Searching for a Solid Proof of Chasles/Mozzi/Cauchy's Theorem

In summary, the conversation is about the search for a proof of Chasles' Theorem, also known as Mozzi's or Cauchy's screw theorem, which states that any rigid body displacement can be broken down into a screw motion. The poster is looking for a solid, algebraic proof and mentions finding a good paper by Dunham Jackson that explains rigid body motions and their decompositions into rotations and translations. The conversation also includes a request for a link to a free copy of the paper.
  • #1
ObsessiveMathsFreak
406
8
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.
 
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  • #2
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.
 
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  • #3
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
EFuzzy said:
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?
Attached is a copy of the paper.
 

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  • Instantaneous Rigid Body Motion.pdf
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  • #5
Thanks!
 

1. What is Chasles/Mozzi/Cauchy's Theorem?

Chasles/Mozzi/Cauchy's Theorem is a fundamental theorem in projective geometry, which states that any projective transformation can be represented as a composition of a finite number of collineations. It was first proposed by mathematicians Jean-Victor Poncelet and Augustin-Louis Cauchy in the 19th century.

2. Why is it important to find a solid proof of this theorem?

The proof of Chasles/Mozzi/Cauchy's Theorem has been a topic of debate among mathematicians for many years. A solid proof would not only validate the theorem itself, but also provide new insights and understanding in the field of projective geometry. It could also potentially lead to new developments and applications in other branches of mathematics.

3. What are the challenges in finding a solid proof of this theorem?

One of the main challenges in proving Chasles/Mozzi/Cauchy's Theorem is the complex nature of projective geometry. The concepts and principles involved are not easily understood and require advanced mathematical knowledge. Additionally, there may be gaps or errors in previous attempts at proving the theorem, making it difficult to build upon existing work.

4. What progress has been made in the search for a solid proof?

Over the years, many mathematicians have attempted to prove Chasles/Mozzi/Cauchy's Theorem, but no universally accepted proof has been found yet. However, there have been significant advancements in related fields, such as using algebraic methods to tackle the problem. Some partial proofs have also been proposed, but they have not been able to fully prove the theorem.

5. What are the potential implications of a solid proof of this theorem?

A solid proof of Chasles/Mozzi/Cauchy's Theorem would have significant implications in the field of projective geometry and mathematics as a whole. It would provide a deeper understanding of the underlying principles and potentially open up new avenues for research. It could also have practical applications in areas such as computer graphics and machine learning.

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