Line at infinity

In projective geometry, the line at infinity is a line which is added to the real (affine) plane in order to give closure to incidence properties of the resulting projective plane.

In projective geometry, any pair of lines always intersect at some point. But 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. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their Y-intercept. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines is parallel.

Every line intersects the line at infinity at some point. The point at which a line intersects the line at infinity determines the slope of the line, but not at all its Y-intercept.

In the affine plane, a line runs off 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 curves: 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 endpoints at all) and so it is actually cyclical.