On the Construction of Generalised Bobillier Formula

TL;DR

This paper extends the Bobillier formula to a generalized complex number system encompassing elliptical, parabolic, and hyperbolic cases, deriving the formula through geometric and kinematic methods for planar motion.

Contribution

It introduces a unified approach to derive the Bobillier formula in generalized complex planes using two distinct methods, broadening its applicability.

Findings

Derived the generalized Bobillier formula for different complex systems.

Presented two methods for deriving the formula: geometric interpretation and kinematic relations.

Unified the treatment of elliptical, parabolic, and hyperbolic cases in planar motion.

Abstract

In this study, we consider the generalized complex number system $C_{\rm{p}} = \left\{ {x + iy\;:\;x,y \in R,\;{i^2} = {\rm{p}} \in R} \right\}$ corresponding to elliptical complex number, parabolic complex number and hyperbolic complex number systems for the special cases of ${\rm{p}} < 0,\;{\rm{p}} = 0,\;{\rm{p}} > 0$, respectively. This system is used to derive Bobillier Formula in the generalized complex plane. In accordance with this purpose we obtain this formula by two different methods for one-parameter planar motion in ${C_{\rm{p}}}$; the first method depends on using the geometrical interpretation of the generalized Euler-Savary formula and the second one uses the usual relations of the velocities and accelerations.