Homogeneous coordinates

Suppose we have a point (*x*,*y*) in the Euclidean plane.
To represent this same point in the projective plane, we simply add a
third coordinate of 1 at the end: (*x*, *y*, 1).^{1}
Overall scaling is unimportant, so the point (*x*,*y*,1) is the same as the
point
,
for any nonzero .
In other
words,

for any (Thus the point (0,0,0) is disallowed). Because scaling is unimportant, the coordinates (

To represent a line in the projective plane, we begin with a standard
Euclidean formula for a line

and use the fact that the equation is unaffected by scaling to arrive at the following:

aX+bY+cW |
= | 0 | |

= | 0, | (1) |

where is the line and is a point on the line. Thus we see that points and lines have the same representation in the projective plane. The parameters of a line are easily interpreted: -

To transform a point in the projective plane back into Euclidean
coordinates, we
simply divide by the third coordinate:
(*x*,*y*) = (*X*/*W*, *Y*/*W*). Immediately
we see that the
projective plane contains more points than the Euclidean plane,
that is, points whose third coordinate is zero. These points
are called *ideal points*, or *points at infinity*. There is a
separate ideal point
associated with each direction in the plane; for example, the points
(1,0,0) and (0,1,0) are associated with the
horizontal and vertical directions,
respectively. Ideal points are considered just like any other point
in
and are given no special treatment.
All the ideal points lie on a line, called the *ideal line*, or
the *line at infinity*, which, once again, is treated just the same as
any other line. The ideal line is represented as (0,0,1).

Suppose we want to find the intersection of two lines.
By elementary algebra, the two
lines
and
are found to
intersect at the point
.
This formula is more easily
remembered as the cross product:
.
If the two lines are parallel, i.e.,
-*a*_{1}/*b*_{1} = -*a*_{2}/*b*_{2}, the point of
intersection is simply
(*b*_{1}*c*_{2}-*b*_{2}*c*_{1},*a*_{2}*c*_{1}-*a*_{1}*c*_{2},0), which is the
ideal point associated with the direction whose slope is -*a*_{1}/*b*_{1}.
Similarly, given two points
and
,
the
equation of the line passing through them is given by
.

Now suppose we want to determine whether three points
,
,
and
lie on the same line. The line joining the first two points
is
.
The third point then lies on the line if
,
or, more succinctly, if the
determinant of the
matrix containing the points is zero:

Similarly, three lines , , and intersect at the same point (i.e., they are concurrent), if the following equation holds:

The concepts of homogeneous coordinates are summarized in Figure 2. For further reading, consult the notes by Guibas [3].

*Example 1*. Given two lines
and
,
the point of intersection is given by:

*Example 2*.
Consider the intersection of the hyperbola *xy*=1 with the horizontal line
*y*=1. To convert these equations to
homogeneous coordinates, recall
that *X*=*Wx* and *Y*=*Wy*, yielding *XY*=*W*^{2} for the hyperbola and *Y*=*W* for
the line. The solution to these two equations is the point (*W*,*W*,*W*),
which is the same as the point (1,1) in the Euclidean plane, the
desired result. Now let us consider the intersection of the same hyperbola
with the horizontal line *y*=0, an intersection which does not exist
in the Euclidean plane. In homogeneous coordinates
the line becomes *Y*=0 which yields the solution (*X*,0,0),
the ideal point associated with the horizontal direction.