Generation of a shape model in terms of spherical harmonics

Solar system bodies such as planets, asteroids, and comets are increasingly becoming targets of satellite missions. These bodies typically exhibit irregular shapes, and generating shape models using spherical harmonics can be valuable for studying their physical properties. Beyond geodesy, spherical harmonic modeling is widely used in other scientific fields, including physics, geophysics, climate and weather science, medical imaging, chemistry, and engineering. In this work, we present two methods for generating shape models, a process commonly referred to as spherical harmonic analysis and synthesis. First, the classical least-squares method is introduced, both in its basic formulation and in combination with auxiliary algebraic techniques. Second, the well-known Neumann method is employed to compute the spherical harmonic coefficients. The aim of this study is to evaluate and compare these methods in terms of precision, computational efficiency, and simplicity. Finally, the performance of both approaches is demonstrated through numerical applications to celestial bodies.