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 $E(u-v)=L(u-v)$ 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 $T(v)=v+w$ .

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

$\displaystyle\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix},$

(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 $E$ .

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 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.

Title	Euclidean transformation
Canonical name	EuclideanTransformation
Date of creation	2013-03-22 15:59:46
Last modified on	2013-03-22 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