# Projective transformation

In geometry, a projective transformation may be a correlation or a collineation. The latter classification includes the case of a perspectivity and a projectivity which is a composition of perspectivities.[1] Another type of projective transformation is a homography which acts on a one-dimensional projective space called a projective line.

Given a projective space, its collection of projectivities forms a group under composition of functions. In fact, such groups have historical importance. For instance, the Möbius group has become a tool of science.

Projective transformations are often expressed in terms of linear transformations on a vector space over a field F. Points of the projective space are represented using homogeneous coordinates from F. A non-singular n × n matrix corresponds to a projective transformation on n − 1 dimensional projective space (due to the homogeneous factor). This context supersedes affine transformation in a hyperplane of the projective space that can be written xMx + v or in terms of column vectors x, y and v of length n − 1:

$\begin{pmatrix}y\\1 \end{pmatrix} = \begin{pmatrix}M & v\\0 & 1 \end{pmatrix} \begin{pmatrix}x\\1 \end{pmatrix} ,$

where M is square and non-singular. The projective linear group PGL(n, F) is formed using the general linear group GL(n, F) and its center ZGL(n, F) consisting of matrices k I where I is the identity matrix and k is in F. Then PGL(n, F) is the quotient group of GL(n, F) by its center.[2]

## References

1. ^ H. S. M. Coxeter (1969) Introduction to Geometry, page 242, John Wiley & Sons
2. ^ Roger Lyndon (1985) Groups and Geometry, page 131, Cambridge University Press ISBN 0-521-31694-4