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10 Space Symmetries or Isometries

A transformation of space (invertible map of the plane to itself) that preserves distances is called an isometry of space. Every isometry of space is of one of the following types:

the identity (which leaves every point fixed);
a translation by a vector v;
a rotation through an angle around a line L;
a screw motion through an angle around a line L, with displacement d;
a reflection in a line P;
a glide-reflection in a line P with displacement vector v.

The identity is a particular case of a translation and of a rotation; rotations are particular cases of screw motions; and reflections are particular cases of glide-reflections. However, as in the plane case, is more intuitive to consider each case separately.

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Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.