# Chasles Theorem

1. Jan 4, 2008

### ObsessiveMathsFreak

I've unsuccessfully been looking for a decent proof of Chasles' Theorem which states that any rigid body displacement whatsoever can be decomposed into a screw motion. In other words, no matter what the displacement is, you can consider it the result of the partiles having moved to their positions by following a circular helixical path about a common axis, with a common angular speed.

I suppose I'm mostly looking for a (linear) algebraic type proof. Most proofs I've encountered have been fairly loose and unconvincing. Does anyone know of a solid proof of this theorem?

Awkwardly, this theorem is also variously known as Mozzi's or Cauchy's screw theorem.

2. Jan 4, 2008

### ObsessiveMathsFreak

In answer to my own question:

Shortly after making this post I stumbled across one of the best papers I've ever read.
http://www.jstor.org/view/00029890/di991259/99p1550p/0 by Dunham Jackson. It's old, but great, and completely and totally explains rigid body motions and their decompositions into rotations and translations and finally into screw motions. He refers to the theorem as Mozzi's theorem.

It's a great paper. I'd seriously recommend anyone to give it an hour of their time. It's an easy read, and the proofs are just brilliant. The only thing I could fault it on is a lack of diagrams.

Last edited by a moderator: Apr 23, 2017
3. Jan 4, 2008

### EFuzzy

Thanks for the link, this is a problem I've been annoyed with for a while too. I've worked out a semi-proof for the 2d case using similar triangles, but I would like to see the 3d case as well. Do you have a link for a free copy of the paper?

4. Jan 5, 2008

### lugita15

Attached is a copy of the paper.

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5. Jan 5, 2008

Thanks!