Pencil (geometry)

Family of geometric objects with a common property
a class of geometric objects

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. 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. 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. 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 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. Two intersecting planes meet in a line in three space, and so, determine the axis and hence all of the planes in the pencil. The four space of quaternions can be seen as an axial pencil of complex planes all sharing the same real line. In fact, quaternions contain a sphere of imaginary units, and a pair of antipodal points on this sphere, together with the real axis, generate a complex plane. The union of all these complex planes constitutes the 4 algebra of quaternions.

Any two circles in the plane have a common radical axis, which is the line consisting of all the points that have the same power with respect to the two circles. A pencil of circles (or coaxial system) is the set of all circles in the plane with the same radical axis. To be inclusive, concentric circles are said to have the line at infinity as a radical axis. There are five types of pencils of circles, the two families of Apollonian circles in the illustration above represent two of them. Each type is determined by two circles called the generators of the pencil. When described algebraically, it is possible that the equations may admit imaginary solutions. The types are:
An elliptic pencil (red family of circles in the figure) is defined by two generators that pass through each other in exactly two points. Every circle of an elliptic pencil passes through the same two points. An elliptic pencil does not include any imaginary circles. A hyperbolic pencil (blue family of circles in the figure) is defined by two generators that do not intersect each other at any point. It includes real circles, imaginary circles, and two degenerate point circles called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil. A parabolic pencil (as a limiting case) is defined where two generating circles are tangent to each other at a single point. It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil. A family of concentric circles centered at a common center (may be considered a special case of a hyperbolic pencil where the other point is the point at infinity). The family of straight lines through a common point; these should be interpreted as circles that all pass through the point at infinity (may be considered a special case of an elliptic pencil).

Another type of pencil of circles can be obtained as follows. Consider a given circle (called the generator circle) and a distinguished point P on the generator circle. The set of all circles that pass through P and have their centers on the generator circle form a pencil of circles. The envelope of this pencil is a cardioid.

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

where 1=C = 0, 1=C′ = 0 are plane curves.

Desargues is credited with inventing the term "pencil of lines" (ordonnance de lignes). An early author of modern projective geometry G. B. Halsted introduced the terms copunctal and flat pencil to define angle: "Straights with the same cross are copunctal." Also "The aggregate of all coplanar, copunctal straights is called a flat pencil" and "A piece of a flat pencil bounded by two of the straights as sides, is called an angle."