Rigid Motions of the Plane

A rigid motion of the plane is a map m: 2 → 2 where m is distance preserving. In our thesis, we explore the group M of all rigid motions of a plane and various subgroups of M. We begin by defining four types of rigid motions: translation, rotation, reflection, and glide reflection. We then prove the theorem: Every rigid motion is either a translation, rotation, reflection, glide reflection, or the identity. Our proof relies on, but is slightly different than, the proof in Artin’s Algebra book. In connection with this theorem Artin states that the composition of rotations about two different points is a rotation about a third point, unless it is a translation: we determine a concrete formula for the center and angle of ration when the final composition results in a rotation. We finish by moving our discussing into three spaces