# 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.

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 revolution.

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

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.

## References

- G. B. Halsted (1906) Synthetic Projective Geometry, page 9, via Internet Archive
- Emil Artin (1957) Geometric Algebra, page 53