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
for any nonzero .
To represent a line in the projective plane, we begin with a standard
Euclidean formula for a line
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., -a1/b1 = -a2/b2, the point of intersection is simply (b1c2-b2c1,a2c1-a1c2,0), which is the ideal point associated with the direction whose slope is -a1/b1. 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
lie on the same line. The line joining the first two points
The third point then lies on the line if
or, more succinctly, if the
determinant of the
matrix containing the points is zero:
Example 1. Given two lines
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=W2 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.