Steiner chain

In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.

This standard configuration allows several properties of Steiner chains to be derived, e.g., its points of tangencies always lie on a circle.

If the two given circles are tangent at a point, the Steiner chain becomes an infinite Pappus chain, which is often discussed in the context of the arbelos (shoemaker's knife), a geometric figure made from three circles.

Because Steiner chain circles are tangent to one another, the distance between their centers equals the sum of their radii, here twice their radius ρ.

The bisector (green in Figure) creates two right triangles, with a central angle of θ = 180°/n.

The sine of this angle can be written as the length of its opposite segment, divided by the hypotenuse of the right triangle Since θ is known from n, this provides an equation for the unknown radius ρ of the Steiner-chain circles The tangent points of a Steiner chain circle with the inner and outer given circles lie on a line that pass through their common center; hence, the outer radius R = r + 2ρ.

These equations provide a criterion for the feasibility of a Steiner chain for two given concentric circles.

Similarly, the n lines tangent to each pair of adjacent circles in the Steiner chain also pass through O.

The centers of the circles of a Steiner chain lie on a conic section.

Let the radius, diameter and center point of the kth circle of the Steiner chain be denoted as rk, dk and Pk, respectively.

All the centers of the circles in the Steiner chain are located on a common ellipse, for the following reason.

Then, the eccentricity e is defined by 2 ae = p, or From these parameters, the semi-minor axis b and the semi-latus rectum L can be determined Therefore, the ellipse can be described by an equation in terms of its distance d to one focus where θ is the angle with the line joining the two foci.

In the former case, this corresponds to a Pappus chain, which has an infinite number of circles.

Soddy's hexlet is a three-dimensional generalization of a Steiner chain of six circles.

Figure 1: A Steiner chain of twelve black circles ( n = 12) . The given circles are shown in blue and red, which are the outermost and innermost circles, respectively.
The radius of the Steiner circles is ρ whereas those of the inner and outer given circles are r and R , respectively. The distance from the center of the inner circle to the center of a Steiner circle is r + ρ (hypotenuse of pink triangle).
If even one closed Steiner chain is possible for two given circles (blue), then infinitely many Steiner chains are possible, all related by rotation. Their points of tangency always fall on a circle (orange). If the two given circles are nested, one inside the other, the centers of the Steiner chain circles (black) fall on an ellipse (red); otherwise, they fall on a hyperbola .
Soddy's hexlet is a three-dimensional analog of the Steiner chain.