## 4 Experiment and Analysis
\begin{table}
\begin{tabular}{l l c c c} \hline \hline
**Satellite** & **Methods** & \(\varepsilon\) & **Rise and Set Times** & **Occlusion Times** \\ \hline \multirow{6}{*}{C01} & \multirow{6}{*}{Brute force method} & 0.1 & 28 & 6 \\ \cline{3-5}  & & 0.01 & 28 & 6 \\ \cline{3-5}  & & 0.001 & 28 & 6 \\ \cline{3-5}  & & 0.1 & 28 & 4 \\ \cline{3-5}  & & 0.01 & 28 & 5 \\ \cline{3-5}  & & 0.001 & 28 & 6 \\ \cline{3-5}  & & 0.1 & 28 & 6 \\ \cline{3-5}  & & 0.001 & 28 & 6 \\ \cline{3-5}  & & 0.001 & 28 & 6 \\ \hline \multirow{6}{*}{C06} & \multirow{6}{*}{Brute force method} & 0.1 & 28 & 3 \\ \cline{3-5}  & & 0.001 & 28 & 3 \\ \cline{3-5}  & & 0.001 & 28 & 3 \\ \cline{3-5}  & & 0.001 & 28 & 3 \\ \cline{3-5}  & & 0.001 & 28 & 3 \\ \hline \multirow{6}{*}{C11} & \multirow{6}{*}{Brute force method} & 0.1 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \cline{3-5}  & & 0.001 & 53 & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Satellite visibility times. the number of step calculations, the number of times the algorithm actually constructs a cubic polynomial for the solution, and the time efficiency improvement (compared with the brute force method) are compared. As seen from these tables, the number of step calculations is always lower than that of the traditional adaptive interpolation algorithm, and the difference in the number of step calculations is more obvious as the fitting accuracy decreases. The algorithm proposed in this paper constructs a cubic polynomial for the number of solutions, and a cubic polynomial is constructed for when there is a solution in that subinterval; however, the traditional adaptive interpolation algorithm requires the construction of cubic polynomials in each subinterval to determine whether the interval has a solution and then calculate it. These tables give the percentage improvements in the time efficiency for adaptive interpolation based on the vertex protection algorithm and traditional adaptive interpolation compared with the brute force method. Under the requirement of a low fitting accuracy \(\varepsilon\)= 0.1, the computational efficiency of the algorithm proposed in this paper and the traditional algorithm are essentially comparable, and both improve by approximately 98% compared with the brute force method. The higher the requirement is, the more obvious the improvement in efficiency. When \(\varepsilon=0.001\) and \(\sigma=0.2\), it can be seen from the following table that, for the C11 MEO satellite, the algorithm proposed in this paper improves the computational efficiency by approximately 30% compared with the traditional adaptive computation. This affects the accuracy of the vertex protection algorithm when seeking the vertex as the endpoint of the fitted subinterval, where the larger \(\sigma\) is, the faster the algorithm is computed under the condition that the overall cycle is computed error-free. To represent the accuracy of the adaptive interpolation algorithm based on vertex protection, the percentage normalization error defined by [19] is listed, where the percentage normalization is specifically defined as follows:  
\[PNE=\frac{|Predicted\ satellite\ rise/set\ time-Acutal\ satellite\ rise/set\ time|}{Actual\ in-view\ period}\times 100 \tag{11}\]  
Figures 7 and 8 show the \(PNE\) of the satellite rise and set times and the \(PNE\) of the Earth occultation start and end times, respectively; the red squares indicate the \(PNE\) of the