436:
for two disjoint circles is a finite cyclic sequence of additional circles, each of which is tangent to the two given circles and to its two neighbors in the chain. Steiner's porism states that if two circles have a
Steiner chain, they have infinitely many such chains. The chain is allowed to wrap
645:, in which specified pairs of circles are tangent to each other. Although less is known about the existence of inversive distance circle packings than for tangent circle packings, it is known that, when they exist, they can be uniquely specified (up to Möbius transformations) by a given
358:
Although transforming the inversive distance in this way makes the distance formula more complicated, and prevents its application to crossing pairs of circles, it has the advantage that (like the usual distance for points on a line) the distance becomes additive for circles in a
592:
661:
at their vertices. However, for manifolds with spherical geometry, these packings are no longer unique. In turn, inversive-distance circle packings have been used to construct approximations to
353:
230:
601:
will support a
Steiner chain. More generally, an arbitrary pair of disjoint circles can be approximated arbitrarily closely by pairs of circles that support Steiner chains whose
457:
whose numerator is the number of circles in the chain and whose denominator is the number of times it wraps around. All chains for the same two circles have the same value of
495:
381:
309:
76:
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128:
108:
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Bowers, Philip L.; Hurdal, Monica K. (2003), "Planar conformal mappings of piecewise flat surfaces", in Hege, Hans-Christian; Polthier, Konrad (eds.),
47:. One pair of circles can be transformed to another pair by a Möbius transformation if and only if both pairs have the same inversive distance.
851:
982:
843:
58:
of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance
523:
823:
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641:) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the
317:
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51:
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31:, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.
622:
268:
835:
44:
637:: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a
650:
642:
980:
Ma, Jiming; Schlenker, Jean-Marc (2012), "Non-rigidity of spherical inversive distance circle packings",
264:(Some authors define the absolute inversive distance as the absolute value of the inversive distance.)
646:
477:. If the inversive distance between the two circles (after taking the inverse hyperbolic cosine) is
403:
as the inversive distance) one of their three pairwise distances will be the sum of the other two.
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1034:
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of the value given above, rather than the value itself. That is, rather than using the number
480:
366:
294:
61:
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a value of −1 for two circles that are tangent to each other, one inside of the other,
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Lester, J. A. (1991), "A Beckman-Quarles type theorem for
Coxeter's inversive distance",
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a value of 1 for two circles that are tangent to each other and both outside each other,
1038:
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more than once around the two circles, and can be characterized by a rational number
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The inversive distance has been used to define the concept of an inversive-distance
638:
896:
Bowers, Philip L.; Stephenson, Kenneth (2004), "8.2 Inversive distance packings",
78:, then it must be a Möbius transformation that preserves all inversive distances.
883:
704:
249:
900:, Memoirs of the American Mathematical Society, vol. 170, pp. 78–82,
653:
can be generalized broadly, to
Euclidean or hyperbolic metrics on triangulated
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1043:
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150:
between their centers, the inversive distance can be defined by the formula
55:
800:
654:
291:
as the inversive distance, the distance is instead defined as the number
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Conversely, every two disjoint circles for which this formula gives a
411:
It is also possible to define the inversive distance for circles on a
412:
28:
363:. That is, if three circles belong to a common pencil, then (using
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941:
260:
and a value less than −1 when one circle contains the other.
245:
a value between −1 and 1 for two circles that intersect,
874:
Deza, Michel Marie; Deza, Elena (2014), "Inversive distance",
649:
and set of
Euclidean or hyperbolic inversive distances. This
587:{\displaystyle p={\frac {\pi }{\sin ^{-1}\tanh(\delta /2)}}.}
39:
The inversive distance remains unchanged if the circles are
927:
Luo, Feng (2011), "Rigidity of polyhedral surfaces, III",
699:, Mathematics and Visualization, Springer, pp. 3–34,
248:
a value of 0 for two circles that intersect each other at
625:
to the value of this formula for the given two circles.
898:
Uniformizing dessins and BelyÄ maps via circle packing
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348:{\displaystyle \delta =\operatorname {arcosh} (I).}
225:{\displaystyle I={\frac {d^{2}-r^{2}-R^{2}}{2rR}}.}
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27:is a way of measuring the "distance" between two
239:a value greater than 1 for two disjoint circles,
267:Some authors modify this formula by taking the
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54:holds true for the inversive distance: if a
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878:(3rd ed.), Springer, p. 369,
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697:Visualization and Mathematics III
787:Canadian Mathematical Bulletin
736:(1966), "Inversive distance",
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1:
16:Concept in inversive geometry
884:10.1007/978-3-662-44342-2_19
517:can be found by the formula
705:10.1007/978-3-662-05105-4_1
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1007:10.1007/s00454-012-9399-3
876:Encyclopedia of Distances
269:inverse hyperbolic cosine
836:New Mathematical Library
415:, or for circles in the
951:10.2140/gt.2011.15.2299
929:Geometry & Topology
623:rational approximations
490:{\displaystyle \delta }
376:{\displaystyle \delta }
304:{\displaystyle \delta }
86:For two circles in the
71:{\displaystyle \delta }
52:Beckman–Quarles theorem
801:10.4153/CMB-1991-079-6
643:circle packing theorem
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45:Möbius transformation
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647:maximal planar graph
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235:This formula gives:
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407:In other geometries
50:An analogue of the
1062:Inversive geometry
1035:Weisstein, Eric W.
832:Geometry Revisited
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663:conformal mappings
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21:inversive geometry
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853:978-0-88385-619-2
734:Coxeter, H. S. M.
651:rigidity property
614:{\displaystyle p}
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510:{\displaystyle p}
470:{\displaystyle p}
450:{\displaystyle p}
396:{\displaystyle I}
361:pencil of circles
284:{\displaystyle I}
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143:{\displaystyle d}
123:{\displaystyle R}
103:{\displaystyle r}
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828:Greitzer, S.L.
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423:Applications
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383:in place of
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250:right angles
234:
85:
49:
38:
24:
18:
621:values are
90:with radii
862:0166.16402
669:References
35:Properties
1044:MathWorld
997:1105.1469
967:119609724
942:1010.3284
768:186215958
744:: 73–83,
655:manifolds
565:δ
559:
553:
545:−
536:π
485:δ
371:δ
331:
322:δ
299:δ
193:−
180:−
66:δ
56:bijection
1056:Category
830:(1967),
41:inverted
1016:2891251
959:2862158
914:2053391
810:1136651
760:0203568
713:2046999
497:, then
29:circles
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413:sphere
328:arcosh
23:, the
992:arXiv
963:S2CID
937:arXiv
764:S2CID
657:with
848:ISBN
556:tanh
110:and
1002:doi
947:doi
902:doi
880:doi
858:Zbl
796:doi
746:doi
701:doi
541:sin
19:In
1058::
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1037:,
1012:MR
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