 Solution

An isometry of the Euclidean plane is a map $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ preserving all Euclidean distances, $\Vert T(x)-T(y)\Vert =\Vert x-y\Vert$. Such a map of the whole plane is automatically bijective, and has the form $T(x)=Rx+b$ with $R$ an orthogonal matrix.

The classification is identity, translations, rotations about a point, reflections in a line, and glide reflections. Here a glide reflection is a reflection followed by a nonzero translation parallel to its mirror. The cases preserving orientation have $\det R=1$ and are translations or rotations; the orientation-reversing cases have $\det R=-1$ and are reflections or glide reflections. The identity is included by allowing a zero translation or rotation angle.