- Glide reflection
In

geometry , a**glide reflection**is a type ofisometry of theEuclidean plane : the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector.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 infinitecyclic 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**, thesymmetry group of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type isFrieze 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 "C"_{2}.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 andtranslational symmetry .In 3D the glide reflection is called a

. It is a reflection in a plane combined with a translation parallel to the plane.glide plane See also:

congruence (geometry) ,similarity (mathematics) ,wallpaper group ,frieze group .**External links*** [

*http://www.cut-the-knot.org/Curriculum/Geometry/GlideReflection.shtml Glide Reflection*] atcut-the-knot

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### Look at other dictionaries:

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