How does duality interchange points and planes in three-dimensional projective geometry?
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 three-dimensional projective geometry, the principle of duality establishes a symmetric relationship between points and planes. This principle allows for the derivation of new geometric theorems by interchanging these two fundamental elements: wherever a theorem refers to a point, its dual counterpart refers to a plane, and vice versa.12
In this spatial context, the line acts as its own dual because it can be defined either by two points or by the intersection of two planes. The interchangeability functions as follows:2
- Points and Planes: A point is dual to a plane. Consequently, a set of points lying on a plane corresponds to a set of planes passing through a point.2
- Pencils: A pencil of planes—defined as all planes passing through a common line (the axis)—is dual to a line of points, which consists of all points lying on a given line.3
By treating these elements as interchangeable within the underlying lattice structure of the geometry, mathematicians can transform one true statement into another, ensuring that all results hold in both the original and the dual formulation.2