Editing Affine transformation (section)

=== Groups ===
As an affine transformation is [[invertible function|invertible]], the [[square matrix]]  <math>A</math> appearing in its [[#Representation|matrix representation]] is [[invertible matrix|invertible]]. The matrix representation of the inverse transformation is thus
:<math>
\left[ \begin{array}{ccc|c} & A^{-1} & & -A^{-1}\vec{b} \ \\ 0 & \ldots & 0 & 1 \end{array} \right].
</math>

The invertible affine transformations (of an affine space onto itself) form the [[affine group]], which has the [[general linear group]] of degree <math>n</math> as subgroup and is itself a subgroup of the general linear group of degree <math>n + 1</math>.

The [[Similarity transformation (geometry)|similarity transformations]] form the subgroup where <math>A</math> is a scalar times an [[orthogonal matrix]]. For example, if the affine transformation acts on the plane and if the [[determinant]] of <math>A</math> is 1 or −1 then the transformation is an [[equiareal map]]ping. Such transformations form a subgroup called the ''equi-affine group''.<ref>[[Oswald Veblen]] (1918) ''Projective Geometry'', volume 2, pp. 105–7.</ref> A transformation that is both equi-affine and a similarity is an [[isometry]] of the plane taken with [[Euclidean distance]].

Each of these groups has a subgroup of ''[[orientability|orientation]]-preserving'' or ''positive'' affine transformations: those where the determinant of <math>A</math> is positive. In the last case this is in 3D the group of [[rigid transformation]]s ([[Improper rotation|proper rotations]] and pure translations).

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.