Isometry

In Euclidean space with the usual distance function, the isometries can be characterized: there are no more than the 'expected' examples generated by rotations, reflections and translations. To put this more accurately, the isometries form a group, that is the semidirect product of the orthogonal group and the group of translations. (This group is sometimes called the Galilean group, at least for three dimensions and in relation with its role in Newtonian mechanics as expressed by permissible changes of frame of reference. See Galilean transformation.)

Within the isometry group of the plane, the product of a rotation and a translation can always be expressed as a single rotation (or translation). On the other hand the product of a reflection and a translation is usually not a reflection: it fixes a system of parallel lines, but is a glide reflection, a combination of a reflection in a line and a translation parallel to that line.