= Solution
An <isometry> of the <Euclidean plane> is a map $T:\mathbb R^2\to\mathbb R^2$ preserving all <Euclidean distances>, $\|T(x)-T(y)\|=\|x-y\|$. 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 \b[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.
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