The circles of Apollonius are any of several sets of circles associated with
Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
, a renowned
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
geometer
A geometer is a mathematician whose area of study is geometry.
Some notable geometers and their main fields of work, chronologically listed, are:
1000 BCE to 1 BCE
* Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra
* ...
. Most of these circles are found in
planar
Planar is an adjective meaning "relating to a plane (geometry)".
Planar may also refer to:
Science and technology
* Planar (computer graphics), computer graphics pixel information from several bitplanes
* Planar (transmission line technologies), ...
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
.
The main uses of this term are fivefold:
# 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
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
. This
Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2.
# The
Apollonian circles
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
are two families of mutually
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis of
bipolar coordinates
Bipolar coordinates are a two-dimensional 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, 1999 Confusingly, the sam ...
.
# The circles of Apollonius of a triangle are three circles, each of which passes through one vertex of the triangle and maintains a constant ratio of distances to the other two. The
isodynamic point
In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
s and
Lemoine line
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This s ...
of a triangle can be solved using these circles 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.
# The
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mat ...
—one of the first
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
Apollonius' definition of a circle
![Apollonius circle definition labels](https://upload.wikimedia.org/wikipedia/commons/2/2a/Apollonius_circle_definition_labels.svg)
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 be defined as the set of points P that have a given ''ratio'' of distances ''k'' = to two given points (labeled A and B in Figure 1). These two points are sometimes called the
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
.
Proof using vectors in Euclidean spaces
Let ''d'', ''d'' be non-equal positive real numbers.
Let C be the internal division point of AB in the ratio ''d'' : ''d'' and D the external division point of AB in the same ratio, ''d'' : ''d''.
:
Then,
:
Therefore, the point P is on the circle which has the diameter CD.
Proof using the angle bisector theorem
![Proof of Apollonius' definition of a circle](https://upload.wikimedia.org/wikipedia/commons/d/d5/Proof_of_Apollonius%27_definition_of_a_circle.svg)
First consider the point
on the line segment between
and
, satisfying the ratio. By the definition
and from the
angle bisector theorem
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the ...
the angles
and
are equal.
Next take the other point
on the extended line
that satisfies the ratio. So
Also take some other point
anywhere on the extended line
. Also by the Angle bisector theorem the line
bisects the exterior angle
. Hence,
and
are equal and
. Hence by
Thales's theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
lies on the circle which has
as a diameter.
Apollonius pursuit problem
The Apollonius pursuit problem is one of finding whether a ship leaving from one point A at speed ''v''
A will intercept another ship leaving a different point B at speed ''v''
B. The minimum time in interception of the two ships is calculated by means of straight-line paths. If the ships' speeds are held constant, their speed ratio is defined by μ. If both ships collide or meet at a future point, ''I'', then the distances of each are related by the equation:
:
Squaring both sides, we obtain:
:
:
:
:
Expanding:
:
Further expansion:
:
Bringing to the left-hand side:
:
Factoring:
:
Dividing by
:
:
Completing the square:
:
Bring non-squared terms to the right-hand side:
:
Then:
:
Therefore, the point must lie on a circle as defined by Apollonius, with their starting points as the foci.
Circles sharing a radical axis
![Apollonian circles](https://upload.wikimedia.org/wikipedia/commons/b/b2/Apollonian_circles.svg)
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
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
intersect each other at
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 ...
s and form the basis of the
bipolar coordinate system
Bipolar coordinates are a two-dimensional 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, 1999 Confusingly, the sam ...
. Within each pencil, any two circles 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 ...
; 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.
Solutions to Apollonius' problem
![Apollonius8ColorMultiplyV2](https://upload.wikimedia.org/wikipedia/commons/8/8c/Apollonius8ColorMultiplyV2.svg)
In
Euclidean plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, Apollonius's problem is to construct
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s that are
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to three given circles in a plane.
Three given circles generically have eight different circles that are tangent to them and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed.
Apollonian gasket
![Apollonian gasket](https://upload.wikimedia.org/wikipedia/commons/e/e6/Apollonian_gasket.svg)
By solving Apollonius' problem repeatedly to find the inscribed circle, the
interstice
An interstitial space or interstice is a space between structures or objects.
In particular, interstitial may refer to:
Biology
* Interstitial cell tumor
* Interstitial cell, any cell that lies between other cells
* Interstitial collagenase, ...
s between mutually tangential circles can be filled arbitrarily finely, forming an
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mat ...
, also known as a ''Leibniz packing'' or an ''Apollonian packing''. This gasket is a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
, being self-similar and having a
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
''d'' that is not known exactly but is roughly 1.3,
which is higher than that of a
regular (or
rectifiable) curve (''d'' = 1) but less than that of a plane (''d'' = 2). The Apollonian gasket was first described by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
in the 17th century, and is a curved precursor of the 20th-century
Sierpiński triangle
The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal curve, fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursion, recu ...
. The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s;
see also
Circle packing theorem
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...
.
Isodynamic points of a triangle
The circles of Apollonius may also denote three special circles
defined by an arbitrary triangle
. The circle
is defined as the unique circle passing through the triangle vertex
that maintains a constant ratio of distances to the other two vertices
and
(cf. Apollonius' definition of the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
above). Similarly, the circle
is defined as the unique circle passing through the triangle vertex
that maintains a constant ratio of distances to the other two vertices
and
, and so on for the circle
.
All three circles intersect the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of the
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
ly. All three circles pass through two points, which are known as the
isodynamic point
In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
s
and
of the triangle. The line connecting these common intersection points is the
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 ...
for all three circles. The two isodynamic points are
inverses of each other relative to the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of the triangle.
The centers of these three circles fall on a single line (the
Lemoine line
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This s ...
). This line is perpendicular to the radical axis, which is the line determined by the isodynamic points.
See also
*
Apollonius point
In Euclidean geometry, the Apollonius point is a triangle center designated as ''X''(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). It is defined as the point of concurrence of the three line segments joining each vertex of the ...
*
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
References
Bibliography
*
Ogilvy, C.S. (1990) ''Excursions in Geometry'', Dover. .
* Johnson, R.A. (1960) ''Advanced Euclidean Geometry'', Dover.
{{Ancient Greek mathematics
Apollonius