## 4 Results and Discussion
### Discussion  
The TTIHS algorithm is applicable to complex polygons, including both convex and non-convex polygons. For cases where the expanded region is much larger than the original region, such as diagonal shapes, consideration can be given to using a quadrilateral fitting method based on this method to obtain the quadrilateral expanded region. The impact of polygon shapes primarily manifests in the second step, the coarse search phase, where the algorithm needs to determine the moments when angles are equal, marking the start or end of the VTW. The logic for this determination is as follows: after obtaining the start point, if the trend of angle relationships changes from less than to greater than and then back to less than until equal, then that point is identified as the end point. This trend represents the process of the satellite passing through the interior of the polygon. Therefore, this logic is unaffected by the shape of the polygon. Regarding the influence of the field of view, since the orbits of low Earth orbit satellites are relatively close to the Earth's surface, the angle of the field of view projected onto the Earth is generally only a few degrees. Hence, this method is suitable for calculating VTWs for ground regions with large latitude and longitude ranges. Further analysis reveals that the time step \(t_{d}\) defined in the coarse search phase has a significant impact on the algorithm's accuracy _acc_. In this paper, an optimization model is established with \(t_{d}\) as the independent variable, and algorithm execution time \(T_{2}\) and algorithm accuracy _acc_ as the objectives. The calculation results are shown in Figure 11. The accuracy calculation formula is given in Formula (10). If the start or end times of the window under the current time step size \(t_{di}\) are consistent with the reference, \(x\) is set to 1; otherwise, it is set to 0. Finally, the sum is divided by the total number of VTW start-end \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**TP Algorithm**} & \multicolumn{2}{c}{**TTIHS Algorithm**} \\ \hline
**Identification Number** & **StartEnd Times** & **Duration/s** & **Start–End Times** & **Duration/s** \\ \hline
1 (Circle 2) & 00:48:29–00:58:54 & 00:10:25 & 00:48:29–00:58:54 & 00:10:25 \\
2 (Circle 3) & 02:29:21–02:30:35 & 00:01:14 & 02:29:20–02:30:34 & 00:01:14 \\
3 (Circle 4) & 04:14:46–04:15:56 & 00:01:10 & 04:14:46–04:15:56 & 00:01:10 \\
4 (Circle 5) & 05:46:31–05:56:50 & 00:10:19 & 05:46:30–05:56:50 & 00:10:20 \\
5 (Circle 6) & 07:19:19–07:33:13 & 00:13:54 & 07:19:18–07:33:12 & 00:13:54 \\
6 (Circle 7) & 08:55:05–09:05:15 & 00:10:10 & 08:55:05–09:05:15 & 00:10:10 \\
7 (Circle 15) & 20:35:08–20:42:43 & 00:07:35 & 20:35:07–20:42:42 & 00:07:35 \\
8 (Circle 16) & 22:06:49–22:19:32 & 00:12:43 & 22:06:49–22:19:31 & 00:12:42 \\
9 (Circle 17) & 23:41:09–23:53:50 & 00:12:41 & 23:41:09–23:53:49 & 00:12:40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: VTW information. times to obtain the percentage, representing the accuracy. Considering real-world tasks, the range of variation for \(t_{d}\) is set to [60 s, 150 s]. \[ \begin{cases}x=1&if\quad S_{i}=T_{si}\quad or\quad E_{i}=T_{ei}\\ x=0&else\end{cases}  \tag{10}\] \[e=\frac{\sum\limits_{x}^{18}x}{18}\times 100\%\]  
From Figure 11, it can be observed that the horizontal axis represents the time step \(t_{d}\), the left vertical axis represents the algorithm execution time \(T_{2}\), and the right vertical axis represents the algorithm accuracy _acc_. The red curve represents the variation of \(T_{2}\) with \(t_{d}\), and the green curve represents the variation of _acc_ with \(t_{d}\). As evident from the curve in the graph, the computational time of this algorithm remains almost constant in the order of 0.1 s, showing stability regardless of changes in the time step. However, as the time step size expands to 90 s, the accuracy rapidly decreases from 100%.