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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.
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.