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2 Plane Symmetries or Isometries

A transformation of the plane (invertible map of the plane to itself) that preserves distances is called an isometry of the plane. Every isometry of the plane 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 point P;
a reflection in a line L;
a glide-reflection in a line L with displacement d.

Although the identity is a particular case of a translation and of a rotation, and reflections are particular cases of glide-reflections, it is more intuitive to consider each case separately.

Next: 2.1 Formulas for Symmetries in Cartesian Coordinates
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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.