# 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**. Reversing the order of combining together with reversing the translation gives the same result.

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. The effect of a reflection combined with *any* translation is a glide reflection, with the component of the translation orthogonal to the line of reflection just causing it to shift to a parallel line.

Frieze group nr. 2 (glide-reflections and translations) is generated by just a glide reflection, with translations being obtained by combining two glide reflections. It is isomorphic to **Z**.

Example pattern with this symmetry group:

+++ + +++ + + + +++ +++ +++ + + + + +++ +

Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about an axis perpendicular to the line of reflection. It is isomorphic to a semi-direct product of **Z** and *C*_{2}.

Example pattern with this symmetry group:

+ + + + + + + + +

In 3D the glide reflection is called a **glide plane**. It is a reflection in a plane combined with a translation parallel to the plane. See also space group.

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

## External link

- Glide Reflection (requires Java)