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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.
Glide reflections, denoted by G c,v,w, where c is a point in the plane, v is a unit vector in R 2, and w is non-null a vector perpendicular to v are a combination of a reflection in the line described by c and v, followed by a translation along w. That is,
Rotations and translations do preserve handedness, which in 3D Plane-based GA implies that they can be written as a composition of an even number of reflections. A rotations can thought of as a reflection in a plane followed by a reflection in another plane which is not parallel to the first (the quaternions, which are set in the context of PGA ...
Now all reflections which map the pattern to itself are of the form a−x where the constant "a" is an integer (the increments of a are 1 again, because we can combine a reflection and a translation to get another reflection, and we can combine two reflections to get a translation). Therefore all isometries can be characterized by an integer ...
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:
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 ...
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.
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]