What is a pencil of planes in three-dimensional projective geometry?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Projective drawingThe 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. — projective geometry Projective drawing
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, a pencil of planes is defined as the set of all planes that pass through a common line. This shared line is referred to as the axis of the pencil.1

This concept is central to the principle of duality in geometry, which describes a symmetry between points and planes. Under this duality, the dual of a pencil of planes is a line of points.1