Pencil (mathematics)

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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.

In affine geometry with the reflexive variant of parallelism, a set of parallel lines forms an equivalence class called a pencil of parallel lines.[1]

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[edit]


  1. ^ Emil Artin (1957) Geometric Algebra, page 53
  • Weisstein, Eric W. "Pencil". MathWorld.