From Quaternions to Möbius Transformations: A Relativistic Framework for Spacecraft Star Tracking

This paper introduces a mathematical framework that generalizes classic Euclidean spacecraft navigation algorithms to account for relativistic effects using Möbius transformations. Traditional star tracker algorithms rely on representations of celestial references as 3D vectors on the unit sphere and use tools such as quaternions, interstar angles, and the Davenport Q-method to estimate spacecraft attitude. These approaches implicitly assume Euclidean geometry and fail to accommodate Lorentz boosts, which become nonnegligible in high-velocity regimes. A novel correspondence between this classical toolkit and an alternative formulation grounded in complex analysis and special relativity is proposed, where the celestial sphere is treated as the Riemann sphere via stereographic projection. In this relativistic setting, Möbius transformations, 2 × 2 complex matrices with unit determinant, serve as analogs to quaternions, and cross-ratios replace interstar angles as projective invariants. The Davenport M-method, MOBEST, and M-TRIAD are presented, which are relativistic counterparts to the Q-method, QUEST, and TRIAD, respectively. These algorithms operate directly in the complex domain and estimate the attitude and velocity of the spacecraft via Möbius transformations. A detailed error analysis of the cross-ratio under focal-length and pixel perturbations is provided, and its robustness relative to interstar angles is demonstrated, particularly in constraining the star identification search space. Furthermore, a bijective mapping between quaternion + velocity pairs and Möbius transformations is derived, offering a bridge between Euclidean and relativistic kinematic models. This work establishes a foundational mathematical and algorithmic framework for relativistic star tracking, potentially enabling higher-accuracy attitude estimation in next-generation high-speed missions.