## 3 Results
### Nadir Point Track and FOV  
Variation of the nadir point is the key factor that induces a temporal variation of ground coverage and revisit cycles and repeat cycles. As one of the most different characteristics between the lunar and halo orbits, it is also the main factor that affects the differences in their Earth observation characteristics, such as the ground coverage and diversity of viewing angle. Figure 3 shows the nadir point tracks of the Moon (left) and a halo orbit (right) at 00:00:00 UTC each day from 20 March 2024 to 20 March 2025 when the lunar orbital inclination reaches the maximum of 28\({}^{\circ}\). Similar to the nominal orbit of \"Magpie Bridge\", whose Z-direction amplitude is around 13,000 km, here the amplitude parameter, which denotes the deviation of the halo orbit from L1, is set at 20,000 km, corresponding to the normalization parameter of 0.3457. Note that since the latitude variation of the nadir point track is relatively small during the daily cycle, the real-time nadir point may reach the whole area with different longitudes and the same latitude. As presented in the figure, the latitude range of the nadir point track of the Moon is limited to around 28\({}^{\circ}\)N and 28\({}^{\circ}\)S, which. together with the period of less than one day, forms one of the Moon's most important features. In fact, as a result of the declination of the Moon varying between 18.3\({}^{\circ}\) and 28.6\({}^{\circ}\) with a period of around 18.6 years, the maximum latitude varies from 18.3\({}^{\circ}\) to 28.6\({}^{\circ}\) both north and south of the equator [36]. In the case of a 20,000-km halo orbit, on the one hand, the variation of nadir points behaves more complicatedly during a period of one month or one year. The maximum latitude is extended to around 42\({}^{\circ}\)N induced by the halo motion, which in a way induces more diversity of viewing angles and is advantageous to the observation of the polar zone with a similar observation performance to the Moon-based platform. On the other hand, as the Moon's orbital period is as long as a sidereal month, the repeat cycle of the Moon is fixed. While the orbital period of the family of halo orbits around L1, it is less than half of a sidereal month and is designable and alterable. As demonstrated in Figure 3, the latitude and longitude variations of the nadir point become more complex induced by the halo motion, while the longitude variation produces a more significant effect. Figure 4 presents the temporal complexity of the latitude variation of the nadir point of the 20,000-km halo orbit (left) and Moon (right). The horizontal axis represents the length of the sliding time window, and the vertical axis represents the range of latitude variation within the time window. Each plotted box counts the changing values of the latitude of the nadir point calculated using one-hour time steps over a one-year period from 20 March 2024 to 20 March 2025 (a,b) and from March 2033 to March 2034 (c,d). Note that (a,b) is during the year when the lunar orbital inclination reaches its maximum, i.e., around 28\({}^{\circ}\), while (c,d) is during the year when the lunar orbital inclination reaches its minimum value, i.e., around 18\({}^{\circ}\). The temporal complexity curve of latitude variation is derived from the concept of Approximate Entropy, which is usually used to characterize the irregularity and complexity of time series and it works for both stochastic processes and deterministic processes (the case of this study). The temporal complexity curve provides a good description of the pattern and complexity of the nadir point's latitude variation with the length of time windows; in the meantime, it can also quantify the distribution of latitude variation values within a certain time window, which is useful for assessing the ground coverage of different latitudes with respect to hemispheric-scale Earth observations.