Steiner Chain

In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual ''closed'' Steiner chains, the first and last (-th) circles are also tangent to each other; by contrast, in ''open'' Steiner chains, they need not be. The given circles and do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.

Steiner chains are named after Jakob Steiner, who defined them in the 19th century and discovered many of their properties. A fundamental result is ''Steiner's porism'', which states:

::If at least one closed Steiner chain of circles exists for two given circles and , then there is an infinite number of closed Steiner chains of circles; and any circle tangent to and in the same way is a member of such a chain.

The method of circle inversion is helpful in treating Steiner chains. Since it preserves tangencies, angles and circles, inversion transforms one Steiner chain into another of the same number of circles. One particular choice of inversion transforms the given circles and into concentric circles; in this case, all the circles of the Steiner chain have the same size and can "roll" around in the annulus between the circles similar to ball bearings. This standard configuration allows several properties of Steiner chains to be derived, e.g., its points of tangencies always lie on a circle. Several generalizations of Steiner chains exist, most notably Soddy's hexlet and Pappus chains.

Definitions and types of tangency

Image:Steiner_chain_7mer.svg, The 7 circles of this Steiner chain (black) are externally tangent to the inner given circle (red) but internally tangent to the outer given circle (blue).
Image:Steiner_chain_7mer_all_external.svg, The 7 circles of this Steiner chain (black) are externally tangent to both given circles (red and blue), which lie outside one another.
Image:Steiner_chain_8mer_all_but_one_external.svg, Seven of the 8 circles of this Steiner chain (black) are externally tangent to both given circles (red and blue); the 8th circle is internally tangent to both.

The two given circles ''α'' and ''β'' cannot intersect; hence, the smaller given circle must lie inside or outside the larger. The circles are usually shown as an annulus, i.e., with the smaller given circle inside the larger one. In this configuration, the Steiner-chain circles are externally tangent to the inner given circle and internally tangent to the outer circle. However, the smaller circle may also lie completely outside the larger one (Figure 2). The black circles of Figure 2 satisfy the conditions for a closed Steiner chain: they are all tangent to the two given circles and each is tangent to its neighbors in the chain. In this configuration, the Steiner-chain circles have the same type of tangency to both given circles, either externally or internally tangent to both. 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. There is no general name for a sequence of circles tangent to two given circles that intersect at two points.

Closed, open and multi-cyclic

Image:Steiner_chain_9mer_annular.svg, Closed Steiner chain of nine circles. The 1st and 9th circles are tangent.
Image:Steiner_chain_open_9mer.svg, Open Steiner chain of nine circles. The 1st and 9th circles overlap.
Image:Steiner_chain_double_17mer.svg, Multicyclic Steiner chain of 17 circles in 2 wraps. The 1st and 17th circles touch.

The two given circles ''α'' and ''β'' touch the ''n'' circles of the Steiner chain, but each circle ''C''_''k'' of a Steiner chain touches only four circles: ''α'', ''β'', and its two neighbors, ''C''_''k''−1 and ''C''_''k''+1. By default, Steiner chains are assumed to be ''closed'', i.e., the first and last circles are tangent to one another. By contrast, an ''open'' Steiner chain is one in which the first and last circles, ''C''₁ and ''C''_''n'', are not tangent to one another; these circles are tangent only to ''three'' circles. Multicyclic Steiner chains wrap around the inner circle more than once before closing, i.e., before being tangent to the initial circle.

Closed Steiner chains are the systems of circles obtained as the circle packing theorem representation of a bipyramid.

Annular case and feasibility criterion

Image:Steiner_chain_3mer_annular.svg,
Image:Steiner_chain_6mer_annular.svg,
Image:Steiner_chain_9mer_annular.svg,
Image:Steiner_chain_12mer_annular.svg,
Image:Steiner_chain_20mer_annular.svg,

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 2''θ'' 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 . The sine of this angle can be written as the length of its opposite segment, divided by the hypotenuse of the right triangle

:$\sin \theta = \frac$

Since ''θ'' is known from ''n'', this provides an equation for the unknown radius ''ρ'' of the Steiner-chain circles

:$\rho = \frac$

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 .

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

:
\frac = 1 + \frac = \frac = \left \sec \theta + \tan \theta \right

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