# Classification of isometries in euclidean plane

Isometries in $\mathbb{R}^2$ are classified as rotations, translations, reflections and glide reflections. Is there a group theory rationale behind this grouping? In principle each of the infinite length preserving bijections is an isometry.

Also, when classifying the subgroups (frieze/wallpaper groups), how is the classification done? Frieze groups p1m1 and p1 are isomorphic. Yet they are categorized separately.

Let $f:\mathbb R^2\to\mathbb R^2$ be an isometry, and let $g(p)=f(p)-f(0,0)$. Note that $g$ is an isometry fixing the origin, so by the polarization identity, $$g(p_1)\cdot g(p_2)=\frac{\|g(p_1)\|^2+\|g(p_2)\|^2-\|g(p_1)-g(p_2)\|^2}{2}=\frac{\|p_1\|^2+\|p_2\|^2-\|p_1-p_2\|^2}{2}=p_1\cdot p_2$$ using the fact that $g$ preserves the Euclidean norm (i.e. distance to $(0,0)$) and preserves the distance from $p_1$ to $p_2$. Hence, $g$ preserves the inner product and is thus an orthogonal linear transformation. Since $O(2)$ consists of rotations and reflections, it follows that $f$ consists of a rotation or reflection followed by a translation. A non-identity rotation followed by a translation is a rotation, a reflection followed by a translation is either a reflection (if the translation is the identity) or a glide reflection, and the identity map (a rotation by $0^\circ$) followed by a translation is a translation.