3. We have thus shown that translations and reflections and compositons thereof are invertible isometries. It is possible to show that a translation is the composition of two reflections. This would enable us to conclude that a translation is an invertible isometry. We will use that technique for rotations in the next section, but in the case of translations, it is much easier to derive these results analytically. The process for breaking up a translation into a composition of two reflections is very much like the process which will be used for rotations in the next section. A translation is essentially a rotation about a point at infinity.

Analytic Foundations of Geometry