What is a pencil of lines in 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 projective geometry, a pencil of lines is defined as the set of all lines in a plane that pass through a single common point.1
This concept is central to the principle of duality in geometry, which describes a symmetry between points and lines. In a plane, the dual of a pencil of lines (a collection of lines passing through a point) is a line of points (a collection of points lying on a line).1
In three-dimensional space, the term "pencil" is also used to describe all the planes that pass through a given line, which is referred to as the axis of the pencil.1