The transformation may be decomposed into a translation vector and a transformation matrix:
2.
However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that.
3.
The algorithm only computes the rotation matrix, but it also requires the computation of a translation vector.
4.
However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that.
5.
The end-to-end vector \ vec { R } is the sum of these translation vectors:
6.
For the translation vector this only causes a change of sign, which has already been described as a possibility.
7.
Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre.
8.
One commonly says that this affine subspace has been obtained by translating ( away from the origin ) the linear subspace by the translation vector.
9.
All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice.
10.
This is because the conjugate of the translation by a glide reflection is the same as by the corresponding reflection : the translation vector is reflected.
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