Editing Line at infinity (section)

==Geometric formulation==
In projective geometry, any pair of lines always intersects at some point, but [[Parallel (geometry)|parallel]] lines do not intersect in the real plane.  The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel.

Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the [[slope]] of the lines, not at all on their [[y-intercept]].

In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity.  Therefore, lines in the projective plane are [[closed curve]]s, i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.