Collineation of a projective plane

Definition

A collineation of a prrojective plane is defined in the following equivalent ways:

1. It is the data of a bijection from its set of points to its set of points and a bijection from its set of lines to its set of lines such that the image of a point is incident to the image of a line iff the point is incident to the line.
2. It is a bijection on the set of points such that three points are collinear (i.e., are incident to a common line) iff their images are collinear.
3. It is a bijection on the set of lines such that three lines are concurrent (i.e., have a common point incident to them) iff their images are concurrent.

The two bijections in (1) are respectively the bijections in (2) and (3).

The notion of collineation is the natural notion of automorphism for a projective plane. The collineations of a projective plane form a group, called the collineation group of a projective plane.

We can also talk of a collineation of an affine plane.