Chasles' theorem (kinematics)
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement.
A direct Euclidean isometry in three dimensions involves a translation and a rotation.
The screw displacement representation of the isometry decomposes the translation into two components, one parallel to the axis of the rotation associated with the isometry and the other component perpendicular to that axis.
The Chasles theorem states that the axis of rotation can be selected to provide the second component of the original translation as a result of the rotation.
This theorem in three dimensions extends a similar representation of planar isometries as rotation.
[1][2] Euclidean geometry is expressed in the complex plane by points
where the last expression shows the mapping equivalent to rotation at 0 and a translation.
as the center for an equivalent rotation, provided that
, that is, provided the direct isometry is not a pure translation.
As stated by Cederberg, "A direct isometry is either a rotation or a translation.
"[3] The proof that a spatial displacement can be decomposed into a rotation and slide around and along a line is attributed to the astronomer and mathematician Giulio Mozzi (1763), in fact the screw axis is traditionally called asse di Mozzi in Italy.
However, most textbooks refer to a subsequent similar work by Michel Chasles dating from 1830.
[4] Several other contemporaries of M. Chasles obtained the same or similar results around that time, including G. Giorgini, Cauchy, Poinsot, Poisson and Rodrigues.
An account of the 1763 proof by Giulio Mozzi and some of its history can be found here.
[5][6] Mozzi considers a rigid body undergoing first a rotation about an axis passing through the center of mass and then a translation of displacement D in an arbitrary direction.
Any rigid motion can be accomplished in this way due to a theorem by Euler on the existence of an axis of rotation.
The displacement D of the center of mass can be decomposed into components parallel and perpendicular to the axis.
The perpendicular (and parallel) component acts on all points of the rigid body but Mozzi shows that for some points the previous rotation acted exactly with an opposite displacement, so those points are translated parallel to the axis of rotation.
Another elementary proof of Mozzi–Chasles' theorem was given by E. T. Whittaker in 1904.
[7] 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 calculation of the commuting translation and rotation from a screw motion can be performed using 3DPGA (
), the geometric algebra of 3D Euclidean space.
representing orthogonal planes through the origin, and one Grassmanian basis vector
from the origin can then be formed as a linear combination
The result is a rotation around their intersection line
, which could also lie on the plane at infinity when the two reflections are parallel, in which case the bireflection
But according to the Mozzi-Chasles' theorem a screw motion can be decomposed into a commuting translation
the commuting translation and rotation can be found using the two formulae above, after which the lines
The Chasles' theorem is a special case of the Invariant decomposition.
