newtonian mechanics - How is Chasles' Theorem, that any rigid displacement can be produced by translating along a line and then rotating about the same line, true?

Chasles' Theorem in its strong form says:

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 same line.

Now, Euler's Theorem simply says that any rigid body displacement can be decomposed into a rotation plus translation. This is easy to visualize. But what Chasles' Theorem says is something much stronger.

Unfortunately, I am just not able to visualize it. Perhaps, I am comprehending it wrong. I mean how is it possible to have the axis of rotation and translation the same (or parallel) for the most general displacement.

I mean, think of this case:

A body is given a finite rotation about the X-axis and then a finite translation about the Z-axis. How can we find that "screw" axis along which both of them can be described?

Answer

Keep in mind that the screw axis does not have to pass through the body. For your example place the axis of rotation parallel to the x axis straight above the cylinder, then rotate the cylinder 180° about it. The result will be equivalent to 180° rotation about the x axis followed by a translation along the z axis by twice the distance from the x axis to the axis of rotation. So for this composition a single rotation along another axis suffices. To get a smaller rotation angle keep the rotation axis parallel to the x axis above the cylinder, but move it along the y axis (into the background of the picture). The further back you move it the smaller the rotation angle about it needed to put the cylinder back on the z axis above its current position. It will end up rotated by that same angle about its own axis.

It is a general observation (of Whittaker's) that composition of a rotation and a translation perpendicular to its axis is a rotation by the same angle about a parallel axis. When rotation and translation axes are not perpendicular we decompose the translation into perpendicular and parallel components relative to the rotation axis. The rotation and the perpendicular translation can then be replaced as above by a single rotation about an axis parallel to the rotation axis. And since translations along parallel axes are equivalent the parallel component can be done along the same axis.

This is Chasles' a.k.a. Mozzi's theorem. See detailed construction and mathematical proof in Jackson's Instantaneous Motion of a Rigid Body.