Pencil (geometry)

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

The Apollonian circles, two orthogonal pencils of circles

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."

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

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

$\lambda C+\mu C'=0\,$

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.

See also

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

References

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