Refining Geodesy and Photogrammetry with Geometric Algebra

Clifford's geometric algebras originally arise from theoretical physics, the most developed mathematically of the empirical sciences, where they provide a natural and elegant description of phenomena, such as classical mechanics and electrodynamics. Unfortunately, however, they are less well known to students nowadays, since already at the end of the 19th century they were replaced by the hollow formalism of vector analysis. Decades later Pauli and Dirac had to reinvent geometric algebras to write their famous equations, but the breakthrough that led to their triumphant return to the epicenter of active research came this time from technology: robotics, navigation, computer vision, animated simulations, etc. In view of these trends, a valuable opportunity opens up in many engineering sciences to upgrade and improve their mathematical description, optimize their algorithms, and significantly simplify the academic curricula. In this paper we approach some of the classical problems in geodesy and photogrammetry from the perspective of geometric algebras, which have proven rather effective in other engineering disciplines, such as robotics and computer vision. More precisely, we tackle problems like self-calibration and 3D reconstruction, trilateration, triangulation, and synchronization of reference frames, using dual quaternions, the Projective geometric algebra (PGA) and the more general Conformal geometric algebra (CGA). As expected, this approach provides a simpler and more elegant mathematical description, more suitable for both pedagogical reasons and algorithm efficiency. The results presented here are also meant to be used as a framework for future research targeting more advanced problems in those areas.