Classifying Isometries

An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry. We then classify all isometries of the plane equipped with the ${\mathrm{\ell }}_p$ metric for any $p\ge 1$ and $p=\mathrm{\infty }$.