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. A more common term for this set is a range of points.
In a plane, let and be two distinct intersecting lines. For concreteness, suppose that has the equation, and has the equation . Then
,represents, for suitable scalars and, any line passing through the intersection of = 0 and = 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 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,[1] 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:
A circle that is orthogonal to two fixed circles is orthogonal to every circle in the pencil they determine.
The circles orthogonal to two fixed circles form a pencil of circles.
Two circles determine two pencils, the unique pencil that contains them and the pencil of circles orthogonal to them. The radical axis of one pencil consists of the centers of the circles of the other pencil. If one pencil is of elliptic type, the other is of hyperbolic type and vice versa.
The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil.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; 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.
Specifically, the equation of a circle of radius centered at a point,
(x-p)2+(y-q)2=r2,
\alpha(x2+y2)-2\betax-2\gammay+\delta=0,
The set of affine combinations of two circles,, that is, the set of circles represented by the quadruple
z(\alpha1,\beta1,\gamma1,\delta1)+(1-z)(\alpha2,\beta2,\gamma2,\delta2)
Another type of pencil of circles can be obtained as follows. Consider a given circle (called the generator circle) and a distinguished point on the generator circle. The set of all circles that pass through and have their centers on the generator circle form a pencil of circles. The envelope of this pencil is a cardioid.
A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.[3] This property is analogous to the property that three non-collinear points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres.[4] Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).[5]
If and are the equations of two distinct spheres then
λf(x,y,z)+\mug(x,y,z)=0
If the pencil of spheres does not consist of all planes, then there are three types of pencils:
All the tangent lines from a fixed point of the radical plane to the spheres of a pencil have the same length.
The radical plane is the locus of the centers of all the spheres that are orthogonal to all the spheres in a pencil. Moreover, a sphere orthogonal to any two spheres of a pencil of spheres is orthogonal to all of them and its center lies in the radical plane of the pencil.
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.[6] The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.
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.[7]
A pencil of conics can be represented algebraically in the following way. Let and be two distinct conics in a projective plane defined over an algebraically closed field . For every pair of elements of, not both zero, the expression:
λC1+\muC2
represents a conic in the pencil determined by and . 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.) Thinking of, say, as a ternary quadratic form, then is the equation of the "conic ". Another concrete realization would be obtained by thinking of as the 3×3 symmetric matrix which represents it. If and have such concrete realizations then every member of the above pencil will as well. 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
λC+\muC'=0
where, 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."