Pencil (geometry)

This article is about a class of geometric objects. For other uses, see Pencil (disambiguation).
"Bundle (geometry)" redirects here. Not to be confused with Bundle (mathematics).
Some lines in the pencil through A

In 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 plane, or the set of circles that pass through two given points in a plane.

Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle.[1] Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a flat pencil.[2]

Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line.[1] A more common term for this set is a range of points.

  1. ^ a b Young 1971, p. 40
  2. ^ Halsted 1906, p. 9