How does the principle of duality interchange points and lines in a projective plane?
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 principle of duality allows for the derivation of new geometric theorems by systematically interchanging the terms "point" and "line" within a statement.12
This principle functions because the projective plane possesses a lattice structure where points and lines are related by inclusion. Under this transformation:2
* "Collinear" points (points lying on a common line) correspond to "concurrent" lines (lines meeting at a common point).3
* The statement "two points determine a line" becomes the dual statement "two lines determine a point," which is always true in projective geometry because the axioms exclude parallel lines.2
This interchange can be applied mechanically to any valid theorem to generate a new, equally valid result. For example, one can associate a point with a line using a conic section; by mapping a point to the line connecting the points of tangency of the lines passing through , one establishes a correspondence where repeating the process returns the original element. This powerful symmetry effectively doubles the number of known theorems in the field.3