What is the difference between a line at infinity and a plane at infinity?
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, the distinction between a line at infinity and a plane at infinity relates to the dimensionality of the space being considered:
- Line at infinity: This concept exists within the two-dimensional projective plane. It is formed by collecting all the points at infinity where parallel lines in the Euclidean plane are said to intersect. It serves to eliminate special cases in geometric theorems by ensuring that any two lines have a unique point of intersection.12
- Plane at infinity: This is the three-dimensional analogue, defined as the aggregate of all points at infinity in a three-dimensional projective space. Just as the line at infinity accounts for the convergence of parallel lines in a plane, the plane at infinity accounts for the "horizon" of parallel lines and planes in three-dimensional space.3
Essentially, these constructs allow mathematicians to treat parallel elements as intersecting at "ideal points," thereby simplifying the study of geometric properties that remain invariant under projection.12