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# Pencil (mathematics)

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 or a sheaf.

See also

Pencil beam

Pencil of circles

Lefschetz pencil

Matrix pencil

References

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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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