 The Euclidean group of all rigid motions (isometries) of the plane (maps such that the Euclidean distance between and equals the distance between and for all and in ) is isomorphic to a semidirect product of the abelian group (which describes translations) and the group of orthogonal matrices (which describes rotations and reflections that keep the origin fixed).Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e.applying the conjugate of the original translation).