One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane $${\mathbb{C}_{J}}$$ C J

Abstract

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) $$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$ where $$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathbb{R}, \quad J^2=\mathfrak{p}\}$$ such that $${-\infty < \mathfrak{p} < \infty}$$ . The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in $${\mathbb{C}_{J}}$$ is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system.