Theoretical analysis demonstrates explicit hyperbolic rotation matrices in three-dimensional space, indicating a consistent algebraic framework for Lorentzian geometry.
This paper presents a geometric and algebraic analysis of hyperbolic hybrid numbers, which combine structural features of the complex, dual, and hyperbolic number systems. Based on the algebraic properties of the hybrid number set, hyperbolic rotations in both the plane and space are investigated. The study constructs mutually orthogonal hyperbolic planes in the hybrid setting and examines the left‐hand, right‐hand, and sandwich multiplication maps that produce hyperbolic rotational transformations. In addition, Rodrigues and Cayley type formulas are formulated within the hybrid number system to obtain explicit hyperbolic rotation matrices in three dimensions. These results show that hyperbolic hybrid numbers provide a consistent algebraic setting for modeling Lorentzian‐type geometries and clarify the relationship between algebraic operations and geometric motions.
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Öztürk et al. (2026) studied this question.
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