Editing Projective plane (section)

===Homography===
{{Main|Projective transformation}}

A '''[[homography]]''' (or ''projective transformation'') of PG(2,&nbsp;''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 {{nowrap|3 × 3}} matrices over ''K'' which act on the points of PG(2,&nbsp;''K'') by {{nowrap|1=''y'' = ''M'' ''x''<sup>T</sup>}}, where ''x'' and ''y'' are points in ''K''<sup>3</sup> (vectors) and ''M'' is an invertible {{nowrap|3 × 3}} matrix over ''K''.<ref>The points are viewed as row vectors, so to make the matrix multiplication work in this expression, the point ''x'' must be written as a column vector.</ref> 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,&nbsp;''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 {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, x<sub>2</sub>) → (''x''<sub>0</sub><sup>''α''</sup>, ''x''<sub>1</sub><sup>''α''</sup>, ''x''<sub>2</sub><sup>''α''</sup>)}} is an automorphic collineation. The [[fundamental theorem of projective geometry]] says that all the collineations of PG(2,&nbsp;''K'') are compositions of homographies and automorphic collineations. Automorphic collineations are planar collineations.