Glide reflection - Definition and Overview

Example of a glide reflection

In geometry, a glide reflection is a type of isometry of the Euclidean plane.

Within the isometry group of the plane, the product of a rotation and a translation can always be expressed as a single rotation (or translation). On the other hand the product of a reflection and a translation is usually not a reflection, but can produce a transformation with no everyday name: a glide reflection.

For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In co-ordinates, it takes

(x, y) to (x + 1, −y).

It fixes a system of parallel lines, but is a combination of a reflection in a line and a translation parallel to that line. If one considers the effect of a reflection combined with any translation, it is a glide reflection with respect to a line parallel to the line of the reflection, as one sees by resolving the translation into components parallel and orthogonal to that line.

See also: congruence (geometry), similarity (mathematics), wallpaper group, frieze group.