On the Construction of Generalised Bobillier Formula

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.