## 9 Analysis of the Results  
The results of the numerical methods used for the algorithm were analyzed by running the simulated flights using different steps to calculate the total aircraft displacement. The procedure is described next. The maximum and minimum aircraft displacement values along the horizontal, flat, and vertical axes (x, y, and z) were calculated for comparison purposes in 4000 steps. This step number is sufficient for a good approach. The results are shown in Table 2. These values trace the aircraft's path inside a fitted area on a computer screen, as shown in Figure 10a,b. Moreover, when the minimum or maximum values are zero, the aircraft moves in any direction from its original position; when the positive and negative values are in any direction--as in flight (1) for the x-axis--the aircraft flies to the south and north. The altitude can never be negative because the lowest altitude defined was (z\({}_{\text{min}}\) = 0), meaning that the aircraft started moving at sea level. However, it is possible to observe a few slight descents during the three flights owing to the defined vertical speed values. Figure 13a-c show the displacement values using 10 to 2000 steps. The analysis was performed by considering the three simulated flights. The displacement fluctuation decreased by over 200 steps; thus, it can be considered rudimentary but acceptable. The solution with 2000 steps can be obtained quickly, and the computational effort is not considerable owing to the data management capacity of personal computers. The aircraft displacement can be quickly calculated because it is not absolutely necessary to include a graphical display to show the results. Thus, these values can be stored, and the computational effort is small. In the beginning, all these curves show significant variation, evidencing a poor approach; as the step time increases to 500 or 1000, the variation decreases. Thus, the invariability in the total aircraft displacement value is a criterion for better \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Flight** & \(\mathbf{x_{max}}\) & \(\mathbf{x_{min}}\) & \(\mathbf{y_{max}}\) & \(\mathbf{y_{min}}\) & \(\mathbf{z_{max}}\) & \(\mathbf{z_{min}}\) \\ \hline
1 & 25,118.12 & \(-\)7583.16 & 15,829.45 & \(-\)11,598.48 & 2659.68 & 0 \\ \hline
2 & 54,463.7 & 0 & 14,862.21 & \(-\)11,984.55 & 4086.10 & 0 \\ \hline
3 & 21,123.13 & \(-\)6508.28 & \(-\)26,463.55 & 0 & 5727.65 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum and minimum displacement values during the simulated flights. results. The curves shown in Figures 13-15 were calculated according to the following ranges, meaning that the displacement values when there were between 10 and 100 steps were calculated for 10, 20, 30, etc., up to 100, and that for 100 to 200 steps, the algorithm was solved for 120, 140, 160, etc. Figure 13: Calculation of the results for the analyzed flights based on the total aircraft displacement values for (**a**) flight (1), (**b**) flight (2), and (**c**) flight (3). From 10 to 100 steps, the steps for the analysis were considered every 10 steps. From 100 to 200 steps, the steps for the analysis were considered every 20 steps. Figure 14: Calculation of the results for the analyzed flights based on the total aircraft displacement values for (**a**) flight (1), (**b**) flight (2), and (**c**) flight (3). From 200 to 1000 steps, the steps for the analysis were considered every 100 steps. From 1000 to 2000 steps, the steps for the analysis were considered every 200 steps. Figure 15: Calculation of the results for the analyzed flights based on the total aircraft displacement values for (**a**) flight (1), (**b**) flight (2), and (**c**) flight (3). The variations in the distances calculated for 500 steps or lower were large; thus, this approach is not good enough. More steps are required for a better approach.