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.
Contents
Pencil of lines
In a plane, let
ℓ
{\displaystyle \ell }
and
ℓ
′
{\displaystyle \ell '}
be two distinct intersecting lines. For concreteness, suppose that
ℓ
{\displaystyle \ell }
has the equation,
a
X
+
b
Y
+
c
=
0
{\displaystyle aX+bY+c=0}
and
ℓ
′
{\displaystyle \ell '}
has the equation
a
′
X
+
b
′
Y
Pencil of planes
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
H
=
{
a
+
b
i
+
c
j
+
d
k
|
a
,
b
,
c
,
d
∈
R
}
{\displaystyle \mathbb {H} =\{a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} |a,b,c,d\in \mathbb {R} \}}
Pencil of circles
Any pair of circles in the plane has a 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).
Properties
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.
Projective space of circles
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.
Specifically, the equation of a circle of radius
r
{\displaystyle r}
centered at a point
(
p
,
q
)
{\displaystyle (p,q)}
,
(
x
−
p
)
2
+
(
y
−
q
)
2
=
r
2
,
{\displaystyle (x-p)^{2}+(y-q)^{2}=r^{2},}
may be rewritten as
Cardioid as envelope of a pencil of circles
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.
Pencil of spheres
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. 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. 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).
If
f
(
x
,
y
,
z
)
=
0
{\displaystyle f(x,y,z)=0}
and
g
(
x
,
y
,
z
)
Pencil of conics
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. 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.
A pencil of conics can be represented algebraically in the following way. Let
C
1
{\displaystyle C_{1}}
and
C
2
{\displaystyle C_{2}}
be two distinct conics in a projective plane defined over an algebraically closed field
K
{\displaystyle K}
. For every pair
(
λ
,
μ
)
{\displaystyle (\lambda ,\mu )}
Pencil of plane curves
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
λ
C
+
μ
C
′
=
0
{\displaystyle \lambda C+\mu C'=0}
where
C
=
0
{\displaystyle C=0}
,
C
′
=
0
{\displaystyle C'=0}
are plane curves.
History
Girard 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."