# Affine transformation

These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that "f preserves parallelism".

This formulation works irrespective of whether any of the domain, codomain and image vector spaces have the same number of dimensions.

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.

The affine transform preserves parallel lines. However, the stretching and shearing transformations warp shapes, as the following example shows:

This is an example of image warping. However, the affine transformations do not facilitate projection onto a curved surface or radial distortions.

Affine transformations do not respect lengths or angles; they multiply area by a constant factor

Transforming the three corner points of the original triangle (in red) gives three new points which form the new triangle (in blue). This transformation skews and translates the original triangle.

In fact, all triangles are related to one another by affine transformations. This is also true for all parallelograms, but not for all quadrilaterals.