shadows - motivation for affine geometry

 Imagine the Sun shining through a window made up of identical rectangular panes. The shadow on the floor will not, in general, be rectangular. As the Sun moves, the shadow will change. The basic problem of affine geometry is to determine what properties do all of the shadows have in common. As the Sun gets lower, shadows lengthen, so we see that lengths are not invariant. Also, experience shows that the shape of the shadow changes, so angles are not invariant. On the other hand, we might observe that the shadows of the rectangular panes are always parallelograms. This suggests that parallelism is preserved. Since the Sun is far from the Earth, a reasonable approximation is to assume that the rays are parallel. This leads us to the idea of a parallel projection. For the moment, we observe that, provided the Sun's rays are not parallel to either the wall or the floor, then each point in the plane of the wall leads to a point in the plane of the floor. In other words, we have a transformation of R2. We will begin our study of affine geometry in the kleinian way, starting with a group of transformations of R2. Later, we will see that our "shadow transformations" actually generate the entire affine group.