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The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection. p11m [∞ +,2] C ∞h Z ∞ ×Dih 1 ∞* jump (THG) Translations, Horizontal reflections, Glide reflections: This group is generated by a translation and the reflection in the horizontal axis.
(A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation.
Combining two equal glide plane operations 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 the group generated by it. For ...
The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances. In 1D, all reflections are in the same class. In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class. In 3D:
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
the conjugation of a translation by a reflection is a translation by a reflected translation vector; Thus the conjugacy class within the Euclidean group E(n) of a translation is the set of all translations by the same distance. The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of all ...
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This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by G L,d, where L is a line in R 2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d.
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