## 1 Introduction
For instance, China's \"Magpie Bridge\" mission in 2018, which orbits in a halo configuration around L2 in cislunar space, serves as a vital relay satellite for communication with Earth. NASA's Artemis mission also has plans to establish the Lunar Orbital Platform-Gateway (LOP-G) around L1 [14; 15; 16]. Presently, research pertaining to the L1 point within the Earth-Moon system primarily centers on lunar station development, space gateway exploration, low-energy transfer orbits, and navigation service systems [7; 17; 18; 19; 20; 21]. Recent endeavors have also explored the potential of Earth observation from the L1 point. For example, [PERSON] et al. introduced a novel Earth observation system featuring a radiometer designed to capture Earth's radiation budget, operating from the L1 point within the Earth-Moon system, yielding simulation results that exhibit high-quality imaging [22]. However, it is worth noting that the three collinear Lagrange points (L1, L2, and L3) along the Earth-Moon line are inherently unstable, requiring spacecraft to employ active control to remain in position, whereas the equilateral Lagrange points (L4, L5) are stable [23]. This inherent instability suggests that spacecraft are more likely to occupy periodic orbits, such as halo orbits, rather than a precise L1 point position [24]. Thus, this study aims to conduct a more scientifically rigorous analysis of the general characteristics of Earth observation, introducing the concept of halo orbits around L1 as opposed to relyingon the simplification that sensors remain static at the L1 point. Additionally, considering the changing distance between the Moon and Earth, which fluctuates from approximately 363,000 km to 405,000 km over time, this study employs ephemeris data to describe the Earth-Moon movement rather than relying on a simplified, uniform circular motion, thereby reducing potential errors in orbit determination [25]. Moreover, we conduct numerical simulations employing various time steps and periods following geometric analysis to effectively demonstrate accurate characteristics. Furthermore, due to the similarity in motion between the Moon and the L1 point within the Earth-Moon system--both revolving around the Earth at the same angular velocity and maintaining a close proximity, approximately 0.15 times the Earth-Moon distance--it is plausible that there are both differences and similarities in their Earth observation capabilities. While a substantial body of the literature has explored Earth observation from Moon-based platforms [6; 26; 27], minimal attention has been devoted to studying the Lagrange points in cisluunar space, which represent unique spatial resources and an ideal location for lunar exploration, offering distinctive potential as a supplementary Earth observation platform. Thus, this paper undertakes a comparative analysis, considering periodic halo orbits around L1 within the Earth-Moon system and elucidating the distinctions in Earth observation characteristics between Moon-based platforms and the L1 point. The remainder of this paper is structured as follows: Section 2 introduces the data used in our analysis, including JPL ephemeris data and methods for calculating periodic halo orbits. The subsequent section elucidates the primary coordinate systems employed in this study, alongside the derivation of transformation matrices that facilitate the calculation of relative motions between the sensor and ground targets. Building upon the geometric model of the halo orbit and the Earth's surface, we derive pertinent geometric parameters. Section 3 conducts a numerical analysis of the variation in radiir point tracks, as characterized by the proposed temporal complexity curve--a critical factor influencing other Earth observation characteristics, such as the distribution of viewing angles and spatiotemporal ground coverage. To facilitate comparison, we also analyze and present data for Moon-based platforms.