Journal of Fixed Point Theory and Applications

, Volume 12, Issue 1, pp 27–34

Chasles’ fixed point theorem for Euclidean motions


DOI: 10.1007/s11784-012-0077-0

Cite this article as:
Palais, B. & Palais, R. J. Fixed Point Theory Appl. (2012) 12: 27. doi:10.1007/s11784-012-0077-0


Chasles’ theorem, a classic and important result of kinematics, states that every orientation-preserving isometry of \({\mathbb{R}^3}\) is a screw motion. We show that this is equivalent to the assertion that each proper Euclidean motion that is not a pure translation, acting on the space of oriented lines, has a unique fixed point (the axis of the screw motion). We use that formulation to derive a simple and novel constructive proof of Chasles’ theorem.

Mathematics Subject Classification



Twistscrew motionChasles

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUtah Valley UniversityOremUSA
  2. 2.Department of MathematicsRH 410H University of California at IrvineIrvineUSA