## 4 Results and Discussion  
### Evaluation of the Algorithm Performance  
This paper utilizes the calculation results obtained through the TP method as a baseline and compares them with both the TP method itself and existing rapid algorithm, specifically focusing on the coarse search and binary search phases [25]. Simulations were conducted to validate the accuracy and superiority of the TTIHS algorithm. The China Space Station (CSS) was chosen as a case study, featuring an external payload platform that supports multiple Earth observation payloads. A randomly chosen twelve-sided polygon served as the ground polygon region, and the experimental parameter settings are detailed in Table 1. The simulation period extended from 00:00 to 24:00 (UTC) on 23 December 2023. The orbit model applied was the SGP4 model, and the Earth observation payload's field of view type was conical with a half angle of 30\({}^{\circ}\). To ensure fairness in the computations, all three algorithms were implemented under identical hardware and software environments. Details of the simulation environment setup are provided in Table 2. The calculation time step for the TP algorithm was denoted as \(t_{a}\) and set to 1 s, while for the existing rapid algorithm, it was denoted as \(t_{b}\) with the coarse search phase set to 90 s and the binary search precision set to 0.0005. The TTIHS algorithm's calculation time step was denoted as \(t_{d}\), with the coarse search phase also set to 90 s and the binary search precision set to 0.0005. \begin{table}
\begin{tabular}{c c} \hline
**Category** & **Parameter** \\ \hline Payload & CSS Earth observation payload \\ TLE [40] & 48274U 21035A 23357.28655182. 00041146 0000 + 0 42316-3 0 9997 \\  & 48274 41.4711 83.9203 0005576 41.6976 318.4288 15.64081887151409 \\ Simulation time & 23 December 2023 00:00:00–23 December 2023 24:00:00 (UTC) \\ Time step & \(t_{a}=1\) s, \(t_{b}=90\) s, \(t_{d}=90\) s \\ Half field of view angle & 30\({}^{\circ}\) \\ Region coordinates & (5\({}^{\circ}\) N, 10\({}^{\circ}\) E), (5\({}^{\circ}\) N, 50\({}^{\circ}\) E), (10\({}^{\circ}\) N, 57\({}^{\circ}\) E), (15\({}^{\circ}\) N, 60\({}^{\circ}\) E), (25\({}^{\circ}\) N, 60\({}^{\circ}\) E), (30\({}^{\circ}\) N, 57\({}^{\circ}\) E), (35\({}^{\circ}\) N, 50\Figure 10 illustrates a schematic diagram of the simulated VTW, with different circles represented by distinct colors. The VTWs for the regional target are denoted by dashed lines, while the invisible intervals are indicated by solid lines. The red dot symbolizes the current position of the space station platform, and the black dashed line represents the ground polygonal region. The figure reveals a total of 17 circles within one day of simulated time. Among them, only Circles 2, 3, 4, 5, 6, 7, 15, 16, and 17 exhibit a VTW. A significant portion of the time lacks a VTW, underscoring the importance of the orbit filtering and coarse search in this method. Table 3 provides details of the simulation computation times for the three algorithms, while Table 4 presents information on the VTWs for the regional target obtained from the simulation. From Table 3, it is evident that, in terms of computation speed, the TTIHS algorithm surpasses both the TP algorithm and the existing rapid algorithm. The TP algorithm, with a time step of 1 s, requires 31.728 s for computation. Using this as a benchmark, the existing algorithm, employing a larger time step of 90 s and a binary search strategy, takes 0.317 s, accounting for 0.999% of the TP time. In contrast, the TTIHS algorithm requires only 0.116 s, representing 0.365% of the TP time. The computational speed is improved by 274 times and 2.73 times compared to the TP algorithm and the existing rapid algorithm, respectively, indicating a significantly higher computational speed of the proposed algorithm. Further analysis reveals that, relative to the 24 h calculation interval, the TTIHS algorithm computes only angle relationships within 1 h, 20 min, and 11 s after orbit filtering.