Affine Transformation - Properties

Properties

An affine transformation preserves:

1. The collinearity relation between points; i.e., points which lie on the same line (called collinear points) continue to be collinear after the transformation.
2. Ratios of vectors along a line; i.e., for distinct collinear points the ratio of and is the same as that of and .
3. More generally barycenters of weighted collections of points.

An affine transformation is invertible if and only if A is invertible. In the matrix representation, the inverse is:

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

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of A is 1 or −1 then the transformation is an equi-areal mapping. Such transformations form a subgroup called the equi-affine group 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 transformations which preserve orientation: those where the determinant of A is positive. In the last case this is in 3D the group of rigid body motions (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 is easier to get an 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. Describing such a transformation for an object tends to make more sense in terms of rotation about an axis through the center of that object, combined with a translation, rather than by just a rotation with respect to some distant point. As an example: "move 200 m north and rotate 90° anti-clockwise", rather than the equivalent "with respect to the point 141 m to the northwest, rotate 90° anti-clockwise".