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]	${\ displaystyle {\ frac {1} {2}} {\ vec {a}}}$	a
	${\ displaystyle [001]; [100]}$	${\ displaystyle {\ frac {1} {2}} {\ vec {b}}}$	b
	${\ displaystyle [100]; [010]}$	${\ displaystyle {\ frac {1} {2}} {\ vec {c}}}$	c
	${\ displaystyle [1 {\ bar {1}} 0]; [110]}$
	${\ displaystyle [100]; [010]; [{\ bar {1}} {\ bar {1}} 0]}$
	${\ displaystyle [1 {\ bar {1}} 0]; [120]; [{\ bar {2}} {\ bar {1}} 0]}$
Diagonal sliding mirror plane	${\ displaystyle [001]; [100]; [010]}$	${\ displaystyle {\ frac {1} {2}} ({\ vec {a}} + {\ vec {b}}) \,; \, {\ frac {1} {2}} ({\ vec { b}} + {\ vec {c}}) \,; \, {\ frac {1} {2}} ({\ vec {a}} + {\ vec {c}})}$	n
	${\ displaystyle [1 {\ bar {1}} 0]; [01 {\ bar {1}}]; [{\ bar {1}} 01]}$	${\ displaystyle {\ frac {1} {2}} ({\ vec {a}} + {\ vec {b}} + {\ vec {c}})}$
	${\ displaystyle [110]; [011]; [101]}$	${\ displaystyle {\ frac {1} {2}} (- {\ vec {a}} + {\ vec {b}} + {\ vec {c}}) \,; \, {\ frac {1} {2}} ({\ vec {a}} - {\ vec {b}} + {\ vec {c}}) \,; \, {\ frac {1} {2}} ({\ vec {a }} + {\ vec {b}} - {\ vec {c}})}$
Diamond sliding mirror plane	${\ displaystyle [001]; [100]; [010]}$	${\ displaystyle {\ frac {1} {4}} (- {\ vec {a}} \ pm {\ vec {b}}) \,; \, {\ frac {1} {4}} ({\ vec {b}} \ pm {\ vec {c}}) \,; \, {\ frac {1} {4}} (\ pm {\ vec {a}} + {\ vec {c}})}$	d
	${\ displaystyle [1 {\ bar {1}} 0]; [01 {\ bar {1}}]; [{\ bar {1}} 01]}$	${\ displaystyle {\ frac {1} {4}} ({\ vec {a}} + {\ vec {b}} \ pm {\ vec {c}}) \,; \, {\ frac {1} {4}} (\ pm {\ vec {a}} + {\ vec {b}} + {\ vec {c}}) \,; \, {\ frac {1} {4}} ({\ vec {a}} \ pm {\ vec {b}} + {\ vec {c}})}$
	${\ displaystyle [110]; [011]; [101]}$	${\ displaystyle {\ frac {1} {4}} (- {\ vec {a}} + {\ vec {b}} \ pm {\ vec {c}}) \,; \, {\ frac {1 } {4}} (\ pm {\ vec {a}} - {\ vec {b}} + {\ vec {c}}) \,; \, {\ frac {1} {4}} ({\ vec {a}} \ pm {\ vec {b}} + - {\ vec {c}})}$

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