In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space
Such involutions are easy to characterize and they can be described geometrically.
[1] To give a linear involution is the same as giving an involutory matrix, a square matrix A such that
It is a quick check that a square matrix D whose elements are all zero off the main diagonal and ±1 on the diagonal, that is, a signature matrix of the form
satisfies (1), i.e. is the matrix of a linear involution.
It turns out that all the matrices satisfying (1) are of the form
Geometrically this means that any linear involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through the origin.
One can easily verify that A represents a linear involution if and only if A has the form
One can check that any affine involution in fact has this form.
Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b. Affine involutions can be categorized by the dimension of the affine space of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix D (see above), i.e., the dimension of the eigenspace for eigenvalue 1.
The two extreme cases for which this always applies are the identity function and inversion in a point.
The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a reflection, and in 3D a rotation about the line by 180°), inversion in a plane (in 3D and up; in 3D this is a reflection in a plane), inversion in a 3D space (in 3D: the identity), etc.