac{\left|z_{i,j}-H _{i,j}\right|}{H_{i,j}} \tag{28}\]For this equation, \(z_{i,j}\) is the micro-scale surface computed via Equation (22), \(H_{i,j}\) is the surface reference, and \(n\cdot m\) is the data number. By computing Equation (28), the relative error for the plastic surface shown in Figure 6a is 0.8216%. From the surface shown in Figure 6a, a spherical surface model is constructed via genetic algorithm, which determines the origin coordinates \((h,k,l)\) as described in Section 2.1. Figure 5: (**a**) Rectangular spherical surface to perform spherical surface modeling. (**b**) Micro laser line projected on the spherical surface to retrieve the surface topography. Figure 6: (**a**) Spherical surface retrieved via micro laser line scanning. (**b**) Spherical surface generated via spherical surface model Equation (1). (**c**) Surface generated via spherical surface model Equation (1) and the surface data over the surface model. (**d**) Surface generated via spherical surface model Equation (1) and surface data under the model. In this way, the genetic algorithm performs the first step to determine the initial population. To carry it out, the search space is defined based on the maximum and minimum of each parameter. For the spherical surface model, the minimum and maximum of the parameter \(h\) are defined as \(x_{0,0}\) and \(x_{n,m}\), respectively. The minimum and maximum of the parameter \(k\) are defined as \(y_{0,0}\) and \(y_{n,m}\), respectively. The minimum and maximum of the parameter \(l\) are defined as \((l+\tilde{R}/2)\) and \((l-\tilde{R})\), respectively. Then, the initial population is determined. Typically, the metaheuristic algorithms determine the initial population from the search space in random form. However, a huge iterations number is employed to achieve the convergence. Therefore, the algorithms employ the best candidates to improve the solution quality and to speed out the convergence. The initial population plays a significant role in the genetic algorithm to provide higher qualities and better convergence [49]. Generally, the genetic algorithms generate the initial population employing the best candidates. These candidates can be obtained via random methods and by sampling the model equations [50; 51]. Based on these statements, the proposed genetic algorithm performs a sampling of the model to obtain the best candidates. The sampling is carried out by solving Equations (2) and (3) to obtain the best candidates to perform the optimization.