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 (http://planetmath.org/OrthogonalTransformation).
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 , $E$ preserves lengths of line segments^{} and angles between two line segments (http://planetmath.org/AngleBetweenTwoLines). 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 $\mathrm{det}(E)=1$, 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
$\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right),$ (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 $\mathrm{det}(E)=1$ instead, then $E$ is a called reflection. Again, in the twodimensional case, a reflection is to a matrix of the form
$\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right),$ (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 $y=x\mathrm{tan}(\frac{\theta}{2})$.
Remarks.

•
In general, an orthogonal transformation can be represented by a matrix of the form
$$\left(\begin{array}{cccc}\hfill {A}_{1}\hfill & \hfill O\hfill & \hfill \mathrm{\cdots}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {A}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill O\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {A}_{n}\hfill \end{array}\right),$$ 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.
Title  Euclidean transformation 
Canonical name  EuclideanTransformation 
Date of creation  20130322 15:59:46 
Last modified on  20130322 15:59:46 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  17 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 15A04 
Classification  msc 51A15 
Synonym  rigid motion 
Defines  translation 
Defines  translate 
Defines  rotation 
Defines  rotate 
Defines  reflection 
Defines  reflect 
Defines  reflexion 
Defines  glide reflection 
Defines  angle of rotation 