Let and be Euclidean vector spaces. A Euclidean transformation is an affine transformation , given by

As an affine transformation, all affine properties, such as incidence and parallelism are preserved by . In addition, since and is an orthogonal linear transformation, preserves lengths of line segments 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

1. translation. If , then is just a translation. Any Euclidean transformation can be decomposed into a product of an orthogonal transformation , followed by a translation .
2. rotation. If , then is just an orthogonal transformation. If , then is called a rotation. The orientation of any basis (of ) is preserved under a rotation. In the case where is two-dimensional, a rotation is conjugate to a matrix of the form
   

   (1)

   
   where . Via this particular (unconjugated) map, any vector emanating from the origin is rotated in the counterclockwise direction by an angle of to another vector emanating from the origin. Thus, if is conjugate to the matrix given above, then is the angle of rotation for .
3. reflection. If but instead, then is a called reflection. Again, in the two-dimensional case, a reflection is conjugate to a matrix of the form
   

   (2)

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

Remarks.

• In general, an orthogonal transformation can be represented by a matrix of the form
  
  where each is either or a rotation matrix (1) (or reflection matrix (2)) given above. When its determinant is -1 (a reflection), it has an invariant subspace of 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.