Revisiting the Hansen Problem: A Geometric Algebra Approach

Jorge Ventura, Fernando Martinez, Isiah Zaplana, Ahmad Hosny Eid, Francisco G. Montoya () and James Smith
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Jorge Ventura: Department of Engineering, University of Almería, 04120 Almeria, Spain
Fernando Martinez: Department of Engineering, University of Almería, 04120 Almeria, Spain
Isiah Zaplana: Institute of Industrial and Control Engineering, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
Ahmad Hosny Eid: Department of Electrical Engineering, Faculty of Engineering, Port-Said University, Port Fouad 42526, Egypt
Francisco G. Montoya: Department of Engineering, University of Almería, 04120 Almeria, Spain
James Smith: Independent Researcher, San Cristóbal de las Casas 29200, Mexico

Mathematics, 2024, vol. 12, issue 13, 1-19

Abstract: The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms.

Keywords: Hansen problem; geometric algebra; resection (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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