Chasles’ fixed point theorem for Euclidean motions

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Abstract

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 

Keywords

Twist screw motion Chasles 

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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

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