Pencil (mathematics)

This article is about a class of geometric objects. For other uses, see Pencil (disambiguation).

The Apollonian circles, two orthogonal pencils of circles.

A pencil in projective geometry 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.

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

λC + μC′ = 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.

See also

• Pencil beam
• Pencil of circles
• Lefschetz pencil
• Matrix pencil

General references

• Weisstein, Eric W., "Pencil", MathWorld.