The projective space can be seen as containing the affine space under the mapping . The plane in can be seen as containing the limit points for , since these points are . This plane is therefore called the plane at infinity . An affine transformation counts 12 independent degrees of freedom. Note, however, that the position of points in the plane at infinity can change under an affine transformation, but that all these points stay within the plane .
All projective properties are a fortiori affine properties.
Lines or planes having their intersection in the plane at infinity are called parallel . A new invariant property for this group is the ratio of lengths along a certain direction . Note that this is equivalent to a cross-ratio with one of the points at infinity.