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| {{Redirect3|Apollonian circle|For a subdivision of this subject, see [[Apollonian circles]]}}
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| The '''circle of Apollonius''' is any of several types of circles associated with [[Apollonius of Perga]], a renowned [[Ancient Greece|Greek]] [[geometer]]. Most of these circles are found in [[plane (mathematics)|planar]] [[Euclidean geometry]], but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through [[stereographic projection]].
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| The main uses of this term are fivefold:
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| * Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ''ratio'' of distances to two fixed points known as [[focus (geometry)|foci]]. This [[Apollonian circles|circle of Apollonius]] is the basis of the Apollonius pursuit problem.
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| * The [[Apollonian circles]] are two families of mutually [[orthogonal]] circles. The first family consists of the circles with all possible distance ratios to two fixed foci, whereas the second family consists of all possible circles that pass through both foci. These circles form the basis of [[bipolar coordinates]].
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| * [[Problem of Apollonius|Apollonius' problem]] is to construct circles that are simultaneously tangent to three specified circles. The solutions to this problem are sometimes called the "circles of Apollonius".
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| * The [[Apollonian gasket]]—one of the first [[fractal]]s ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
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| * The [[isodynamic point]]s and [[Lemoine line]] of a triangle can be solved using three circles, each of which passes through one vertex of the triangle and maintains a constant ratio of distances to the other two.
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| ==Apollonius' definition of a circle==
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| [[File:Apollonius circle definition labels.svg|thumb|right|250px|Figure 1. Apollonius' definition of a circle.]]
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| {{main|Apollonian circles}}
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| A circle is usually defined as the set of points '''P''' at a given distance ''r'' (the circle's radius) from a given point (the circle's center). However, there are other, equivalent definitions of a circle. Apollonius discovered that a circle could also be defined as the set of points '''P''' that have a given ''ratio'' of distances ''k'' = ''d''<sub>1</sub>/''d''<sub>2</sub> to two given points (labeled '''A''' and '''B''' in Figure 1). These two points are sometimes called the [[focus (geometry)|foci]].
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| ===Apollonius pursuit problem===
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| The Apollonius pursuit problem is one of finding where a ship leaving from one point '''A''' at speed ''v''<sub>1</sub> will intercept another ship leaving a different point '''B''' at speed ''v''<sub>2</sub>. By assumption, the ships travel in straight lines and the ratio of their speeds is denoted as ''k'' = ''v''<sub>1</sub>/''v''<sub>2</sub>. At the point they meet, the first ship will have traveled a ''k''-fold longer distance than the second ship. Therefore, the point must lie on a circle as defined by Apollonius, with their starting points as the foci.
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| [[File:Apollonian circles.svg||thumb|left|250px|Figure 2: A set of Apollonian circles. Every blue circle intersects every red circle at a right angle, and vice versa. Every red circle passes through the two foci, which correspond to points '''A''' and '''B''' in Figure 1.]]
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| ==Circles sharing a radical axis==
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| {{main|Apollonian circles}}
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| The circles defined by the Apollonian pursuit problem for the same two points '''A''' and '''B''', but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a ''hyperbolic pencil''. Another family of circles, the circles that pass through both '''A''' and '''B''', are also called a pencil, or more specifically an ''elliptic pencil''. These two pencils of [[Apollonian circles]] intersect each other at [[right angle]]s and form the basis of the [[Bipolar coordinates|bipolar coordinate system]]. Within each pencil, any two circles have the same [[radical axis]]; the two radical axes of the two pencils are perpendicular, and the centers of the circles from one pencil lie on the radical axis of the other pencil.
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| ==Solutions to Apollonius' problem==
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| [[File:Apollonius8ColorMultiplyV2.svg|right|thumb|300px|Figure 3:[[Apollonius' problem]] may have up to eight solutions. The three given circles are shown in black, whereas the solution circles are colored.]]
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| {{main|Problem of Apollonius}}
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| ==Apollonian gasket==
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| {{main|Apollonian gasket}}
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| [[File:Apollonian gasket.svg|thumb|left|Figure 4: A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor [[Gottfried Leibniz]].]]
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| By solving Apollonius' problem repeatedly to find the inscribed circle, the [[wikt:interstice|interstice]]s between mutually tangential circles can be filled arbitrarily finely, forming an [[Apollonian gasket]], also known as a ''Leibniz packing'' or an ''Apollonian packing''.<ref>{{cite journal | author = Kasner, E., and Supnick, F. | year = 1943 | title = The Apollonian packing of circles | journal = Proceedings of the National Academy of Sciences USA | volume = 29 | issue = 11 | pages = 378–384 | doi = 10.1073/pnas.29.11.378 | pmc = 1078636 | pmid=16588629}}</ref> This gasket is a [[fractal]], being self-similar and having a [[Hausdorff dimension|dimension]] ''d'' that is not known exactly but is roughly 1.3,<ref name="boyd_1973">{{cite journal | author = Boyd, D.W. | year = 1973 | title = Improved Bounds for the Disk Packing Constants | journal = Aequationes Mathematicae | volume = 9 | pages = 99–106 | doi = 10.1007/BF01838194}}<br />{{cite journal | author = Boyd, D.W. | year = 1973 | title = The Residual Set Dimension of the Apollonian Packing | journal = Mathematika | volume = 20 | pages = 170–174 | doi = 10.1112/S0025579300004745 | issue = 2}}<br />{{cite journal|last=McMullen|first= Curtis, T.|title= Hausdorff dimension and conformal dynamics III: Computation of dimension|url=http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf|journal=American Journal of Mathematics | volume=120|year=1998|pages=691–721|format=PDF|doi=10.1353/ajm.1998.0031|issue=4}}</ref> which is higher than that of a [[regular curve|regular]] (or [[rectifiable curve|rectifiable]]) curve (''d''=1) but less than that of a plane (''d''=2). The Apollonian gasket was first described by [[Gottfried Leibniz]] in the 17th century, and is a curved precursor of the 20th-century [[Sierpiński triangle]].<ref>{{cite book | author = [[Benoit Mandelbrot|Mandelbrot, B.]] | year = 1983 | title = The Fractal Geometry of Nature | publisher = W.H. Freeman | location = New York | isbn = 978-0-7167-1186-5 | page = 170}}<br />{{cite book | author = Aste, T., and [[Denis Weaire|Weaire, D.]] | year = 2008 | title = In Pursuit of Perfect Packing | edition = 2nd | publisher = Taylor and Francis | location = New York | isbn = 978-1-4200-6817-7 | pages = 131–138}}</ref> The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of [[Kleinian group]]s.<ref>{{cite book | author = [[David Mumford|Mumford, D.]], Series, C., and Wright, D. | year = 2002 | title = Indra's Pearls: The Vision of Felix Klein | publisher = Cambridge University Press | location = Cambridge | isbn = 0-521-35253-3 | pages = 196–223}}</ref>
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| ==Isodynamic points of a triangle==
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| '''Circles of Apollonius''' may be used as a technical term to denote three special circles <math>\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}</math> defined by an arbitrary triangle <math>\mathrm{A_{1}A_{2}A_{3}}</math>. The circle <math>\mathcal{C}_{1}</math> is defined as the unique circle passing through the triangle vertex <math>\mathrm{A_{1}}</math> that maintains a constant ratio of distances to the other two vertices <math>\mathrm{A_{2}}</math> and <math>\mathrm{A_{3}}</math> (cf. Apollonius' definition of the [[circle]] above). Similarly, the circle <math>\mathcal{C}_{2}</math> is defined as the unique circle passing through the triangle vertex <math>\mathrm{A_{2}}</math> that maintains a constant ratio of distances to the other two vertices <math>\mathrm{A_{1}}</math> and <math>\mathrm{A_{3}}</math>, and so on for the circle <math>\mathcal{C}_{3}</math>.
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| All three circles intersect the [[circumcircle]] of the [[triangle]] [[orthogonal]]ly. All three circles pass through two points, denoted as the [[isodynamic point]]s <math>S</math> and <math>S^{\prime}</math> of the triangle. The line connecting these common intersection points is the [[radical axis]] for all three circles. The two isodynamic points are [[circle inversion|inverses]] of each other relative to the [[circumcircle]] of the triangle.
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| The centers of these three circles fall on a single line (the '''Lemoine line'''). This line is perpendicular to the radical axis defined by the [[isodynamic point]]s <math>S</math> and <math>S^{\prime}</math>.
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| ==See also==
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| *[[Apollonius point]]
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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| {{commons category|Circles of Apollonius}}
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| * Ogilvy, C.S. (1990) ''Excursions in Geometry'', Dover. ISBN 0-486-26530-7.
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| * Johnson, R.A. (1960) ''Advanced Euclidean Geometry'', Dover.
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| {{DEFAULTSORT:Circles Of Apollonius}}
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| [[Category:Circles|Apollonius]]
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Nice to meet you, my title is Figures Held although I don't truly like being called like that. Hiring is her working day occupation now and she will not change it whenever soon. One of the issues she enjoys most is to do aerobics and now she is attempting to make cash with it. South Dakota is exactly where me and my husband reside.
my page - xrambo.com