Pencil (geometry)

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 defined by a parameter whose value can be determined from any two of its members. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a flat pencil.

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. A more common term for this set is a range of points.

In a plane, let ${\displaystyle \ell }$ and ${\displaystyle \ell '}$ be two distinct intersecting lines. For concreteness, suppose that ${\displaystyle \ell }$ has the equation, ${\displaystyle aX+bY+c=0}$ and ${\displaystyle \ell '}$ has the equation ${\displaystyle a'X+b'Y+c'=0}$. Then

represents, for suitable scalars ${\displaystyle \lambda }$ and ${\displaystyle \mu }$, any line passing through the intersection of ${\displaystyle \ell =0}$ and ${\displaystyle \ell '=0}$. This set of lines passing through a common point is called a pencil of lines. The common point of a pencil of lines is called the vertex of the pencil.

In an affine plane with the reflexive variant of parallelism, a set of parallel lines forms an equivalence class called a pencil of parallel lines. This terminology is consistent with the above definition since in the unique projective extension of the affine plane to a projective plane a single point (point at infinity) is added to each line in the pencil of parallel lines, thus making it a pencil in the above sense in the projective plane.

A pencil of half-lines, also called a half-pencil, is the set of half-lines sharing a common vertex. It's used in optics as a model for pencil beams.

A pencil of planes, is the set of planes through a given straight line in three-space, called the axis of the pencil. The pencil is sometimes referred to as a axial-pencil or fan of planes or a sheaf of planes. For example, the meridians of the globe are defined by the pencil of planes on the axis of Earth's rotation.