Glide reflection
In 2-dimensional geometry, a glide reflection (or transflection) is a type of opposite isometry of the Euclidean plane: the composition of a reflection in a line and a translation along that line.
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 S2∞, Coxeter notation as [∞+,2+], and orbifold notation as ∞×.
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
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 p11g.
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 a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C2.
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