Example of a glide reflection

The combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. However, a glide reflection cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide reflection, with as special case just a reflection. These are the two kinds of indirect isometries in 2D.

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

(x, y) to (x + 1, −y).

It fixes a system of parallel lines.

The isometry group generated by just a glide reflection is an infinite cyclic group.

Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group.

In the case of glide reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type is Frieze group nr. 2.

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 C₂.

Example pattern with this symmetry group: + + + + + + + + +

For any symmetry group containing some glide reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry.

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

External links

• Glide Reflection at cut-the-knot

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