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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 (''n''th) 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: ::If at least one closed Steiner chain of ''n'' circles exists for two given circles ''α'' and ''β'', then there is an infinite number of closed Steiner chains of ''n'' circles; and any circle tangent to ''α'' and ''β'' in the same way is a member of such a chain. "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.〔Ogilvy, p. 60.〕 ==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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Steiner chain」の詳細全文を読む スポンサード リンク
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