Glide Plane - Formal Treatment

Formal Treatment

In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector.

The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane. However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide plane operation in the general sense, with as special case just a reflection. The glide plane operation in the strict sense and the pure reflection are two of the four kinds of indirect isometries in 3D.

The isometry group generated by just a glide plane operation is an infinite cyclic group. Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide plane operation, so the even powers of the glide plane operation form a translation group.

In the case of glide plane symmetry, the symmetry group of an object contains a glide plane operation, and hence the group generated by it. 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. If the translation vector of a glide plane operation is itself an element of the translation group, then the corresponding glide plane symmetry reduces to a combination of reflection symmetry and translational symmetry.

See also lattice.