Apollonian Circle
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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Apollonian circles are two families (
pencils A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion (mechanical), abrasi ...
) of
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s such that every circle in the first family intersects every circle in the second family
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
ly, and vice versa. These circles form the basis for
bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal coordinates, orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 19 ...
. They were discovered by
Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
, a renowned ancient Greek geometer.


Definition

The Apollonian circles are defined in two different ways by a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
denoted . Each circle in the first family (the blue circles in the figure) is associated with a positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, and is defined as the locus of points such that the ratio of distances from to and to equals , \left\. For values of close to zero, the corresponding circle is close to , while for values of close to , the corresponding circle is close to ; for the intermediate value , the circle degenerates to a line, the perpendicular bisector of . The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points. Each circle in the second family (the red circles in the figure) is associated with an angle , and is defined as the locus of points such that the
inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an ...
equals , \left\. Scanning from 0 to ''π'' generates the set of all circles passing through the two points and . The two points where all the red circles cross are the limiting points of pairs of circles in the blue family.


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 that sees points under a given oriented angle of vectors i.e. \operatorname(\theta) = \left\. Such an arc is contained into a red circle and is bounded by points . The remaining part of the corresponding red circle is . When we really want the whole red circle, a description using oriented angles of straight lines has to be used: \text = \left\


Pencils 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 (). 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.


Radical axis and central line

Any two of these circles within a pencil have the same
radical axis In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...
, and all circles in the pencil have
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
centers. Any three or more circles from the same family are called coaxial circles or coaxal circles. The elliptic pencil of circles passing through the two points (the set of red circles, in the figure) has the line as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of . The hyperbolic pencil defined by points (the blue circles) has its radical axis on the perpendicular bisector of line , and all its circle centers on line .


Inversive geometry, orthogonal intersection, and coordinate systems

Circle inversion In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...
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 In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
. Inversion of the blue Apollonian circles with respect to a circle centered on point results in a pencil of concentric circles centered at the image of point . The same inversion transforms the red circles into a set of straight lines that all contain the image of . Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a
polar coordinate system In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from ...
. Obviously, the transformed pencils meet at right angles. Since inversion is a
conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles. Alternatively, the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point on the radical axis of a pencil the lengths of the tangents from to each circle in are all equal. It follows from this that the circle centered at with length equal to these tangents crosses all circles of perpendicularly. The same construction can be applied for each on the radical axis of , forming another pencil of circles perpendicular to . 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.


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 In physics, polaritons are bosonic quasiparticles resulting from strong coupling of electromagnetic waves (photon) with an electric or magnetic dipole-carrying excitation (state) of solid or liquid matter (such as a phonon, plasmon, or an exc ...
waves. The trajectories arise from the Rabi rotation of the
Bloch sphere In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system ( qubit), named after the physicist Felix Bloch. Mathematically each quantum mechanical syst ...
and its
stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
on the real space where the observation is made.


See also

*
Apollonius quadrilateral In geometry, an Apollonius quadrilateral is a quadrilateral ABCD such that the two products of opposite side lengths are equal. That is, \overline\cdot\overline=\overline\cdot\overline. An equivalent way of stating this definition is that the ...


Notes


References

*. *. Dover reprint, 1979, .


Further reading

*. *. *. Dover reprint, 1990, .


External links

* * David B. Surowski
''Advanced High-School Mathematics''
p. 31 {{Ancient Greek mathematics Circles Elementary geometry Euclidean plane geometry Greek mathematics