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A triple of real numbers (*x*:*y*:*t*), with *t*0, is a set of
**homogeneous coordinates** for the point *P* with cartesian
coordinates (*x*/*t*, *y*/*t*). Thus the same point has many sets of
homogeneous coordinates: (*x*:*y*:*t*) and (*x'*:*y'*:*t'*) represent the same
point if and only if there is some real number such that
*x'*=*x*, *y'*=*y*, *t'*=*t*.
If *P* has cartesian coordinates (*x*,*y*), one set of homogeneous
coordinates for *P* is (*x*,*y*,1).

When we think of the same triple of numbers as the cartesian
coordinates of a point in three-dimensional space
(Section 9.1), we write it (*x*,*y*,*t*) instead of
(*x*:*y*:*t*). The connection between the point in space with cartesian
coordinates (*x*,*y*,*t*) and the point in the plane with homogeneous
coordinates (*x*:*y*:*t*) becomes apparent when we consider the plane
*t*=1 in space, with cartesian coordinates given by the first two
coordinates *x*, *y* of space (Figure 1). The point
(*x*,*y*,*t*) in space can be connected to the origin by a line *L* that
intersects the plane *t*=1 in the point with cartesian coordinates
(*x*/*t*, *y*/*t*), or homogeneous coordinates (*x*:*y*:*t*).

**Figure 1:** The point *P* with spatial coordinates (*x*,*y*,*t*) projects to
the point *Q* with spatial coordinates (*x*/*t*, *y*/*t*, 1). The plane
cartesian coordinates of *Q* are (*x*/*t*, *y*/*t*), and (*x*:*y*:*t*) is one
set of homogeneous coordinates for *Q*. Any point on the line *L*
(except for the origin *O*) would also project to *P'*.

Projective coordinates are useful for several reasons, one the most
important being that they allow one to unify all symmetries of the
plane (as well as other transformations) under a single umbrella. All
these transformations can be regarded as linear maps in the space of
triples (*x*:*y*:*t*), and so can be expressed in terms of matrix
multiplications. (See Section 2.2).

If we consider triples (*x*:*y*:*t*) such that at least one of *x*, *y*, *t* is
nonzero, we can name not only the points in the plane but also
points ``at infinity''. Thus (*x*:*y*:*t*) represents the point at
infinity in the direction of the line with slope *y*/*x*.

*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.