Inversion transformations with respect to conics in hybrid number planes

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.