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A transformation that maps lines to lines (but does not necessarily
preserve parallelism) is a **projective transformation**. Any plane
projective transformation can be expressed by an invertible 3×3
matrix in homogeneous coordinates; conversely, any invertible
3×3 matrix defines a projective transformation of the plane.
Projective transformations (if not affine) are not defined on all of
the plane, but only on the complement of a line (the missing line is
``mapped to infinity'').

A common example of a projective transformation is given by a
**perspective transformation** (Figure 1). Strictly
speaking this gives a transformation from one plane to another, but if
we identify the two planes by (for example) fixing a cartesian system
in each, we get a projective transformation from the plane to itself.

**Figure 1:**
A perspective transformation with center *O*, mapping the plane *P* to
the plane *Q*. The transformation is not defined on the line *L*,
where *P* intersects the plane parallel to *Q* and going throught *O*.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.