Glide reflection
A sliding mirror or shear mirror is understood to mean a special congruence mapping in geometry . In the plane is the sequential execution of a parallel displacement and a straight mirroring , in which the displacement takes place parallel to the straight line. In a general vector space V a glide is defined as the sequential execution of a parallel displacement and a reflection at a hyperplane H defined, in which the translation vector parallel to H is.
As congruence images, sliding reflections are given lengths, ie a “sliding mirrored” segment is just as long as the original. Glide reflections are therefore isometrics . However, glide reflections do not get the orientation of a figure.
Glide reflections play a role especially in discrete geometry, for example when classifying isometrics in dimensions 2 and 3 or when examining ribbon ornament groups .
In crystallography, gliding mirror 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 two-dimensional geometry, a sliding reflection is a reflection on an affine straight line combined with a translation parallel to this straight line:
Isometrics in Euclidean vector spaces of dimension 2 can be classified according to geometrical criteria. Within this classification, glide reflection is one of a total of 5 types. Other types are:
Dimension 3
In third-dimensional spaces, an affine hyperplane is a plane. A sliding reflection reflects an object here on a plane and shifts the result parallel to this.
Isometrics can also be classified geometrically in Euclidean vector spaces of dimension three. The sliding reflection is one of a total of 7 types. One further distinguishes:
The glide plane as an element of a space group
A space group can only contain glide mirror planes that are compatible with the translation grid of the group. The double execution of a pure reflection results in the identity. From this it follows that the double execution of a glide mirror must result in a pure translation compatible with the grating. There are therefore only the following options for the combinations of mirroring and translation:
description | Direction perpendicular to the mirror plane | Translation vector | Hermann Mauguin symbol |
---|---|---|---|
Axial gliding mirror plane | [010]; [001] | a | |
b | |||
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c | ||
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Diagonal sliding mirror plane | n | ||
Diamond sliding mirror plane | d | ||
In the case of the axial and diagonal gliding mirror planes, it is obvious that the 2-fold translation vector leads again to a grid point.
Diamond sliding mirror planes only exist in orthorhombic F-centered, tetragonal I-centered and cubic I- and F-centered Bravais gratings. The double translation vector gives the vector describing the centering.
literature
- Hans Schupp: Elementarge Geometry , UTB Schoeningh (1977)
- Schwarzenbach D. Kristallographie , Springer Verlag, Berlin 2001, ISBN 3-540-67114-5