Glide Reflection

In geometry, a glide reflection is a type of isometry of the Euclidean 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 coirect 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 a point on 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.

Glide reflection symmetry with respect to two parallel lines with the same translation implies that there is also translational symmetry in the direction perpendicular to these lines, with a translation distance which is twice the distance between glide reflection lines. This corresponds to wallpaper group pg; with additional symmetry it occurs also in pmg, pgg and p4g.

If there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. A glide reflection line parallel to a true reflection line already implies this situation. This corresponds to wallpaper group cm. The translational symmetry is given by oblique translation vectors from one point on a true reflection line to two points on the next, supporting a rhombus with the true reflection line as one of the diagonals. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m and p6m.

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.

Glide symmetry can be observed in nature among certain fossils of the Ediacara biota; the machaeridians; and certain palaeoscolecid worms.

Read more about Glide Reflection:  Glide Reflection in Cellular Automata

Other articles related to "glide reflection, glide, reflection, reflections":

Glide Reflection in Cellular Automata
... Glide reflection is common in Conway's Game of Life ... glider, LWSS, MWSS and HWSS all move using glide reflection ... these spaceships appear to have been glide reflected to their new positions ...
Transformation (function) - Glide Reflection
... A glide reflection is a type of isometry of the Euclidean plane the combination of a reflection in a line and a translation along that line ... Depending on context, we may consider a simple reflection (without translation) as a special case where the translation vector is the zero vector ...
Cartesian Coordinate System - Cartesian Formulas For The Plane - Euclidean Transformations - Glide Reflection
... A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line ... of these operations does not matter (the translation can come first, followed by the reflection) ...
Euclidean Plane Isometry - Classification of Euclidean Plane Isometries
... a rotation around the origin is performed, followed by a translation Reflections, or mirror isometries, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2 ... The line L is called the reflection axis or the associated mirror ... the dot product to find the component t of p − c in the v direction, and then we obtain the reflection of p by subtraction, The combination of rotations about the origin and reflections about a line through ...

Famous quotes containing the words reflection and/or glide:

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