Euclidean transformation

Let $V$ and $W$ be Euclidean vector spaces. A Euclideantransformation is an affine transformation $E:V\to W$, given by

$$E(v)=L(v)+w$$

such that $L$ is an orthogonal lineartransformation (http://planetmath.org/OrthogonalTransformation).

As an affine transformation, all affine properties, such asincidence 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 linesegments (http://planetmath.org/AngleBetweenTwoLines). Because of this, a Euclidean transformation is also calleda rigid motion, which is a popular term used in mechanics.

Types of Euclidean transformations

There are three maintypes of Euclidean transformations:

1. translation. If $L=I$, then $E$ is just a translation. AnyEuclidean transformation can be decomposed into a product of anorthogonal transformation $L(v)$, followed by atranslation $T(v)=v+w$.

2. rotation. If $w=0$, then $E$ is just an orthogonal transformation. If $\det (E)=1$, then $E$ is called a rotation. Theorientation of any basis (of $V$) is preserved under a rotation. In thecase where $V$ is two-dimensional, a rotation is conjugate to a matrix of the form

   (1)

   where $\theta \in ℝ$. Via this particular (unconjugated) map, any vector emanating from the origin is rotatedin 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 $\det (E)=-1$ instead, then $E$ is a calledreflection. Again, in the two-dimensional case, a reflection is to a matrix of the form

   (2)

   where $\theta \in ℝ$. Any vector is reflected by this particular (unconjugated) map to anotherby 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 theform

  where each $A_i$ is either $\pm 1$ or a rotation matrix (1) (or reflectionmatrix (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.