In this study, we will introduce arithmetical operations on dual hyperbolic numbers w x y ju jv      and hyperbolic complex numbers w x iy ju ijv     which forms a commutative ring. Then, we will investigate dual hyperbolic number valued functions and hyperbolic complex number valued functions. One can see that these functions have similar properties.

We expressed the higher-order velocities, accelerations, and poles under the one-parameter planar hyperbolic motions and their inverse motions. The higher-order accelerations and poles are also presented by considering the rotation angleφas a parameter of the motion and its inverse motion.

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