View source for Apollonian circles

{{Short description|Circles in two perpendicular families}}
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{{About|a family of circles sharing a [[radical axis]], and the corresponding family of orthogonal circles|other circles associated with [[Apollonius of Perga]]|circles of Apollonius}}
[[File:Apollonian circles.svg|thumb|upright=1.0|Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points {{mvar|C, D}}, and every blue circle separates the two points.]]

In [[geometry]], '''Apollonian circles''' are two families ([[Pencil (geometry)|pencils]]) of [[circle]]s such that every circle in the first family intersects every circle in the second family [[orthogonal]]ly, and vice versa. These circles form the basis for [[bipolar coordinates]]. They were discovered by [[Apollonius of Perga]], a renowned [[Greek mathematics|ancient Greek geometer]].

==Definition==
[[File:Apolloniuskreis en.svg|thumb|upright=1.25|Apollonian circle, the angle bisectors in X yield <math>\frac{|CT_i|}{|DT_i|}=\frac{|CT_o|}{|DT_o|}=r</math>, due <math>\angle T_iXT_o = \frac{180^\circ}{2}=90^\circ </math> and [[Thales's theorem]] X is located on a half circle with diameter <math>CD</math>]]
The Apollonian circles are defined in two different ways by a [[line segment]] denoted {{mvar|CD}}.

Each circle in the first family (the blue circles in the figure) is associated with a positive [[real number]] {{mvar|r}}, and is defined as the locus of points {{mvar|X}} such that the ratio of distances from {{mvar|X}} to {{mvar|C}} and to {{mvar|D}} equals {{mvar|r}},
<math display=block>\left\{X \ \Biggl| \ \frac{d(X,C)}{d(X,D)} = r\right\}.</math>
For values of {{mvar|r}} close to zero, the corresponding circle is close to {{mvar|C}}, while for values of {{mvar|r}} close to {{math|∞}}, the corresponding circle is close to {{mvar|D}}; for the intermediate value {{math|1=''r'' = 1}}, the circle degenerates to a line, the perpendicular bisector of {{mvar|CD}}. The equation defining these circles as a locus can be generalized to define the [[Fermat–Apollonius circle]]s of larger sets of weighted points.

Each circle in the second family (the red circles in the figure) is associated with an angle {{math|θ}}, and is defined as the locus of points {{mvar|X}} such that the [[inscribed angle]] {{math|∠''CXD''}} equals {{mvar|θ}},
<math display=block>\left\{X \ \Bigl| \ C\hat{X}D = \theta\right\}.</math>

Scanning {{mvar|θ}} from 0 to ''π'' generates the set of all circles passing through the two points {{mvar|C}} and {{mvar|D}}.

The two points where all the red circles cross are the [[limiting point (geometry)|limiting points]] of pairs of circles in the blue family.

==Bipolar coordinates==
{{main|Bipolar coordinates}}
A given blue circle and a given red circle intersect in two points. In order to obtain bipolar '''coordinates''', a method is required to specify which point is the right one. An isoptic arc is the locus of points {{mvar|X}}  that sees points {{mvar|C, D}} under a given oriented angle of vectors i.e. 
<math display=block>\operatorname{isopt}(\theta) = \left\{X \ \Biggl| \ \measuredangle \biggl( \overrightarrow{XC}, \overrightarrow{XD} \biggr) = \theta +2k\pi \right\}.</math>
Such an arc is contained into a red circle and is bounded by points {{mvar|C, D}}. The remaining part of the corresponding red circle is {{math|isopt(''&theta;'' + ''&pi;'')}}. When we really want the whole red circle, a description using oriented angles of straight lines has to be used:
<math display=block>\text{full red circle} = \left\{X \ \Biggl| \ \measuredangle \biggl( \overrightarrow{XC}, \overrightarrow{XD} \biggr) = \theta + k\pi \right\}</math>

==Pencils of circles==
{{main|Pencil (mathematics)#Pencil of circles}}

Both of the families of Apollonian circles are '''pencils of circles'''. Each is determined by any two of its members, called ''generators'' of the pencil. Specifically, one is an ''elliptic pencil'' (red family of circles in the figure) that is defined by two generators that pass through each other in exactly '''two''' points ({{mvar|C, D}}). The other is a ''hyperbolic pencil'' (blue family of circles in the figure) that is defined by two generators that do not intersect each other at '''any''' point.<ref>{{harvtxt|Schwerdtfeger|1962|pp=8–10}}.</ref>

===Radical axis and central line===

Any two of these circles within a pencil have the same [[radical axis]], and all circles in the pencil have [[collinear]] centers. Any three or more circles from the same family are called '''coaxial circles''' or '''coaxal circles'''.<ref>MathWorld uses “coaxal,” while {{harvtxt|Akopyan|Zaslavsky|2007}} prefer “coaxial.”</ref>

The elliptic pencil of circles passing through the two points {{mvar|C, D}} (the set of red circles, in the figure) has the line {{mvar|CD}} as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of {{mvar|CD}}.
The hyperbolic pencil defined by points {{mvar|C, D}} (the blue circles) has its radical axis on the perpendicular bisector of line {{mvar|CD}}, and all its circle centers on line {{mvar|CD}}.

==Inversive geometry, orthogonal intersection, and coordinate systems==
[[Circle inversion]] transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a parabolic pencil is another parabolic pencil.

It is relatively easy to show using inversion that, in the Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a [[right angle]].  Inversion of the blue Apollonian circles with respect to a circle centered on point {{mvar|C}} results in a pencil of concentric circles centered at the image of point {{mvar|D}}.  The same inversion transforms the red circles into a set of straight lines that all contain the image of {{mvar|D}}.  Thus, this inversion transforms the [[bipolar coordinates|bipolar coordinate system]] defined by the Apollonian circles into a [[polar coordinates|polar coordinate system]].
Obviously, the transformed pencils meet at right angles. Since inversion is a [[conformal map|conformal transformation]], it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles.

Alternatively,<ref>{{harvtxt|Akopyan|Zaslavsky|2007}}, p. 59.</ref> the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point {{mvar|X}} on the radical axis of a pencil {{mvar|P}} the lengths of the tangents from {{mvar|X}} to each circle in {{mvar|P}} are all equal. It follows from this that the circle centered at {{mvar|X}} with length equal to these tangents crosses all circles of {{mvar|P}} perpendicularly. The same construction can be applied for each {{mvar|X}} on the radical axis of {{mvar|P}}, forming another pencil of circles perpendicular to {{mvar|P}}.

More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.<ref>{{harvtxt|Schwerdtfeger|1962|pp=30–31, Theorem A}}.</ref>

==Physics==
Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined pseudospin states in some physical systems involving interferential or coupled fields, such photonic or coupled [[polariton]] waves.<ref>{{Cite journal | title = Full-Bloch beams and ultrafast Rabi-rotating vortices | author = Dominici | display-authors=etal | journal = Physical Review Research | volume = 3 | pages = 013007  | year = 2021 | issue = 1 | doi = 10.1103/PhysRevResearch.3.013007 | arxiv = 1801.02580 | bibcode = 2021PhRvR...3a3007D | doi-access = free }}</ref> The trajectories arise from the [[Rabi cycle|Rabi rotation]] of the [[Bloch sphere]] and its [[stereographic projection]] on the real space where the observation is made.

== See also ==
* [[Apollonius quadrilateral]]

== Notes ==
{{reflist}}

== References ==
*{{citation
 | last1 = Akopyan | first1 = A. V.
 | last2 = Zaslavsky | first2 = A. A.
 | title = Geometry of Conics
 | publisher = [[American Mathematical Society]]
 | series = Mathematical World | volume = 26
 | year = 2007 | isbn = 978-0-8218-4323-9
 | pages = 57–62}}.
*{{citation
 | last=Schwerdtfeger | first=Hans | author-link = Hans Schwerdtfeger
 | year=1962
 | title=Geometry of Complex Numbers
 | title-link = Geometry of Complex Numbers
 | publisher=University of Toronto Press
 }}. Dover reprint, 1979, {{isbn|0-486-63830-8}}.

==Further reading==
*{{citation
 | last1=Pfeifer | first1=Richard E. 
 | last2=Van Hook | first2=Cathleen 
 | year=1993
 | title=Circles, Vectors, and Linear Algebra
 | journal=Mathematics Magazine
 | volume=66 | issue=2
 | pages=75–86
 | jstor=2691113 | doi=10.2307/2691113}}.
*{{citation
 | last=Samuel | first=Pierre
 | year=1988
 | title=Projective Geometry
 | publisher=Springer
 | pages=40–43 }}.
*{{citation
 | last = Ogilvy | first = C. Stanley | author-link = C. Stanley Ogilvy
 | year = 1969
 | title = Excursions in Geometry
 | publisher = Oxford University Press
 | url = https://archive.org/details/excursionsingeom0000ogil_w9f7/page/13/ |url-access = limited
 | at = esp. Ch. 2 "Harmonic division and Apollonian circles", {{pgs|13–23}}
 }}. Dover reprint, 1990, {{isbn|0-486-26530-7}}.

== External links ==
* {{mathworld|title=Coaxal Circles|urlname=CoaxalCircles}}
* David B. Surowski: [https://www.math.ksu.edu/~dbski/writings/further.pdf ''Advanced High-School Mathematics'']. p. 31

{{Ancient Greek mathematics}}

[[Category:Circles]]
[[Category:Elementary geometry]]
[[Category:Euclidean plane geometry]]