Projective Transformation

a one-to-one mapping of the projective plane or projective space into itself such that collinear points are carried into collinear points (for this reason, a projective transformation is sometimes called a collineation). A projective transformation of a projective line is a one-to-one mapping of the projective line into itself such that the harmonicity of points of the line is preserved.

The simplest example of a projective transformation—and the one most important in applications—is the homology, which is the projective transformation that leaves invariant a line and a point not on the line. An example of a projective transformation of space is the perspective transformation, whereby a figure F in the plane Π is projected from a point S into a figure F′ in the plane Π′. Any projective transformation can be obtained by a finite sequence of perspective transformations. The projective transformations form a group whose fundamental invariant is the cross ratio of four points on a line. Theories of the invariants of groups of projective transformations that leave unchanged some figure are metric geometries.

The fundamental projective-transformation theorem for the projective plane can be stated as follows. Let A, B, C, and D be any four points in the plane Π such that no three of them are collinear. If A′, B′, C′, and D′ are also four points in this plane such that no three of them are collinear, then there exists one, and only one, projective transformation that carries the points A, B, C, and D into the points A′, B′, C′, and D′, respectively. This theorem is made use of in nomography and aerial photographic survey. A similar theorem is true in projective space. In this case, a projective transformation is determined by five points, no four of which are coplanar. This theorem is equivalent to Pappus’ theorem.

In homogeneous coordinates a projective transformation is expressed as a homogeneous linear transformation whose matrix has a nonzero determinant. Projective transformations of the Euclidean plane or space are also studied. In Cartesian coordinates, they are expressed by linear-fractional functions; the mappings, however, are not one-to-one in this case.