## 3 Results of Micro-Scale Spherical and Cylindrical Surface Modeling
Thus, the number of generations that obtain the surface model is 136. The optimized origin coordinates are \(h=1688.3\) um, \(k=1409.4\) um, \(l=-3685.6\) um, and \(R=4390.9\) um. These coordinates are substituted in Equation (1) to obtain the spherical surface model \(z_{i,j}=-3685.6+[4390.9^{2}-(x_{i,j}-1688.3)^{2}-(y_{i,j}-1409.4)^{2}]^{1/2}\), which produces the spherical surface shown in Figure 6b. The relative error produced by the spherical model is determined by the next expression  
\[error\%=\frac{100}{n\cdot m}\sum_{i=0}^{n}\sum_{j=0}^{m}\frac{\left|M_{i,j}-z _{i,j}\right|}{z_{i,j}} \tag{29}\]  
For this equation, \(M_{i,j}\) is the micro-scale surface computed via Equation (1), \(z_{i,j}\) is the surface retrieved via micro laser line projection, and \(n\cdot m\) is the data number. Thus, Equation (29) determines the optimality gap of the genetic algorithm [53]. By computing Equation (29), the relative error between the spherical surface model and the plastic surface shown in Figure 6b is 2.8631%. Additionally, Figure 6c shows the surface generated via the spherical surface model and the surface data over the surface model. Figure 6d shows the surface generated via the spherical surface model and the surface data under the surface model. The fitness variation with respect to the generation number is depicted by the graphic shown in Figure 7, where, the variation decreases when the generation number increases. Thus, the optimality gap of the algorithm for the spherical surface model is 2.8631%. The solution quality is deduced based on the running time to obtain the optimal solution [54]. The running time is defined by the generation number and the computer speed. Thus, the 136 generations are computed in 22.87 s by employing a PC to 1.8 GHz. The fitness variation in each generation is shown in Figure 7. In this case, the variation decreases when the generation number increases. Thus, the spherical surface modeling is performed by the metaheuristic algorithm. The micro-scale cylindrical surface modeling is performed for the metallic surface shown in Figure 8a. The scale of this figure is represented in millimeters in the \(x\)-axis. The micro laser line projected on the metallic surface is shown in Figure 8b. Thus, the metallic 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 (22) and (23) to compute the micro-scale surface (\(z_{i,j}\), \(y_{i,j}\)). The surface coordinate \(x_{i,j}\) is provided by the slider device. In this way, one hundred and eighty images are processed to retrieve the micro-scale cylindrical surface shown in Figure 9a. In this figure, the \(x\)-axis, \(y\)-axis, and \(z\)-axis are indicated in microns. The accuracy of this surface is determined via relative error in Equation (28). Where \(z_{i,j}\) 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 metallic surface shown in Figure 9a is 0.8783%. Figure 7: Fitness variation based on the number of generations. Figure 8: (**a**) Rectangular cylindrical surface to perform cylindrical surface modeling. (**b**) Micro laser line projected on the cylindrical surface to retrieve the surface topography. Figure 9: (**a**) Rectangular cylindrical surface to perform cylindrical surface modeling. (**b**) Micro laser line projected on the cylindrical surface to retrieve the surface topography. (**c**) Surface generated via cylindrical surface model and the surface data over the surface model. (**d**) Surface generated via cylindrical surface model and the surface data under the surface model. From the surface shown in Figure 9a, a cylindrical surface model is performed by the genetic algorithm via surface coordinates (\(x_{i,j}\), \(y_{i,j}\), \(z_{i,j}\)).