One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

TL;DR

This paper extends the concept of one-parameter homothetic motion from the complex plane to the hyperbolic plane, deriving velocity, acceleration, and Euler-Savary formulas in this new geometric context.

Contribution

It introduces a new framework for homothetic motion in the hyperbolic plane and derives related kinematic properties and the Euler-Savary formula within this setting.

Findings

Velocity and acceleration vector properties in hyperbolic homothetic motion

Characterization of pole curves in the hyperbolic plane

Euler-Savary formula adapted to hyperbolic geometry

Abstract

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite{Mul}. One-parameter planar homothetic motion was defined in the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion is defined in the Hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale $h$ identically equal to 1, the results given in \cite{Yuc} are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.