More projective groups

The projective group PGL(2, C).

This is the group that acts on the complex projective line P(C²) = CP¹.
As in the last example the elements of this group can be regarded as the set of rational analytic functions:
z ↦ (az + b)/(cz + d) with a, b, c, d ∈ C with ad - bc ≠0,
from C ∪ {∞} to itself.

Remarks

1. These transformations are sometimes called Möbius functions (after the German mathematician August Möbius (1790 to 1868) best known for his results in Number Theory and for the so-called Möbius band).

2. Complex variable theory shows that these mappings are conformal (angle preserving) transformations at most points.

3. As in the last example, we may define the cross-ratio of four points (a complex number this time) and this is preserved by these transformations.
   

This is the group that acts on the real projective plane P(R³) = RP².
One may show that there is a unique element of this group taking any four points in RP² (no three on a line) into any other four points in RP².
As in the projective line case, one may take standard reference points to be: [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]. These are the points at infinity on the x and y axes, the origin and the point (1, 1). We may manipulate these as in the earlier case.
We may define the cross-ratio of four points on a line and these transformations preserve that.