Journal of Fixed Point Theory and Applications

, Volume 12, Issue 1, pp 27-34

First online:

Chasles’ fixed point theorem for Euclidean motions

  • Bob PalaisAffiliated withDepartment of Mathematics, Utah Valley University Email author 
  • , Richard PalaisAffiliated withDepartment of Mathematics, RH 410H University of California at Irvine

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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

55M20 01A50


Twist screw motion Chasles