Glide reflection

Under a glide reflection or push mirroring refers to the geometry of a special congruence. In the plane is the sequential execution of a parallel shift and a straight reflection, in which the displacement occurs parallel to the line. In a general vector space V a glide is defined as the sequential execution of a parallel displacement, and to a reflection on a hyperplane H in which the translation vector is parallel to H.

As Kongruenzabbildungen get glide reflections lengths, ie a " gleitgespiegelte " route is the same length as the original. Glide reflections are therefore isometries. However, glide reflections do not get the orientation of a figure.

Glide reflections play in discrete geometry a role, such as the classification of isometries in dimension 2 and 3, or in the investigation of band ornament groups.

In crystallography glide planes are possible symmetry elements of a space group.

Examples

Dimension 2

An affine hyperplane in the plane of the drawing is a straight line. In the two-dimensional geometry, a glide reflection is a mirror that is linked to an affine straight line with a translation parallel to this straight:

Isometries in Euclidean vector spaces of dimension 2 can be classified based on geometrical considerations. Within this classification, the glide is a total of five types. Other types are:

• Identity
• Translation
• Rotation
• Reflection

Dimension 3

The third dimension in space affine hyperplane is a plane. A glide that is reflected at an object plane, and here, the result is shifted in parallel thereto.

Also in euklidisichen vector spaces of dimension three can classify geometric isometries. The glide is here a total of seven types. A distinction continues:

• Reflection
• Translation
• Rotation
• Screwing
• Plane reflection
• Rotation-reflection

The glide plane as an element of a space group

In a space group only glide can occur, which are compatible with the translation lattice of the group. The two fold applying a pure reflection yields the identity. It follows that the two fold applying a glide reflection must result in a pure compatible with the lattice translation. For the combinations of reflection and translation, there are therefore only the following options:

In the case of axial and diagonal glide, it is obvious that the 2-fold translation vector leads back to a grid point.

Diamantgleitspiegelebenen exist only in orthorhombic F- centered tetragonal I- centered and cubic I and F- centered Bravais lattices. Twice the translation vector yields the vector describing the centering.