Euclidean isometry

A Euclidean isometry has the form $\mathbf{x}\mapsto \mathbf{a}+R\mathbf{x}$, where $R$ is orthogonal. It preserves inner products of displacement vectors, distances, angles, and arc length.

The vertex set of a triangle with three distinct side lengths has trivial setwise stabilizer subgroup in the plane isometry group. Each vertex is characterized by the unordered pair of lengths of its incident edges, so a setwise-preserving Euclidean isometry fixes every vertex. An isometry fixing three noncollinear plane points is the identity: squared Euclidean distances to them determine every other point, since subtracting two of the Euclidean distance equations gives two independent linear equations. For example, $(0,0),(1,0),(0,2)$ have side lengths $1,2,\sqrt{5}$.

In Euclidean geometry, a rotation about the origin is an orientation-preserving orthogonal linear map, represented by an element of the special orthogonal group. About another center $a$, the map has form $x\mapsto a+R(x-a)$. In three dimensions a nonidentity rotation fixes an axis.

The translation subgroup of a group of Euclidean isometries consists of its translations. It is a normal subgroup, because conjugation by $x\mapsto Ax+b$ takes translation by $v$ to translation by $Av$. Its vectors form an additive subgroup of Euclidean space; for a plane crystallographic group they form a rank-two translation lattice.

A translation lattice is the additive group generated by linearly independent vectors $v_1,\dots ,v_r$ in a Euclidean space. It defines a discrete subgroup of translations. A full-rank translation lattice has a compact parallelepiped as a fundamental region and therefore a cocompact group action. This geometric usage is distinct from an order-theoretic lattice.

A discrete subgroup of Euclidean isometries is a frieze group when its translation subgroup is infinite cyclic and has finite index. Translations along a line alone give $\mathbb{Z}$; adding perpendicular mirrors gives the infinite dihedral group, while adding a mirror along the translation direction gives $\mathbb{Z}\times C_2$. These groups describe patterns periodic in one planar direction.

A glide reflection of the plane composes reflection in a line with a nonzero translation parallel to that line. It reverses orientation and has no fixed point. It cannot be one reflection, because a reflection fixes a line, or a product of two reflections, because such a product preserves orientation. The finite reflection decomposition of a Euclidean isometry therefore shows that it requires exactly three reflections.

Every Euclidean isometry of ${\mathbb{R}}^n$ is a product of at most $n+1$ operations of reflection in a hyperplane. An isometry fixing zero is orthogonal: the polarization identity preserves inner products, and its values on an orthonormal basis determine its linear action. An orthogonal transformation is a product of at most $n$ linear reflections, by reflecting the image of the first basis vector back to that vector and inducting on its orthogonal complement. For an arbitrary isometry, first reflect its image of zero back to zero, if necessary, and apply the orthogonal result. Affine reflection hyperplanes, rather than only hyperplanes through zero, are essential here.

A product of $m$ reflections has $\mathrm{rank}(I-Q)\le m$, by the rank inequality applied successively to $I-AB=(I-A)+A(I-B)$. Conversely an orthogonal transformation fixes the orthogonal complement of its moved space; successive Householder reflections fix one new basis direction at each step in that moved space. At most its dimension $\mathrm{rank}(I-Q)$ are required. Thus central inversion in three dimensions needs exactly three reflections.