= { P1 } L3 = { P1 } ... Lm = { P1 } 2) For any number of points P1, ..., Pn, and lines L1, ..., Ln, (same number of points as lines) L1 = { P2, P3, ..., Pn } L2 = { P1, P2 } L3 = { P1, P3 } ... Ln = { P1, Pn } Collineations A collineation of a projective plane is a bijective map of the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that if σ is a bijection and point P is on line m, then Pσ is on mσ. If σ is a collineation of a projective plane, a point P with P = Pσ is called a fixed point of σ, and a line m with m = mσ is called a fixed line of σ. The points on a fixed line need not be fixed points, their images under σ are just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form a closed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane. Collineations whose fixed structure forms a plane are called planar collineations. Homography A homography (or projective transformation) of PG(2, K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible matrices over K which act on the points of PG(2, K) by , where x and y are points in K3 (vectors) and M is an invertible matrix over K. Two matrices represent the same projective transformation if one is a constant multiple of the other. Thus the group of projective transformations is the quotient of the general linear group by the scalar matrices called the projective linear group. Another type of collineation of PG(2, K) is induced by any automorphism of K, these are called automorphic collineations. If α is an automorphism of K, then the collineation given by is an automorphic collineation. The fundamental theorem of projective geometry says that all the collineations of PG(2, K) are