On the instantaneous acceleration of points in a rigid body

This work re-examines some classical results in the kinematics of points in space using modern vector–matrix methods. In particular, some very simple Lie theory allows the velocities and accelerations of points to be found in terms of the instantaneous twist of the motion and its derivative. From these results many of the classical results follow rather simply.Although most of the results are well known, some new material is presented. In particular, the discriminant curve that separates cases with one or three real acceleration axes is found and plotted. Another new result concerns the chords to the cubic of inflexion points. It is shown that for points on such a chord the osculating planes of the point's trajectories are parallel. Also a new result is found which distinguishes between cases where the Bresse hyperboloid of points whose velocities and accelerations are perpendicular, has one or two sheets.

Research highlights
► Old subject of instantaneous kinematics revisited with matrix–vector formalism. Classical results found in simple manner.
► New results found include characterisation of motion where one or three acceleration axes exist.
► Connexion between the chords to the inflexion cubic and the osculating planes to the trajectories of points also new.
► Several other new results including conditions for the Bresse hyperboloid to have one or two sheets.