If you fix a general quadrilateral, you fix the coordinates of every point in the affine chart of that quadrilateral. See Hartshorne, Foundations of Projective Geometry. If you permute the order of points in the quadrilateral, you move around which points lie in the affine chart, covering the whole plane.

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. If you permute the order of points in the quadrilateral, you move around which points lie in the affine chart, covering the whole plane.