Pencil (mathematics)

In projective geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane.

The Apollonian circles, two orthogonal pencils of circles

For instance, in the development of G. B. Halsted, "Straights with the same cross are copunctal." Also "The aggregate of all coplanar, copunctal straights is called a flat-pencil" and "A piece of a flat-pencil bounded by two of the straights as sides, is called an angle."[1]

"The aggregate of all planes on a straight is called an axial-pencil." For example, the meridians of the globe are defined by the pencil of planes on the axis of Earth's rotation.

In affine geometry with the reflexive variant of parallelism, a set of parallel lines forms an equivalence class called a pencil of parallel lines.[2]

More generally, a pencil is the special case of a linear system of divisors in which the parameter space is a projective line. Typical pencils of curves in the projective plane, for example, are written as

${\displaystyle \lambda C+\mu C'=0\,}$

where C = 0, C′ = 0 are plane curves.

A pencil of planes, the family of planes through a given straight line, is sometimes referred to as a fan or a sheaf.