If you fix a general quadrilateral, you fix the coordinates of every point in the affine chart of that quadrilateral. See Bumcroft, **Modern Projective Geometry**, chapter 3, section I or Hartshorne, [**Foundations of Projective Geometry**][1]. If you permute the order of points in the quadrilateral, you move around which points lie in the affine chart, covering the whole plane.


  [1]: http://www.math.rug.nl/~piter/KR/Hartshorne.pdf