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Euclidean transformation
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 $E(uv)=L(uv)$ 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
There are three main 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 twodimensional, a rotation is conjugate to a matrix of the form
$\displaystyle\begin{pmatrix}\cos\theta&\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix},$ (1) 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 twodimensional case, a reflection is conjugate to a matrix of the form
$\displaystyle\begin{pmatrix}\cos\theta&\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix},$ (2) 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.
Mathematics Subject Classification
51A10 no label found15A04 no label found51A15 no label found Forums
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