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A transformation that maps lines to lines (but does not necessarily
preserve parallelism) is a **projective transformation**. Any
spatial projective transformation can be expressed by an invertible
4×4 matrix in homogeneous coordinates; conversely, any
invertible 4×4 matrix defines a projective transformation of
the plane. Projective transformations (if not affine) are not defined
on all of space, but only on the complement of a plane (the missing
plane is ``mapped to infinity'').

The following particular case is often useful, especially in computer
graphics, in **projecting a scene** from space to the plane. Suppose
an observer is at the point *E*=(*x*,*y*,*z*) of space, looking toward
the origin *O*=(0,0,0). Let *P*, the **screen**, be the plane
through *O* and perpendicular to the ray *EO*. Place a rectangular
coordinate system on *P* with origin at *O* and such that
the positive -axis lies in the half-plane determined by *E* and
the positive *z*-axis of space (that is, the *z*-axis is pointing
``up'' as seen from *E*). Then consider the transformation that
associates to a point *X*=(*x*,*y*,*z*) the triple (,,), where
(,) are the coordinates of the point where the line *EX*
intersects *P* (the **screen coordinates** of *X* as seen from *E*),
and is the inverse of the signed distance from *X* to *E*
along the line *EO* (the **depth** of *X* as seen from *E*). This is
a projective transformation, given by the matrix

with and .

*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.