The Euler-Savary Formula for One-Parameter Planar Hyperbolic Motion

Abstract

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-parameter planar motions in the Euclidean plane $\mathbb{E}^2$ and the Euler-Savary formula in one-parameter planar motions were given by M$\ddot{\texttt{u}}$ller, \cite{Mul}. In the present article, one hyperbolic plane moving relative to two other hyperbolic planes, one moving and the other fixed, was taken into consideration and the relation between the absolute, relative and sliding velocities of this movement was obtained. In addition, a canonical relative system for one-parameter hyperbolic planar motion was defined. Euler-Savary formula, which gives the relationship between the curvature of trajectory curves, was obtained with the help of this relative system.