## 4 Experiment and Analysis  
Since the period of the Moon's revolution around the Earth is approximately 27.32 days, the period of a geosynchronous Earth orbit (GEO) satellite is approximately 24 h, the period of an inclined geosynchronous orbit (IGSO) satellite is approximately 24 h, and the period of a medium Earth orbit (MEO) satellite is approximately 12 h. Satellite data with a period of 28 days for the Moon's revolution are used for the analysis. The data used here are  
Figure 4: Flowchart of the adaptive interpolation based on the vertex protection algorithm. from the multisystem precision ephemeris file provided by the Analysis Centre of Wuhan University for 31 December-30 January 2021, which contains the coordinates of each BDS, GLONASS, and Galileo satellite at five-minute intervals in the geocentric geostationary coordinate system. The lunar coordinates were calculated using a simplified model [18]. Figures 5 and 6 show a comparison between the adaptive interpolation algorithm based on vertex protection and the traditional adaptive interpolation algorithm [5] in fitting the elevation angle function and the Earth occlusion function from the C01 satellite to the Moon when \(\varepsilon=0.1\) and \(\sigma=0.2\). From the two figures, it can be seen that the adaptive interpolation algorithm based on vertex protection can make the local extreme points of the multi-hump functions, such as the elevation angle function and the Earth occlusion function, the endpoints of the interpolation sub-interval (to the greatest extent possible), and the fit is good. The period of visibility from the satellite to the Moon is calculated as the true value using the brute force method, including the solution for the rise and set moments of the satellite with respect to the Moon and the period of the Earth's line-of-sight obscuration between the satellite and the Moon. Figure 5: Effects of vertex protection algorithms on fitting of elevation angle functions. Figure 6: Effects of vertex protection algorithms on fitting of Earth occultation functions. Table 1 shows a comparison between the adaptive interpolation based on the vertex protection algorithm, the conventional adaptive interpolation algorithm, and the brute force method in solving the number of periods of visibility between the satellite and the Moon; that is, the times of the satellite's rise and set with respect to the Moon and the times of the Earth's occultation. For example, for the C01 satellite at fitting errors \(\varepsilon=0.1\) and \(\varepsilon=0.01\), the C06 satellite at fitting error \(\varepsilon=0.1\), and the C11 satellite at fitting error \(\varepsilon=0.1\), the solved Earth occlusion times do not match the real times, which are errors that cannot occur in practical applications. Adaptive interpolation based on vertex protection can accurately solve the Earth occlusion count for satellites C01, C06, and C11 when the search factor in the protection vertex algorithm is \(\sigma=0.2\) for the fitting errors \(\varepsilon=0.1\), \(\varepsilon=0.01\), and \(\varepsilon=0.001\). The efficiency of the adaptive interpolation method based on vertex protection proposed in this paper is compared to that of the traditional adaptive interpolation method when applied to the satellite-to-Moon visibility calculation in Tables 2-4. For satellites C01, C06, and C11, under the limits of interpolation accuracies of 0.1, 0.01, and 0.001, respectively,