A novel geometric method based on conformal geometric algebra applied to the resection problem in two and three dimensions

• Published in: Journal of geodesy 2024-06, Vol.98 (6), Article 47
• Main Authors: Ventura, Jorge, Martinez, Fernando, Manzano-Agugliaro, Francisco, Návrat, Aleš, Hrdina, Jaroslav, Eid, Ahmad H., Montoya, Francisco G.
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Earth and Environmental Science; Earth Sciences; Geodesy; Geophysics/Geodesy; Navigation; Numerical simulations; Original Article
• ISSN: 0949-7714; 1432-1394
• DOI: 10.1007/s00190-024-01854-1

This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric understanding of the problem, in contrast to existing purely algebraic methods. Several numerical examples are presented to demonstrate the efficacy of the proposed method and to compare its validity with established techniques in the field. Numerical simulations indicate that our vector geometric algebra implementation is faster than the best-known algorithms to date, suggesting that the proposed GA-based methods can provide a more efficient and comprehensible solution to the two- and three-dimensional resection problem, paving the way for further applications and advances in geodesy research. Furthermore, the method’s emphasis on graphical and geometric representation makes it particularly suitable for educational purposes, allowing the reader to grasp the concepts and principles of resection more effectively. The proposed method has potential applications in a wide range of other fields, including surveying, robotics, computer vision, or navigation.