Let $V$ and $W$ be Euclidean vector spaces. A Euclidean transformation is an affine transformation $E:V\to W$ given by $$E(v)=L(v)+w$$ such that $L$ is an orthogonal linear transformation.

As an affine transformation, all affine properties, such as incidence and parallelism are preserved by $E$ In addition, since $E(u-v)=L(u-v)$ and $L$ is an orthogonal linear transformation, $E$ preserves lengths of line segments and angles between two line segments. Because of this, a Euclidean transformation is also called a rigid motion, which is a popular term used in mechanics.

Types of Euclidean transformations

1. translation. If $L=I$ then $E$ is just a translation. Any Euclidean transformation can be decomposed into a product of an orthogonal transformation $L(v)$ followed by a translation $T(v)=v+w$
2. rotation. If $w=0$ then $E$ is just an orthogonal transformation. If $\operatorname{det}(E)=1$ then $E$ is called a rotation. The orientation of any basis (of $V$ is preserved under a rotation. In the case where $V$ is two-dimensional, a rotation is conjugate to a matrix of the form \begin{eqnarray} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, \end{eqnarray}where $\theta\in \mathbb{R}$ Via this particular (unconjugated) map, any vector emanating from the origin is rotated in the counterclockwise direction by an angle of $\theta$ to another vector emanating from the origin. Thus, if $E$ is conjugate to the matrix given above, then $\theta$ is the angle of rotation for $E$
3. reflection. If $w=0$ but $\operatorname{det}(E)=-1$ instead, then $E$ is a called reflection. Again, in the two-dimensional case, a reflection is conjugate to a matrix of the form \begin{eqnarray} \begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{pmatrix}, \end{eqnarray}where $\theta\in \mathbb{R}$ Any vector is reflected by this particular (unconjugated) map to another by a ``mirror'', a line of the form $y=x\tan(\frac{\theta}{2})$

Remarks.

• In general, an orthogonal transformation can be represented by a matrix of the form $$ \begin{pmatrix} A_1 & O & \cdots & O \\ O & A_2 & \cdots & O \\ \vdots & \vdots & \ddots & \vdots \\ O & O & \cdots & A_n \end{pmatrix}, $$ where each $A_i$ is either $\pm 1$ or a rotation matrix (1) (or reflection matrix (2)) given above. When its determinant is -1 (a reflection), it has an invariant subspace of $V$ of codimension 1. One can think of this hyperplane as the mirror.
• Another common rigid motion is the glide reflection. It is a Euclidean transformation that is expressible as a product of a reflection, followed by a translation.