Glide reflection

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File:Glide reflection.png
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. 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 C2.

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.

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