See Also

Glide reflection

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a glide reflection is a type of isometry of the Euclidean plane Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek [i] mathematician [i] Euclid [i] ... 

: the combination of a reflection in a line and a translation Translation

Translation is an activity comprising the interpretation [i] of the meaning [i] of a text in on ... 

 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.

Discussions

  Discussion Features

   Ask a question about 'Glide reflection'

   Start a new discussion about 'Glide reflection'

   Answer questions about 'Glide reflection'

   'Glide reflection' discussion forum


Encyclopedia


In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a glide reflection is a type of isometry of the Euclidean plane Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek [i] mathematician [i] Euclid [i] ... 

: the combination of a reflection in a line and a translation Translation

Translation is an activity comprising the interpretation [i] of the meaning [i] of a text in on ... 

 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

to .


It fixes a system of parallel lines.

The isometry group generated by just a glide reflection is an infinite cyclic group Cyclic group

In group theory [i], a cyclic group or monogenous group is a group [i] that can be generated [i] ... 

.

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 Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ... 

 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 Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repet... 

 nr. 2.

Example pattern with this symmetry group:
+++ + +++
+ + +
+++ +++ +++
+ + +
+ +++ +
Frieze group nr. 6 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:
+ + + +
+ + + + +

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 Reflection symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry [i] ... 

 and translational symmetry.

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: congruence , similarity , wallpaper group Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the... 

, frieze group Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repet... 

.

External link


  • at cut-the-knot




Categories: