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.

See also


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