Euclidean transformation (original) (raw)

As an affine transformation, all affine properties, such as incidence and parallelism are preserved by E. In addition, sinceE⁢(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 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 anorthogonal transformation L⁢(v), followed by a translation T⁢(v)=v+w.
2. 1. 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 the case where V is two-dimensional, a rotation is conjugate to a matrix of the form

      (cos⁡θ-sin⁡θsin⁡θcos⁡θ),	(1)
      where θ∈ℝ. 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. 1. 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

      (cos⁡θsin⁡θsin⁡θ-cos⁡θ),	(2)
      where θ∈ℝ. Any vector is reflected by this particular (unconjugated) map to another by a “mirror”, a line of the form y=x⁢tan⁡(θ2).