## 3 Results of the Geometric Method
### Comparative Results of the Error with the Iterative Process  
The geometric method was implemented in an iterative process to increase the accuracy and then compared with the method developed by [PERSON] et al. [7], which also follows an iterative process. The results shown below were obtained for a wedge angle \(\alpha\) of 10\({}^{\circ}\). The mean computation time and the number of iterations that each method took to reach the required error threshold were calculated, with the results shown in Figure 8. Each point represented in the graphs in Figure 8 is the mean time obtained for a set of 40,000 points scattered all over the region that the Risley system can reach. The left \(y\)-axis shows the mean time taken by each method, and the right \(y\)-axis shows the number of iterations taken. The graphs in Figure 8a,c show the results obtained for the iterative process with, respectively, the circumference and ellipse approximations of the geometric method. The graph in Figure 8b shows the results obtained with the iterative method proposed by [PERSON] et al. [7]. The dashed red lines are the \(D2\) values taken as reference to show the mean number of iterations in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Geometric Method**} \\ \cline{2-4}  & **Circumference** & **Ellipse** & **Two-Step Method** \\  & **Approximation** & **Approximation** & \\ \hline Mean time (\(\upmu\)s) & 2.7257 & 4.3378 & 9.6737 \\ Deviation (\(\upmu\)s) & 0.4754 & 0.5015 & 7.3917 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean time and deviation of the geometric method and the two-step method obtained from a total set of 78,400 points. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Geometric Method**} \\ \cline{2-4}
**D2 (m)** & \(\Delta\) (m) & \begin{tabular}{c} **Circumference** \\ **Approximation** \\ \end{tabular} & \
\begin{tabular}{c} **Ellipse** \\ **Approximation** \\ \end{tabular} &
\begin{tabular}{c} **Two-Step** \\ **Method** \\ \end{tabular} \\ \hline
0.5 & & 3.00 & 3.21 & 5.12 \\
8 & \(1\cdot 10^{-7}\) & 3.76 & 3.78 & 3.30 \\
128 & \(4.06\) & 4.04 & 38.10 \\
1024 & & 4.72 & 4.64 & 48.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean number of iterations of each method for different distances \(D2\) and a specific error threshold. Figure 7: Mean error as percentage of \(D2\) for each value of the prism angle \(\alpha\). Figure 8 shows how the iterative process has a shorter computation time when using the geometric method proposed in this paper than the method developed by [PERSON] et al. [7] for any distance \(D2\) and error threshold. The shortest time taken by the iterative process with the geometric method to obtain the final rotation angles is 4 \(\upmu\)s, whereas the corresponding value for the two-step method is 7 \(\upmu\)s. This time is equivalent to three iterations performed using the iterative process with the geometric method and about four iterations with the two-step method. The maximum number of iterations that the iterative process can reach is 50, a value that was established after several tests found that if the method is unable to reach the required threshold error in 50 iterations, it will never do so. The graph in Figure 8b shows that with [PERSON] et al. method [7] for large distances (more than 100 m) and for very small error thresholds (1 nm) the results are obtained on average in 100 \(\upmu\)s. This is because the method has reached the maximum number of iterations (i.e., the two-step method does not achieve error thresholds smaller than 1 nm). In contrast, an error threshold of up to 1 pm can be achieved for any distance \(D2\) up to 16.384 km with the circumference approximation (Figure 8a), while a margin of error of 1 nm is reached for any distance \(D2\) with the ellipse approximation (Figure 8c). Table 2 shows the average number of iterations each method needed to reach a margin of error of 1\(\cdot\)10\({}^{-7}\) m, the least accurate margin of error, and for different distances \(D2\).