Year: 2025
Type: article
Abstract: This paper presents a comprehensive study of geometric inversion with respect to central conics in hybrid number planes, which unify complex, hyperbolic, and dual numbers within a single algebraic structure. By employing the hybrid scalar product and the associated pseudo-Euclidean metric, the hybridian planes were classified as elliptic, hyperbolic, or parabolic. Explicit inversion formulas were derived for points, lines, and conics in each plane type. It was shown that lines passing through the inversion center remain invariant, while others transform into conics. Homothetic conics preserve their type under inversion, whereas non-homothetic conics yield cubic or quartic curves depending on their relation to the inversion center. These results extend classical inversion geometry into a unified hybrid setting, providing a new framework for geometric transformations in generalized number systems.
Source: AIMS Mathematics
DOI: 10.3934/math.2025651
Authors: İskender Öztürk, Hasan Çakır, Mustafa Özdemir
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,200
OpenAlex ID: W4411607133