Steiner chain

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where n 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 (nth) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β 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.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).The 7 circles of this Steiner chain (black) are externally tangent to both given circles (red and blue), which lie outside one another.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.Closed Steiner chain of nine circles. The 1st and 9th circles are tangent.Open Steiner chain of nine circles. The 1st and 9th circles overlap.Multicyclic Steiner chain of 17 circles in 2 wraps. The 1st and 17th circles touch.n = 3n = 6n = 9n = 12n = 20Two circles (pink and cyan) that are internally tangent to both given circles and whose centers are collinear with the center of the given circles intersect at the angle 2θ.Under inversion, these lines and circles become circles with the same intersection angle, 2θ. The gold circles intersect the two given circles at right angles, i.e., orthogonally.The circles passing through the mutual tangent points of the Steiner-chain circles are orthogonal to the two given circles and intersect one another at multiples of the angle 2θ.The circles passing through the tangent points of the Steiner-chain circles with the two given circles are orthogonal to the latter and intersect at multiples of the angle 2θ.Steiner chain with the two given circles shown in red and blue.Same set of circles, but with a different choice of given circles.Same set of circles, but with yet another choice of given circles. In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where n 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 (nth) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β 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: 'Tangent in the same way' means that the arbitrary circle is internally or externally tangent in the same way as a circle of the original Steiner chain. A porism is a type of theorem relating to the number of solutions and the conditions on it. Porisms often describe a geometrical figure that cannot exist unless a condition is met, but otherwise may exist in infinite number; another example is Poncelet's porism. 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 α and β 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. 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. The two given circles α and β touch the n circles of the Steiner chain, but each circle Ck of a Steiner chain touches only four circles: α, β, and its two neighbors, Ck−1 and Ck+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, C1 and Cn, are not tangent to one another; these circles are tangent only to three circles. Multicyclic Steiner chains wrap around the inner circle several times before closing, i.e., before being tangent to the initial circle. 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.