Chasles' theorem (kinematics)

A screw axis. Chasles' theorem says that every Euclidean motion is a screw displacement along some screw axis.

In kinematics, Chasles' theorem says that the most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about that line.^[1][2]

History

The proof that a spatial displacement can be decomposed into a rotation and slide around and along a line in space is attributed to Michel Chasles in 1830.^[3] Recently the work of Giulio Mozzi has been identified as presenting a similar result in 1763.^[4][5]

Proof

An elementary proof of Chasles theorem was given by E. T. Whittaker in 1904.^[6] Suppose A is to be transformed into B. Whittaker suggests that line AK be selected parallel to the axis of the given rotation, with K the foot of a perpendicular from B. The appropriate screw displacement is about an axis parallel to AK such that K is moved to B. The method corresponds to Euclidean plane isometry where a composition of rotation and translation can be replaced by rotation about an appropriate center. In Whittaker's terms, "A rotation about any axis is equivalent to a rotation through the same angle about any axis parallel to it, together with a simple translation in a direction perpendicular to the axis."

References