Affine transformations

[Pascual2005]

Sorry, can somebody explain about Affine transformations:
when can we used them?
What do they do?
What changes and what remains invariant?
What figures can we obtain from others via these transformations?

Thanks

[grobber]

We can use them whenever the properties we want to prove depend only on incidence between lines and points and ratios in which points divide segments. They preserve parallelism, incidence relations and ratios in which a point divides a segment.

If we denote a point of the plane by its homogenous coordinates , the transformation has the form . This is necessary because we want the line at infinity to be invariant, i.e. the points having their -coordinate equal to have to turn into points with the same property.

In fact, this is why the transformation preserves parallelism: two parallel lines concur on the line at infinity, so the images of the two lines must concur on the image of the line at infinity, which is also the line at infinity, so the images of two parallel lines must also be parallel.