Euclidean transformation

Let V and W be Euclidean vector spaces. A Euclidean transformation is an affine transformationPlanetmathPlanetmath E:VW, given by


such that L is an orthogonalMathworldPlanetmathPlanetmathPlanetmath linear transformation (

As an affine transformation, all affine properties, such as incidence and parallelismPlanetmathPlanetmathPlanetmath are preserved by E. In additionPlanetmathPlanetmath, since E(u-v)=L(u-v) and L is an , E preserves lengths of line segmentsMathworldPlanetmath 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. 1.

    translation. If L=I, then E is just a translation. Any Euclidean transformation can be decomposed into a product of an orthogonal transformationMathworldPlanetmath L(v), followed by a translation T(v)=v+w.

  2. 2.

    rotation. If w=0, then E is just an orthogonal transformation. If 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 conjugatePlanetmathPlanetmath to a matrix of the form

    (cosθ-sinθsinθcosθ), (1)

    where θ. Via this particular (unconjugated) map, any vector emanating from the origin is rotated in the counterclockwise direction by an angle of θ to another vector emanating from the origin. Thus, if E is conjugate to the matrix given above, then θ is the angle of rotation for E.

  3. 3.

    reflection. If w=0 but 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

    (cosθsinθsinθ-cosθ), (2)

    where θ. Any vector is reflected by this particular (unconjugated) map to another by a “mirror”, a line of the form y=xtan(θ2).


  • In general, an orthogonal transformation can be represented by a matrix of the form


    where each Ai is either ±1 or a rotation matrixMathworldPlanetmath (1) (or reflection matrix (2)) given above. When its determinantMathworldPlanetmath is -1 (a reflection), it has an invariant subspacePlanetmathPlanetmath of V of codimension 1. One can think of this hyperplaneMathworldPlanetmathPlanetmath 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