Glide reflection

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 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

:("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 an axis perpendicular to the line of reflection. It is isomorphic to a semi-direct product of Z and "C"2.

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.

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 (geometry), similarity (mathematics), wallpaper group, frieze group.

External links

* [http://www.cut-the-knot.org/Curriculum/Geometry/GlideReflection.shtml Glide Reflection] at cut-the-knot


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • glide-reflection plane — veidrodinio slydimo plokštuma statusas T sritis fizika atitikmenys: angl. glide reflection plane vok. Gleitebene, f; Gleitspiegelebene, f rus. плоскость зеркального скольжения, f; плоскость скользящего отражения, f pranc. plan de réflexion… …   Fizikos terminų žodynas

  • glide reflection — …   Useful english dictionary

  • Glide plane — In crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged. Glide planes are noted by a , b or c , depending on which axis… …   Wikipedia

  • glide plane — noun : a crystallographic plane of symmetry that requires identity of the structure of the crystal with its original configuration following the combination of reflection of the crystal across the plane with movement of the structural… …   Useful english dictionary

  • glide plane — Crystall. a symmetry element of a space group such that a reflection of the lattice with respect to the plane and a translation of the lattice parallel to the plane bring the lattice back to its original position. [1890 95] * * * …   Universalium

  • false glide slope — A characteristic of the glide slope portion of the ILS (instrument landing system), in which one or more false glide slopes at different angles to the horizontal occur well above the true glide slope. This is because of a radiation pattern of the …   Aviation dictionary

  • Скользящая симметрия — Скользящая симметрия  изометрия евклидовой плоскости. Скользящей симметрией называют композицию симметрии относительно некоторой прямой и переноса на вектор, параллельный (этот вектор может быть и нулевым). Скользящую симметрию можно… …   Википедия

  • Wallpaper group — A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and… …   Wikipedia

  • Frieze group — A frieze group is a mathematical concept to classify designs on two dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The… …   Wikipedia

  • Euclidean plane isometry — In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections,… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”