For any symmetry group containing some glide reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group.
12.
For any symmetry group containing glide plane symmetry, the translation vector of any glide plane operation is one half of an element of the translation group.
13.
Where \ mathbf { R } is a 3 \ times 3 rotation matrix and \ mathbf { t } is a 3-dimensional translation vector.
14.
The rotation matrix \ mathbf { R } and the translation vector \ mathbf { t } have three degrees of freedom each, in total six.
15.
If each unit in a polymer is represented by a point in space, translation vectors \ vec { r } _ i connect between these joints.
16.
Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space.
17.
For comparison, the six parameters that define a spatial displacement can also be given by three Euler Angles that define the rotation and the three components of the translation vector.
18.
For example, in rigid registration, the output is a scale, a rotation matrix \ mathbf { R }, and a translation vector \ mathbf { t }.
19.
If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry.
20.
Combining two equal glide reflections 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.
