Apollonian Circles - Pencils of Circles

Both of the families of Apollonian circles are called pencils of circles. More generally, 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 called a pencil.

Specifically, the equation of a circle of radius r centered at a point (p,q),

may be rewritten as

where α = 1, β = p, γ = q, and δ = p2 + q2 − r2. However, in this form, multiplying the 4-tuple (α,β,γ,δ) by a scalar produces a different four-tuple that represents the same circle; thus, these 4-tuples may be considered to be homogeneous coordinates for the space of circles. Straight lines may also be represented with an equation of this type in which α = 0 and should be thought of as being a degenerate form of a circle. When α ≠ 0, we may solve for p = β/α, q = γ/α, and r =√((−δ − β2 − γ2)/α2); note, however, that the latter formula may give r = 0 (in which case the circle degenerates to a point) or r equal to an imaginary number (in which case the 4-tuple (α,β,γ,δ) is said to represent an imaginary circle).

The set of affine combinations of two circles (α₁,β₁,γ₁,δ₁), (α₂,β₂,γ₂,δ₂), that is, the set of circles represented by the four-tuple

for some value of the parameter z, forms a pencil; the two circles are called generators of the pencil. There are three types of pencil:

• An elliptic pencil (red family of circles in the figure) is defined by two generators that pass through each other in exactly two points (C and D). At these points, the defining formula has zero value, and therefore will also equal zero for any affine combination. Thus, 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 (here C and D) called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil. forms a pencil of this type.
• Finally, 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 single focus C forms a special case of a hyperbolic pencil, in which the other focus is the point at infinity of the complex projective line. The corresponding elliptic pencil consists of the family of straight lines through C; these should be interpreted as circles that all pass through the point at infinity.