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.

Analogously, a set of geometric objects that are determined by any three of its members is called a bundle.

The common ones are lines, planes, circles, conics, spheres, and general curves.

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

[4] 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.

[8] For example, the meridians of the globe are defined by the pencil of planes on the axis of Earth's rotation.

can be seen as an axial pencil of complex planes all sharing the same real line.

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.

[9] To be inclusive, concentric circles are said to have the line at infinity as a radical axis.

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.

Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear.

There is a natural correspondence between circles in the plane and points in three-dimensional projective space(see below); a line in this space corresponds to a one-dimensional continuous family of circles, hence a pencil of points in this space is a pencil of circles in the plane.

by a scalar produces a different quadruple that represents the same circle; thus, these quadruples may be considered to be homogeneous coordinates for the space of circles.

[14] Straight lines may also be represented with an equation of this type in which

, that is, the set of circles represented by the quadruple for some value of the parameter

More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.

[17] A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.

In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points.

Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.

be two distinct conics in a projective plane defined over an algebraically closed field

, not both zero, the expression: represents a conic in the pencil determined by

This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.)

Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.

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

Desargues is credited with inventing the term "pencil of lines" (ordonnance de lignes).

[21] 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.