Editing Glide reflection (section)

{{short description|Geometric transformation combining reflection and translation}}
[[File:Glide reflection.svg|right|thumb|A glide reflection is the composition of a reflection across a line and a translation parallel to the line.]]
[[File:Krok_6.svg|thumb|This footprint trail has glide-reflection symmetry. Applying the glide reflection maps each left footprint into a right footprint and vice versa.]]

In [[geometry]], a '''glide reflection''' or '''transflection''' is a [[geometric transformation]] that consists of a [[reflection (mathematics)|reflection]] across a [[hyperplane]] and a [[translation (geometry)|translation]] ("glide") in a direction [[parallel (geometry)|parallel]] to that hyperplane, combined into a single transformation. Because the distances between [[point (geometry)|points]] are not changed under glide reflection, it is a [[motion (geometry)|motion]] or [[isometry]]. When the context is the two-dimensional [[Euclidean plane]], the hyperplane of reflection is a [[straight line]] called the ''glide line'' or ''glide axis''. When the context is [[three-dimensional space]], the hyperplane of reflection is a plane called the ''glide plane''. The [[displacement vector]] of the translation is called the ''glide vector''.

When some geometrical object or configuration appears unchanged by a transformation, it is said to have [[symmetry (geometry)|symmetry]], and the transformation is called a [[symmetry operation]]. ''Glide-reflection symmetry'' is seen in [[frieze group]]s (patterns which repeat in one dimension, often used in decorative borders), [[wallpaper group]]s (regular [[tessellation]]s of the plane), and [[space group]]s (which describe e.g. [[crystal]] symmetries). Objects with glide-reflection symmetry are in general not [[reflection symmetry|symmetrical under reflection]] alone, but two applications of the same glide reflection result in a double translation, so objects with glide-reflection symmetry always also have a simple [[translational symmetry]].

When a reflection is composed with a translation in a direction perpendicular to the hyperplane of reflection, the composition of the two transformations is a reflection in a parallel hyperplane. However, when a reflection is composed with a translation in any other direction, the composition of the two transformations is a glide reflection, which can be uniquely described as a reflection in a parallel hyperplane composed with a translation in a direction parallel to the hyperplane.

A single glide is represented as [[frieze group]] p11g. A glide reflection can be seen as a limiting [[rotoreflection]], where the rotation becomes a translation.  It can also be given a [[Schoenflies notation]] as S<sub>2∞</sub>, [[Coxeter notation]] as [∞<sup>+</sup>,2<sup>+</sup>], and [[orbifold notation]] as ∞×.