How are collinear points related by duality to concurrent lines?
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 relationship between collinear points and concurrent lines is defined by the principle of duality. This principle states that any true geometric theorem remains true if the terms "point" and "line" are interchanged throughout the statement.12
Because of this underlying lattice structure, these two concepts are dual to one another:
- Collinear points: A set of points is collinear if they all lie on a single common line.
- Concurrent lines: A set of lines is concurrent if they all pass through a single common point.
The duality arises because the axioms of projective geometry treat points and lines as interchangeable elements. For instance, the statement "two points determine a unique line" is the dual of "two lines determine a unique point". Consequently, the property of points being collinear is the dual of the property of lines being concurrent, as both describe the incidence relationship between these fundamental geometric entities.2