## 3 Results of the Geometric Method
### Comparison of the Errors Obtained with the Geometric Method and the Two-Step Method  
The accuracy in the direction of the laser beam to any point in space varies radially and also depends on the wedge angle \(\alpha\). To analyze the accuracy of the method, the error is calculated, where the error is the difference between the target point and the point that has been reached with the method used. The two-step method [9] developed by [PERSON] is used as a reference in this paper to show the improvement provided by the geometric method.  
For the three methods (geometric method-circumference, geometric method-ellipse and the two-step method), as can be seen in Figure 6a,c,e, the error increases with the wedge angle. Three wedge angle values were also selected to observe their profile as a function of the radius, Figure 6b,d,f. The two-step method [9] (Figure 6e,f) has radial behavior, with the error increasing with the radius. However, in the geometric method (Figure 6a-d), the minimum error is obtained at the minimum and maximum radius. This is because the laser beam does not actually describe a circumference but is closer to an ellipse, and the radius of the circumference is the same as the semi-axis \(b\) of the ellipse. Moreover, in the ellipse approximation (Figure 6c,d), the error is minimized such that when the radius is 65% of the maximum radius, a minimum error occurs.  
It can be seen from Figure 7 that the error obtained by the ellipse approximation is smaller than the error obtained by the circumference approximation. The error of the circumference approximation is approximately twice that of the ellipse approximation. However, both approximations of the geometric method have a smaller error than the two-step method.  
Another important aspect to consider is the calculation time required to obtain the desired data, the results of which are shown in Table 1. For this, we used a set of 4900 points, which contains points for seventy different angles of the whole possible region and seventy different radii between the minimum and maximum values. This set of points was repeated sixteen times, giving a total of 78,400 points to calculate the mean time and the deviation.  
In all three methods, the error depends on the wedge angle of the prisms \(\alpha\). However, the error as a percentage of the distance \(D2\) does not change even if the distance increases. Considering a target distance (\(D2\)) of 1 m, Figure 7 shows the mean error as a percentage of the distance \(D2\) and the shadow regions correspond to the standard deviation of the mean error obtained for each wedge angle of the prisms. The mean error was obtained from a set of eighty points. This set of points was in turn obtained on a straight line for an azimuth angle of 45\({}^{\circ}\). As the error is radial, it does not matter which angle is considered.  
Figure 6: Error of the two approximations for different values of wedge angle \(\alpha\) and radius of the target point. (**a**) represents a heat map with the error obtained by the circumference approximation; (**b**) represents the error profile obtained by the circumference approximation; (**c**) represents a heat map with the error obtained by the ellipse approximation; (**d**) represents the error profile obtained by the ellipse approximation; (**e**) represents a heat map with the error obtained by the two-step method [9]; (**f**) represents the error profile obtained by the two-step method [9].  
The ellipse approximation had a longer mean computation time than the circumference approximation. The circumference approximation was about 1.6 times faster than the ellipse approximation. However, both approximations of the geometric method were faster than the two-step method.