Pencil (geometry)
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.
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
- 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
References
- ^ Emil Artin (1957) Geometric Algebra, page 53