Electromagnetic curves and Rytov curves based on the hyperbolic split quaternion algebra

This paper aims to investigate the hyperbolically the motion of the polarization plane traveling along the linearly polarized light wave in the optical fiber via the hyperbolic split quaternion algebra. The motion of the electric field (polarization vector) about an axis on the general hyperboloid is described by the Lorentzian scalar product space Ra1,a2,a32,1. The hyperbolic geometric (Berry) phase models are generated through the pseudo-spheres of Ra1,a2,a32,1. Then, the parametric representations of the Rytov curves are obtained via the hyperbolic split quaternion product and one-parameter homothetic motion. Then, the hyperbolically motion of the electric field is expressed by the Fermi–Walker parallel transportation law. Moreover, the electromagnetic curves (EM-curves) associated with the electric field E are determined via the hyperbolic geometric phase models in the optical fiber. Furthermore, some motivating examples are presented by using the MAPLE program.