Inversive distance: Difference between revisions
usually? Bowers & Hurdal use InvDIst, not delta |
it's the inverse hyperbolic cosine: cosh delta = frac{...} not delta = cosh frac{...}. Add another ref that explains why the cosh is a useful transformation here. Sectionize. Add applications. |
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In [[inversive geometry]], the '''inversive distance''' is a way of measuring the "[[distance]]" between two [[circle]]s, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other. <ref name="bh"/> |
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==Properties== |
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The inversive distance remains unchanged if the circles are [[inversive geometry|inverted]], or transformed by a [[Möbius transformation]].<ref name="bh"/><ref name="ampa"/> 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.<ref name="bh"/> |
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==Distance formula== |
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For two circles in the [[Euclidean plane]] with radii <math>r</math> and <math>R</math>, and distance <math>d</math> between their centers, the inversive distance can be defined |
For two circles in the [[Euclidean plane]] with radii <math>r</math> and <math>R</math>, and distance <math>d</math> between their centers, the inversive distance can be defined |
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by the formula<ref name="bh">{{citation |
by the formula<ref name="bh">{{citation |
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This formula gives a value of 1 for two circles that are tangent to each other, less than 1 for two circles that cross, and greater than one for two disjoint circles. |
This formula gives a value of 1 for two circles that are tangent to each other, less than 1 for two circles that cross, and greater than one for two disjoint circles. |
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Some authors modify this formula by taking the [[hyperbolic cosine]] of the value given above, rather than the value itself.<ref>{{ |
Some authors modify this formula by taking the [[inverse hyperbolic cosine]] of the value given above, rather than the value itself.<ref name="ampa"/><ref>{{citation|title = Geometry Revisited| author1-link = Harold Scott MacDonald Coxeter | last1=Coxeter | first1=H.S.M. | author2-link=S. L. Greitzer | last2=Greitzer | first2=S.L. | year = 1967| publisher = [[Mathematical Association of America]]| location = [[Washington, D.C.]] | series=[[New Mathematical Library]] | volume=19 | isbn = 978-0-88385-619-2| zbl=0166.16402 | pages=123–124 }}</ref> Although transforming the inversive distance in this way makes the distance formula more complicated, it has the advantage that (like the usual distance for points on a line) the distance becomes additive for circles in a [[pencil of circles]]. That is, if three circles belong to a common pencil, one of their three distances will be the sum of the other two.<ref name="ampa">{{citation |
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| last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter |
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| doi = 10.1007/BF02413734 |
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| journal = Annali di Matematica Pura ed Applicata |
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| mr = 0203568 |
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| pages = 73–83 |
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| title = Inversive distance |
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| volume = 71 |
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| year = 1966}}.</ref> |
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==In other geometries== |
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It is also possible to define the inversive distance for circles on a [[sphere]], or for circles in the [[hyperbolic plane]].<ref name="bh"/> |
It is also possible to define the inversive distance for circles on a [[sphere]], or for circles in the [[hyperbolic plane]].<ref name="bh"/> |
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==Applications== |
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The inversive distance has been used to define the concept of an inversive-distance [[circle packing]]: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a [[planar graph]] ) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the [[circle packing theorem]], in which specified pairs of circles are tangent to each other.<ref name="bh"/><ref>{{citation |
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| last1 = Bowers | first1 = Philip L. |
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| last2 = Stephenson | first2 = Kenneth |
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| contribution = 8.2 Inversive distance packings |
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| doi = 10.1090/memo/0805 |
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| mr = 2053391 |
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| pages = 78–82 |
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| series = Memoirs of the American Mathematical Society |
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| title = Uniformizing dessins and Belyĭ maps via circle packing |
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| volume = 805 |
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| year = 2004}}.</ref> 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 [[maximal planar graph]] and set of Euclidean or hyperbolic inversive distances; this [[Rigidity (mathematics)|rigidity property]] can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated [[manifold]]s.<ref>{{citation |
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| last = Luo | first = Feng |
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| doi = 10.2140/gt.2011.15.2299 |
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| issue = 4 |
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| journal = Geometry & Topology |
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| mr = 2862158 |
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| pages = 2299–2319 |
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| title = Rigidity of polyhedral surfaces, III |
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| volume = 15 |
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| year = 2011}}.</ref> However, for manifolds with spherical geometry, these packings are no longer unique.<ref>{{citation |
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| last1 = Ma | first1 = Jiming |
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| last2 = Schlenker | first2 = Jean-Marc |
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| doi = 10.1007/s00454-012-9399-3 |
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| issue = 3 |
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| journal = Discrete Comput. Geom. |
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| mr = 2891251 |
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| pages = 610–617 |
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| title = Non-rigidity of spherical inversive distance circle packings |
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| volume = 47 |
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| year = 2012}}.</ref> In turn, inversive-distance circle packings have been used to construct approximations to [[conformal mapping]]s.<ref name="bh"/> |
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== See also == |
== See also == |
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[[Category:Inversive geometry]] |
[[Category:Inversive geometry]] |
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{{elementary-geometry-stub}} |
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Revision as of 00:56, 15 June 2014
In inversive geometry, the inversive distance is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other. [1]
Properties
The inversive distance remains unchanged if the circles are inverted, or transformed by a Möbius transformation.[1][2] 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.[1]
Distance formula
For two circles in the Euclidean plane with radii and , and distance between their centers, the inversive distance can be defined by the formula[1]
This formula gives a value of 1 for two circles that are tangent to each other, less than 1 for two circles that cross, and greater than one for two disjoint circles.
Some authors modify this formula by taking the inverse hyperbolic cosine of the value given above, rather than the value itself.[2][3] Although transforming the inversive distance in this way makes the distance formula more complicated, it has the advantage that (like the usual distance for points on a line) the distance becomes additive for circles in a pencil of circles. That is, if three circles belong to a common pencil, one of their three distances will be the sum of the other two.[2]
In other geometries
It is also possible to define the inversive distance for circles on a sphere, or for circles in the hyperbolic plane.[1]
Applications
The inversive distance has been used to define the concept of an inversive-distance circle packing: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a planar graph ) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the circle packing theorem, in which specified pairs of circles are tangent to each other.[1][4] 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 maximal planar graph and set of Euclidean or hyperbolic inversive distances; this rigidity property can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated manifolds.[5] However, for manifolds with spherical geometry, these packings are no longer unique.[6] In turn, inversive-distance circle packings have been used to construct approximations to conformal mappings.[1]
See also
References
- ^ a b c d e f g Bowers, Philip L.; Hurdal, Monica K. (2003), "Planar conformal mappings of piecewise flat surfaces", in Hege, Hans-Christian; Polthier, Konrad (eds.), Visualization and Mathematics III, Mathematics and Visualization, Springer, pp. 3–34, doi:10.1007/978-3-662-05105-4_1, MR 2046999.
- ^ a b c Coxeter, H. S. M. (1966), "Inversive distance", Annali di Matematica Pura ed Applicata, 71: 73–83, doi:10.1007/BF02413734, MR 0203568.
- ^ Coxeter, H.S.M.; Greitzer, S.L. (1967), Geometry Revisited, New Mathematical Library, vol. 19, Washington, D.C.: Mathematical Association of America, pp. 123–124, ISBN 978-0-88385-619-2, Zbl 0166.16402
- ^ Bowers, Philip L.; Stephenson, Kenneth (2004), "8.2 Inversive distance packings", Uniformizing dessins and Belyĭ maps via circle packing, Memoirs of the American Mathematical Society, vol. 805, pp. 78–82, doi:10.1090/memo/0805, MR 2053391.
- ^ Luo, Feng (2011), "Rigidity of polyhedral surfaces, III", Geometry & Topology, 15 (4): 2299–2319, doi:10.2140/gt.2011.15.2299, MR 2862158.
- ^ Ma, Jiming; Schlenker, Jean-Marc (2012), "Non-rigidity of spherical inversive distance circle packings", Discrete Comput. Geom., 47 (3): 610–617, doi:10.1007/s00454-012-9399-3, MR 2891251.