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

TL;DR

This paper extends the Euler-Savary formula to one-parameter hyperbolic planar motion, analyzing velocity relationships and defining a canonical system in hyperbolic geometry.

Contribution

It introduces a new formulation of the Euler-Savary formula for hyperbolic motions and establishes velocity relations in hyperbolic plane movements.

Findings

Derived the relation between absolute, relative, and sliding velocities in hyperbolic motion.

Defined a canonical relative system for hyperbolic planar motion.

Established the Euler-Savary formula in the context of hyperbolic geometry.

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.