Pencil (mathematics)

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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


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.

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