§ 3. Line and Plane at Infinity.—Having arrived at the notion of replacing all points at infinity in a line by one ideal point, there is no difficulty in replacing all points at infinity in a plane by one ideal line.
To make this clear, let us suppose that a line p, which cuts two fixed lines a and b in the points A and B, moves parallel to itself to a greater and greater distance. It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane.
All points at infinity in a plane lie in a line, which is called the line at infinity in the plane.
It follows that parallel planes must be considered as planes having a common line at infinity, for any other plane cuts them in parallel lines which have a point at infinity in common.
If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, and they are therefore all in one plane.
All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity.
§ 4. Parallelism.—We have now the following definitions:—
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.