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A theorem on circle configurations

Profile image of Jerzy KocikJerzy Kocik

Abstract

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.

FAQs

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AI

What does the new theorem reveal about circle configurations?add

The theorem provides a generalization beyond Descartes configurations, allowing relations between arbitrary circle curvatures and positions with specific conditions.

How does the Pedoe map contribute to the analysis of circles?add

The Pedoe map facilitates viewing circles as vectors in Minkowski space, linking traditional and modern geometrical concepts.

What implications does the theorem have for geometric configurations?add

The theorem establishes that certain configurations, like four mutually perpendicular circles, are impossible due to the properties of the configuration matrix.

How does the theorem apply to the problem of Apollonius?add

The theorem offers analytical solutions for the Apollonian problem by generating a specific configuration matrix applicable for varying tangencies.

What historical significance does the theorem hold in geometry?add

The theorem builds on classical results, such as those by Apollonius and Descartes, reinterpreting their applications in higher-dimensional configurations.

Figures (19)
Figure 1.1: Four circles ; (a) as a solution to an Apollonian problem; (b) in Descartes configuration (c) in a general configuration where a=1/r,, b=1/r2, etc. are the curvatures of the corresponding circles (reciprocals of radii). For the history of this formula, rediscovered a number of times, and its higher-dimensional generalizations, see [Des, Ped3, Coxt69, Sodd, Gos] .
Figure 1.1: Four circles ; (a) as a solution to an Apollonian problem; (b) in Descartes configuration (c) in a general configuration where a=1/r,, b=1/r2, etc. are the curvatures of the corresponding circles (reciprocals of radii). For the history of this formula, rediscovered a number of times, and its higher-dimensional generalizations, see [Des, Ped3, Coxt69, Sodd, Gos] .
Figure 2.2: (a) Power of a point. Geometric interpretation of the Darboux product of (a) intersecting circles; (c) distant circles. If the circles intersect, the product equals C;*C, = rjmcos @ , where @ is the angle made by the circles (see Figure 2.2a). Let us note that in the case of more distant circles, the product C\*C, equals the square of the segment constructed in Figure 2.2c.
Figure 2.2: (a) Power of a point. Geometric interpretation of the Darboux product of (a) intersecting circles; (c) distant circles. If the circles intersect, the product equals C;*C, = rjmcos @ , where @ is the angle made by the circles (see Figure 2.2a). Let us note that in the case of more distant circles, the product C\*C, equals the square of the segment constructed in Figure 2.2c.
Figure 2.2: The Pedoe map carries circles to rays in the Minkowski space Note that the Pedoe map may be extended to points (improper circles) so that point P = (x, y) has image m(P) =span { [ 1, x+y", x, y]"}, which is a ray in the light cone, as In(v)I° =0, Vvenm(P). It is easy to see that the rays that represent proper circles are space-like, i.e., they lie in M outside the light cone. For them, we may restrict the Pedoe map to the unit vectors. Definition 2: The Pedoe special map is a specification of 7: Q—PM, i.e., is map
Figure 2.2: The Pedoe map carries circles to rays in the Minkowski space Note that the Pedoe map may be extended to points (improper circles) so that point P = (x, y) has image m(P) =span { [ 1, x+y", x, y]"}, which is a ray in the light cone, as In(v)I° =0, Vvenm(P). It is easy to see that the rays that represent proper circles are space-like, i.e., they lie in M outside the light cone. For them, we may restrict the Pedoe map to the unit vectors. Definition 2: The Pedoe special map is a specification of 7: Q—PM, i.e., is map
(Note that the second term of the Pedoe vector is determined by the other three and by the requirement of normalization). Remark on the geometric meaning: The curvature b has the standard meaning. The c
(Note that the second term of the Pedoe vector is determined by the other three and by the requirement of normalization). Remark on the geometric meaning: The curvature b has the standard meaning. The c
Init circle centered at the origin. As for the reduced position (x,y), one can think of it as the curvature b happens to equal the curvature of the circle that is the image of C via inversion in the
Init circle centered at the origin. As for the reduced position (x,y), one can think of it as the curvature b happens to equal the curvature of the circle that is the image of C via inversion in the
Remark: Thus, the following vectors represent the same circle: where the first vector is the set of coefficients of Eq. (2.6), the second is scaled by setting a=/, and the third, scaled by the radius of the circle, is the Pedoe vector. The first parameterization of vectors has the advantage that choosing a = 0 admits lines into the formalism as special circles. Points (x,y) go to [1, x’+y’, x, y], corresponding to light-like rays; hence they do not permit Pedoe vectors.
Remark: Thus, the following vectors represent the same circle: where the first vector is the set of coefficients of Eq. (2.6), the second is scaled by setting a=/, and the third, scaled by the radius of the circle, is the Pedoe vector. The first parameterization of vectors has the advantage that choosing a = 0 admits lines into the formalism as special circles. Points (x,y) go to [1, x’+y’, x, y], corresponding to light-like rays; hence they do not permit Pedoe vectors.
Although the orthonormal basis side is more “suggestive” for those accustomed to relativistic physics (see e.g. [Sod]), the path to this form leads from the Darboux product in a way that is not as intuitive and clear with respect to the isotropic basis and coordinates. Figure 2.4 shows the bases and corresponding variables symbolically, squeezing 4 dimensions into a 3-dimensional picture. One of the axes carries variables x and y (the position of the circle’s center). (The figure represents exactly the special case of n=1, where spheres are pairs of points on a line).
Although the orthonormal basis side is more “suggestive” for those accustomed to relativistic physics (see e.g. [Sod]), the path to this form leads from the Darboux product in a way that is not as intuitive and clear with respect to the isotropic basis and coordinates. Figure 2.4 shows the bases and corresponding variables symbolically, squeezing 4 dimensions into a 3-dimensional picture. One of the axes carries variables x and y (the position of the circle’s center). (The figure represents exactly the special case of n=1, where spheres are pairs of points on a line).
Circles correspond to the rays (vectors) outside the cone. Points in R", which may be viewed as circles of null radius, correspond to light-like rays. Similarly, lines, which may be viewed as circles of zero curvature, are rays in the horizontal hyper-plane b = 0.
Circles correspond to the rays (vectors) outside the cone. Points in R", which may be viewed as circles of null radius, correspond to light-like rays. Similarly, lines, which may be viewed as circles of zero curvature, are rays in the horizontal hyper-plane b = 0.
Let us try to understand the benefits of this equation. First, we introduce four vectors that represent the data on the circles in a way dual to what we used so far, namely we consider Vector b will be called the curvature vector of the circle configuration, b the co-curvature vector, and x, and y are the reduced position vectors. The data matrix constructed by superposing these eolumneis AW Vector b will be called the curvature vector of the circle configuration, b the co-curvature vector, Now, by multiplying both sides on the left and right by A and A’, respectively, we get (3.3) as desired. [1
Let us try to understand the benefits of this equation. First, we introduce four vectors that represent the data on the circles in a way dual to what we used so far, namely we consider Vector b will be called the curvature vector of the circle configuration, b the co-curvature vector, and x, and y are the reduced position vectors. The data matrix constructed by superposing these eolumneis AW Vector b will be called the curvature vector of the circle configuration, b the co-curvature vector, Now, by multiplying both sides on the left and right by A and A’, respectively, we get (3.3) as desired. [1
Among the special cases of Theorem 3.1 is the original Descartes formula and its extension, discovered by J. Lagarias et al. [LMW]. Let us look at some details. Four circles are said to be in a Descartes configuration if all pairs of circles are mutually tangent at distinct points (see Figure 4.1 for possible arrangements). Figure 4.1: Four circles of the Descartes configuration.
Among the special cases of Theorem 3.1 is the original Descartes formula and its extension, discovered by J. Lagarias et al. [LMW]. Let us look at some details. Four circles are said to be in a Descartes configuration if all pairs of circles are mutually tangent at distinct points (see Figure 4.1 for possible arrangements). Figure 4.1: Four circles of the Descartes configuration.
(Notice that f? = 4 J; hence f' = 1/4 f readily follows). Thus our master equation AFA‘ = G is —after a convenient multiplication of both sides by 4 — simply: First, consider the “all-external-tangencies” configuration (Fig. 4.1a). Construct the configuration matrix f and calculate its inverse F:
(Notice that f? = 4 J; hence f' = 1/4 f readily follows). Thus our master equation AFA‘ = G is —after a convenient multiplication of both sides by 4 — simply: First, consider the “all-external-tangencies” configuration (Fig. 4.1a). Construct the configuration matrix f and calculate its inverse F:
Matrix D = 4f~' (consisting of negative ones on the diagonal and ones everywhere else) is often called the Descartes matrix. The particular vector equation b'F b = G, part of (4.1), is equivalent to the original Descartes law:
Matrix D = 4f~' (consisting of negative ones on the diagonal and ones everywhere else) is often called the Descartes matrix. The particular vector equation b'F b = G, part of (4.1), is equivalent to the original Descartes law:
where D is the Descartes matrix, and R = diag(1, 1, 1, -1). The Descartes formula for curvatures becomes
where D is the Descartes matrix, and R = diag(1, 1, 1, -1). The Descartes formula for curvatures becomes
The quadratic relations may be read from these matrices. For the curvatures, one may arrive at the following concise forms: Let a, b, and c be three pair-wise orthogonal circles and let d be a circle tangent to each of them. There are four distinct realizations, presented here in columns:
The quadratic relations may be read from these matrices. For the curvatures, one may arrive at the following concise forms: Let a, b, and c be three pair-wise orthogonal circles and let d be a circle tangent to each of them. There are four distinct realizations, presented here in columns:
Let a and b be a pair of orthogonal circles. Add two mutually tangent circles c and d each of which is also tangent to a and to b. Then we have these four cases: We encounter the same problem with signs as in Example 2. (Note that the signs inside the squares may be altered when one seeks a unifying formula for all four cases). Here is a solution:
Let a and b be a pair of orthogonal circles. Add two mutually tangent circles c and d each of which is also tangent to a and to b. Then we have these four cases: We encounter the same problem with signs as in Example 2. (Note that the signs inside the squares may be altered when one seeks a unifying formula for all four cases). Here is a solution:
For the Apollonian problem, the configuration matrix is of the following type: The problem of Apollonius is to find a circle that is simultaneously tangent to three geometric objects. If we choose these objects to be three circles, we may expect eight possible solutions, which differ according to the type of each of the tangencies — external versus internal (see Fig. 4.2). A geometric (constructive) solution is known (see [Coxt68]). Let us see how Theorem 3.3 provides analytic solutions.
For the Apollonian problem, the configuration matrix is of the following type: The problem of Apollonius is to find a circle that is simultaneously tangent to three geometric objects. If we choose these objects to be three circles, we may expect eight possible solutions, which differ according to the type of each of the tangencies — external versus internal (see Fig. 4.2). A geometric (constructive) solution is known (see [Coxt68]). Let us see how Theorem 3.3 provides analytic solutions.
In the standard basis, the Minkowski metric g in M and its inverse are represented by (n+3)x(n+3 matrices: The special Pedoe map sends proper spheres into normed future-oriented vectors
In the standard basis, the Minkowski metric g in M and its inverse are represented by (n+3)x(n+3 matrices: The special Pedoe map sends proper spheres into normed future-oriented vectors
where N is a matrix with | in every entry, and / is the unit matrix. Its inverse is F = (1/2n)(N — nD); hence, after multiplying both sides of (5.1) by 2n, we get: which agrees with the formula discovered by Soddy [Sod] for n = 3, generalized to arbitrary dimension by Gossett [Gos].
where N is a matrix with | in every entry, and / is the unit matrix. Its inverse is F = (1/2n)(N — nD); hence, after multiplying both sides of (5.1) by 2n, we get: which agrees with the formula discovered by Soddy [Sod] for n = 3, generalized to arbitrary dimension by Gossett [Gos].
Figure 5.1: Bounded and unbounded disc Figure 5.2: Descartes configurations reconsidered Remark 5.4 (on bends, circles and disks): In order to account for different cases of the Descartes configuration, it is customary to introduce the notion of the bend of a circle: the curvature with a negative sign if the circle contains the other three circles (see e.g., [LMW]). We propose an alternative solution: replace circles by generalized disks. Any circle (n-sphere) is the boundary of either of two discs, one bounded and one unbounded (see Figure 5.1). The radius of the unbounded one is negative. Then each case of Descartes configuration may be represented in terms of external tangencies, as in Figure 5.2. The definition of the Pedoe map 7 extends to such discs: if D and D’ are mutual complements, then we define z(D')=-7(D).
Figure 5.1: Bounded and unbounded disc Figure 5.2: Descartes configurations reconsidered Remark 5.4 (on bends, circles and disks): In order to account for different cases of the Descartes configuration, it is customary to introduce the notion of the bend of a circle: the curvature with a negative sign if the circle contains the other three circles (see e.g., [LMW]). We propose an alternative solution: replace circles by generalized disks. Any circle (n-sphere) is the boundary of either of two discs, one bounded and one unbounded (see Figure 5.1). The radius of the unbounded one is negative. Then each case of Descartes configuration may be represented in terms of external tangencies, as in Figure 5.2. The definition of the Pedoe map 7 extends to such discs: if D and D’ are mutual complements, then we define z(D')=-7(D).

Related papers

Apollonian circle packings: geometry and group theory I. The Apollonian group
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Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system M D consisting of those 4 × 4 real matrices W with W T Q D W = Q W where Q D is the matrix of the Descartes quadratic form Q D = x 2 1 + x 2 2 + x 2 3 + x 2 4 − 1 2 (x 1 + x 2 + x 3 + x 4) 2 and Q W of the quadratic form Q W = −8x 1 x 2 + 2x 2 3 + 2x 2 4. On the parameter space M D the group Aut(Q D) acts on the left, and Aut(Q W) acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group O(3, 1). The right action of Aut(Q W) (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space R 2 while the left action of Aut(Q D) is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of Aut(Q D), which we call the Apollonian group. This group consists of 4 × 4 integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in Aut(Q D), the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer 4 × 4 matrices. We show these groups are hyperbolic Coxeter groups.

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N T ] 1 1 Se p 20 00 Apollonian Circle Packings : Number Theory
Colin Mallows

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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: 2(x2 + y2 + z2 + w2) − (x + y + z + w)2 = 0. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We determine asymptotics for the number of root quadruples of size below T . We study which integers occur in a given integer packing, and determine con...

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N T ] 1 7 O ct 2 00 3 Apollonian Circle Packings : Number Theory
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An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles. In this paper we consider the basic building blocks of these rules, irreducible Apollonian configurations. Our main result is to show how to find a small field that can realize such a configuration and also give a method to relate the bends of the new circles to the bends of the circles forming the curvilinear triangle.

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N T / 0 00 91 13 v 2 1 7 O ct 2 00 3 Apollonian Circle Packings : Number Theory
Colin Mallows

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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 + y2 + z2 + w2 = 12(x + y + z + w) 2. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in...

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The Six-Point Circle Theorem
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Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$, $\aangle F=\gamma$, (iii) $\Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $\Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $\alpha,\beta,\gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1,....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.

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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x 2 + y 2 + z 2 + w 2 = 1 2 (x + y + z + w) 2 . Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.

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Colin Mallows

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This paper gives n-dimensional analogues of the Apollonian circle packings in parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of n-dimensional Descartes configurations, which consist of n + 2 mutually touching spheres. We work in the space Mn D of all n-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those (n + 2) × (n + 2) real matrices W with WTQD,nW = QW,n where QD,n = x2 1 +... + x2 1 n+2 − n (x1 + · · · + xn+2) 2 is the n-dimensional Descartes quadratic form, QW,n = −8x1x2 + 2x2 3 + · · · + 2x2n+2, and QD,n and QW,n are their corresponding symmetric matrices. On the parameter space Mn D of ACC-matrices the group

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Colin Mallows

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Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object ...

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Ömer Avcı

2021

Apollonius of Perga, showed that for two given points A,B in the Euclidean plane and a positive real number k 6= 1, geometric locus of the points X that satisfies the equation |XA| = k|XB| is a circle. This circle is called Apollonius circle. In this paper we generalize the definition of the Apollonius circle for two given circles Γ1,Γ2 and we show that geometric locus of the points X with the ratio of the power with respect to the circles Γ1,Γ2 is constant, is also a circle. Using this we generalize the definition of Apollonius Circle, and generalize some results about Apollonius Circle. 1 Preliminaries Theorem 1.1 (Apollonius Theorem) For points A,B in the Euclidean plane, and a positive real number k 6= 1, the points X which satisfies the equation |XA| = k|BX| forms a circle. When k = 1, they form the line perpendicular to AB at the middle point of [AB] [1]. Definition 1.1 For three different points A,B,C in Euclidean plane such that A is not on the perpendicular bisector of the ...

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