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 $R_{a_1,a_2,a_3}^{2,1}$. The hyperbolic geometric (Berry) phase models are generated through the pseudo-spheres of $R_{a_1,a_2,a_3}^{2,1}$. 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 ($\mathbf{E}$M-curves) associated with the electric field $\mathbf{E}$ are determined via the hyperbolic geometric phase models in the optical fiber. Furthermore, some motivating examples are presented by using the MAPLE program.

本文旨在通过双曲分裂四元数代数研究沿线偏振光波在光纤中传播的偏振面的双曲线运动。电场（极化矢量）绕一般双曲面轴的运动由洛伦兹标量积空间描述${\mathrm{电阻}}_{{\mathrm{一种}}_1,{\mathrm{一种}}_2,{\mathrm{一种}}_3}^{2,1}$. 双曲几何 (Berry) 相位模型是通过伪球体生成的${\mathrm{电阻}}_{{\mathrm{一种}}_1,{\mathrm{一种}}_2,{\mathrm{一种}}_3}^{2,1}$. 然后，通过双曲分裂四元数积和单参数相似运动获得 Rytov 曲线的参数表示。然后，电场的双曲线运动由费米-沃克平行输运定律表示。此外，电磁曲线（$\mathbf{乙}$M 曲线）与电场相关 $\mathbf{乙}$由光纤中的双曲线几何相位模型确定。此外，还通过使用 MAPLE 程序提供了一些激励示例。