# Projective Transformation

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## 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.