Steiner Chain - Annular Case and Feasibility Criterion

Annular Case and Feasibility Criterion

The simplest type of Steiner chain is a closed chain of n circles of equal size surrounding an inscribed circle of radius r; the chain of circles is itself surrounded by a circumscribed circle of radius R. The inscribed and circumscribed given circles are concentric, and the Steiner-chain circles lie in the annulus between them. By symmetry, the angle 2θ between the centers of the Steiner-chain circles is 360°/n. 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. A closed Steiner chain of n circles requires that the ratio of radii R/r of the given circles equal exactly

As shown below, this ratio-of-radii criterion for concentric given circles can be extended to all types of given circles by the inversive distance δ of the two given circles. For concentric circles, this distance is defined as a logarithm of their ratio of radii

Using the solution for concentric circles, the general criterion for a Steiner chain of n circles can be written

If a multicyclic annular Steiner chain has n total circles and wraps around m times before closing, the angle between Steiner-chain circles equals

In other respects, the feasibility criterion is unchanged.