How does adding points at infinity simplify geometric theorems?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
In projective geometry, adding points at infinity simplifies geometric theorems by eliminating the need to treat parallel lines as special cases. In Euclidean space, parallel lines never intersect, which often requires theorems to be modified or stated with exceptions when lines are parallel. By postulating that every straight line has an added point at infinityâand that all parallel lines share a common point at infinityâmathematicians can treat these lines as intersecting, allowing for a more unified and consistent formulation of geometric principles.12
This framework allows theorems to be stated simply in terms of the intersections of lines and the collinearity of points, without requiring references to distance, angle, or congruence, which are not preserved under projection. For example, Desarguesâs theorem can be formulated without special modifications for parallel sides, and the projective approach effectively unites different types of conic sections (circles, ellipses, parabolas, and hyperbolas) as projective images of one another.12