Circles in Mathematics: Theory & Applications

• Circles are closed curves defined by a constant distance from a center, with extensions to non-Euclidean, analytic, and arithmetic frameworks.
• Researchers analyze circle arrangements using incidence graphs, intersection bounds, and dynamic chain methodologies to reveal combinatorial and topological properties.
• Applications include graph representations, convex geometries, and enumerative combinatorics, linking classical constructions with modern analytic techniques.

Circles occupy a central position in contemporary geometry, combinatorics, and discrete mathematics, but recent research treats them in several distinct formal senses. In the Euclidean plane they appear as incidence objects in arrangements, as carriers of graph and convex-geometry representations, and as separators of finite point configurations; in arithmetic and constant-curvature settings they acquire modified distance laws and spectral invariants; and in homogeneous geometry the term “circle” denotes a distinguished class of curves defined without reference to a Riemannian metric (Carmesin et al., 2021, Hungerbühler, 10 Feb 2025, Karpenkov et al., 2024, Csima, 14 May 2026, Salvai, 2019). The resulting theory is correspondingly plural: it includes extremal bounds for intersection graphs, rigidity and closure theorems for circle chains, representation theorems for planar graphs and convex geometries, and specialized families such as Omega and Ajima circles.

1. Analytic definitions and generalized meanings

Several recent works begin by replacing the informal notion of a circle with explicit analytic or structural conditions. For an arrangement of Euclidean circles, orthogonality of two circles $A,B$ with centers $C_A,C_B$ and radii $r_A,r_B$ is equivalent to

$|C_AC_B|^2=r_A^2+r_B^2.$

If the circles meet at an angle $\theta\le \pi/2$, then

$$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$

hence $|C_AC_B|^2\le r_A^2+r_B^2$ (Carmesin et al., 2021).

A different analytic model appears in the generalized Apollonius problem in constant-curvature geometries. In the Euclidean plane the classical locus $PA/PB=k$ is a circle for $k\neq 1$; in the unified formulation for the Euclidean, spherical, and hyperbolic planes, the generalized Apollonius curve of two points $A,B$ is defined by

$C_A,C_B$0

where

$C_A,C_B$1

In affine coordinates with $C_A,C_B$2, $C_A,C_B$3, the locus is a real conic with equation

$C_A,C_B$4

reducing in the Euclidean limit to the classical Apollonius circle (Csima, 14 May 2026).

In integer geometry, a circle is defined relative to the lattice $C_A,C_B$5 and the integer distance

$C_A,C_B$6

For a lattice point $C_A,C_B$7 and integer $C_A,C_B$8, the integer circle of radius $C_A,C_B$9 is

$r_A,r_B$0

This notion is invariant under integer affine transformations, and it leads to integer and rational circumscribed spectra for finite subsets of $r_A,r_B$1 (Karpenkov et al., 2024).

The most abstract usage occurs for self-dual symmetric $r_A,r_B$2-spaces. If $r_A,r_B$3 are pairwise opposite points and $r_A,r_B$4 in an opposite chart, then

$r_A,r_B$5

is called the circle through $r_A,r_B$6. Different triples with the same image differ by a fractional-linear reparametrization. Here “circle” denotes a distinguished $r_A,r_B$7-invariant curve rather than a planar metric locus (Salvai, 2019).

2. Arrangements, intersections, and extremal structure

One major research direction studies finite arrangements of circles through their intersection combinatorics. For an arrangement $r_A,r_B$8, the intersection graph $r_A,r_B$9 has one vertex per circle and an edge whenever two circles meet. When no two circles are nested, the straight-line drawing obtained by placing each vertex at the center of its circle is crossing-free for arrangements of orthogonal circles. The same holds when all intersections occur at an angle of at most $|C_AC_B|^2=r_A^2+r_B^2.$0. Consequently, in the nonnested case the intersection graph is planar (Carmesin et al., 2021).

The nested case is subtler. For $|C_AC_B|^2=r_A^2+r_B^2.$1 circles, the maximal number of edges in an intersection graph of an arrangement of orthogonal circles lies between

$|C_AC_B|^2=r_A^2+r_B^2.$2

The lower bound is realized by the “nested wheels” construction $|C_AC_B|^2=r_A^2+r_B^2.$3: for $|C_AC_B|^2=r_A^2+r_B^2.$4, one places a hub circle $|C_AC_B|^2=r_A^2+r_B^2.$5 together with a ring of $|C_AC_B|^2=r_A^2+r_B^2.$6 satellite circles $|C_AC_B|^2=r_A^2+r_B^2.$7, each orthogonal to $|C_AC_B|^2=r_A^2+r_B^2.$8 and to its two near-neighbors, then nests the wheels so that each satellite on ring $|C_AC_B|^2=r_A^2+r_B^2.$9 meets exactly two satellites on ring $\theta\le \pi/2$0. With $\theta\le \pi/2$1, the edge count is

$\theta\le \pi/2$2

for $\theta\le \pi/2$3. The same construction improves the lower bound for the number of triangular cells to

$\theta\le \pi/2$4

The paper also records open problems, including tightening the coefficient $\theta\le \pi/2$5, fixed-parameter or approximation algorithms for dense subgraphs, higher-dimensional analogues for orthogonal spheres in $\theta\le \pi/2$6, and analogous extremal questions for a fixed acute angle $\theta\le \pi/2$7 (Carmesin et al., 2021).

This extremal viewpoint is complemented by a topological description of the regions surrounded by circle arrangements. For a normally-inductive arrangement $\theta\le \pi/2$8, one fixes a coordinate projection

$\theta\le \pi/2$9

and collapses each fiber $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$0 to a point. The quotient is a finite graph $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$1, with vertices corresponding to selected poles or circle intersections and edges corresponding to interval fibers. The resulting structure is a $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$2-digraph whose vertex order is induced by the projection values (Kitazawa, 21 Feb 2025).

3. Regions, labels, and local modification of circle arrangements

The Poincaré–Reeb framework refines the combinatorial study of arrangements by encoding which circle arcs appear along monotone fibers. For an NI arrangement, every edge $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$3 lifts to a strip in $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$4 bounded by two smooth arcs $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$5 lying on circles $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$6. The edge label is a pair of finite sets of $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$7-arcs on these circles. Vertex labels are more elaborate: depending on degree $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$8, $$|C_AC_B|^2=r_A^2+r_B^2-2r_Ar_B\cos\theta, \qquad \cos\theta\ge 0,$$9, or $|C_AC_B|^2\le r_A^2+r_B^2$0, one records a finite sequence of finite arc-sets, one for each sector around the vertex. The labeling function

$|C_AC_B|^2\le r_A^2+r_B^2$1

is well defined, and the admissible labels are forced by combinatorial rules relating tangent-vector signs to allowable quarter-arc data (Kitazawa, 21 Feb 2025).

This labeling theory supports explicit local change theorems. If an edge label contains a single $|C_AC_B|^2\le r_A^2+r_B^2$2-arc of a circle $|C_AC_B|^2\le r_A^2+r_B^2$3, then adding a sufficiently small new circle $|C_AC_B|^2\le r_A^2+r_B^2$4 at a point $|C_AC_B|^2\le r_A^2+r_B^2$5 in the interior of that arc changes the graph locally by splitting the edge into three edges, inserting two new vertices, and attaching a new leaf edge to the middle vertex. If the chosen label contains two adjacent arcs, one can realize either of the two basic local changes by a suitable choice of chord position. These operations give a controlled calculus for modifying circle arrangements while tracking the induced Poincaré–Reeb graph (Kitazawa, 21 Feb 2025).

Examples illustrate the formalism. For the unit disk $|C_AC_B|^2\le r_A^2+r_B^2$6, $|C_AC_B|^2\le r_A^2+r_B^2$7 is a single edge with two end-vertices, and the edge and vertex labels can be written explicitly in terms of arc-sets. For the annulus between two concentric circles, the quotient graph has three edges and four vertices, and adding a small circle near a specified quarter-arc produces one of the local modifications described above. The same paper notes that $|C_AC_B|^2\le r_A^2+r_B^2$8 is realized as the image of a real algebraic map generalizing natural projections of spheres (Kitazawa, 21 Feb 2025).

4. Circle-based representations of graphs and convex geometries

Circles also function as representational primitives for discrete structures. For a connected simple $|C_AC_B|^2\le r_A^2+r_B^2$9-regular planar graph $PA/PB=k$0, a circle-representation is a drawing together with a set $PA/PB=k$1 of circles such that every vertex is realized as either a crossing point or a tangency point of two circles, and every edge is drawn along an arc lying on one of the defining circles. If only tangencies occur, the representation is a touching-circle representation. For $PA/PB=k$2, the minimum number $PA/PB=k$3 of circles in a circle-representation satisfies

$PA/PB=k$4

with both bounds attained by infinite families. The upper bound comes from the fact that each circle contains at least $PA/PB=k$5 vertices and each vertex lies on exactly $PA/PB=k$6 circles; the lower bound follows from $PA/PB=k$7 (Bekos et al., 2019).

The principal positive realization theorem states that every $PA/PB=k$8-connected simple $PA/PB=k$9-regular planar graph admits a touching-circle representation. The proof uses the dual graph, a face $k\neq 1$0-coloring, an incidence-line graph $k\neq 1$1, and Koebe’s circle-packing theorem. By contrast, the non-$k\neq 1$2-connected case contains explicit obstructions: the paper constructs an infinite family of simple connected and biconnected $k\neq 1$3-regular planar graphs that admit no circle-representation. It also leaves open the decision problem for arbitrary $k\neq 1$4-regular planar graphs (Bekos et al., 2019).

A parallel but logically distinct program studies convex geometries represented by circles. For a finite set $k\neq 1$5 of circles in the Euclidean plane, one defines

$k\neq 1$6

where $k\neq 1$7 is the disk occupied by $k\neq 1$8. Then $k\neq 1$9 is a convex geometry, meaning a closure system satisfying the anti-exchange axiom. Such circle convex geometries obey the Weak $A,B$0-Carousel rule: for every triple $A,B$1 and every $A,B$2, there exist two of the three points, say $A,B$3, such that

$A,B$4

The rule is proved by reducing to the case of three point-circles, projecting two interior circles to the sides of a triangle, and eliminating nonrealizable overlap patterns by a detailed case analysis. Not every finite convex geometry satisfies this rule, so not every finite convex geometry is strongly representable by circles in the plane; the paper notes counterexamples already of convex dimension $A,B$5 and formulates further representation problems for circles and balls in higher dimension (Adaricheva et al., 2016).

5. Enumeration, separation, and topological classes

The combinatorics of circles includes exact enumeration problems for separating circles. Let $A,B$6 be a set of $A,B$7 dots in general position in the plane or on the sphere. Two types of separating circles are distinguished: incident circles, which pass through exactly three dots, and avoidant circles, which pass through none of the dots, considered up to equivalence by the induced bipartition. For incident circles, the number of circles separating the remaining $A,B$8 dots into parts of sizes $A,B$9 and $C_A,C_B$00 with $C_A,C_B$01 is

$C_A,C_B$02

For avoidant circles, the number of equivalence classes separating the $C_A,C_B$03 dots into parts of sizes $C_A,C_B$04 and $C_A,C_B$05 with $C_A,C_B$06 is

$C_A,C_B$07

Both counts are independent of the configuration. The avoidant-circle count is derived from the $C_A,C_B$08th-order spherical Voronoi decomposition, whose $C_A,C_B$09-cells correspond to oriented avoidant circles. As the dots move continuously and a cocircular quadruple appears, the Voronoi decomposition changes by one of three local moves, identified with Postnikov’s plabic-graph moves; hence the associated cluster algebra depends only on $C_A,C_B$10, not on the configuration (Beyer et al., 28 May 2025).

A different enumerative theory studies topologically distinct sets of circles in the plane. Well-formed parenthesis words, Dyck paths, and non-intersecting circles on a line are equivalent classical models, yielding the Catalan recurrence

$C_A,C_B$11

and generating function

$C_A,C_B$12

When the order of side-by-side factors is ignored, one counts topologically distinct plane packings of non-intersecting circles by the numbers $C_A,C_B$13, equivalently unlabeled rooted forests. Their generating function satisfies the Pólya-type equation

$C_A,C_B$14

with initial values

$C_A,C_B$15

The same species-theoretic framework extends to one intersecting pair of circles, arbitrary many disjoint intersecting pairs, and families allowing triple intersections, with corresponding implicit functional equations and asymptotic growth of the form $C_A,C_B$16 (Mathar, 2016).

These two enumeration programs are complementary. One counts circles that separate prescribed finite sets of points; the other counts ambient isotopy classes of circle configurations themselves. This suggests a useful distinction between incidence enumeration and topological enumeration.

6. Circle chains, closure, and periodicity

Classical theorems about chains of tangent or intersecting circles have recently been recast in dynamical form. In the revisited Six Circles Theorem, one starts with a triangle $C_A,C_B$17 having angles $C_A,C_B$18, then constructs a sequence $C_A,C_B$19 where $C_A,C_B$20 is inscribed in the angle at $C_A,C_B$21, $C_A,C_B$22 is tangent to $C_A,C_B$23, and at each step the smaller tangent circle is chosen. If all circles touch the sides of the triangle rather than their extensions, the chain is $C_A,C_B$24-periodic. More generally, if at least one tangency point of the initial circle lies on a side of the triangle, then the chain is eventually $C_A,C_B$25-periodic, but the pre-period may be arbitrarily long (Ivanov et al., 2013).

The proof converts the tangency conditions into an explicit piecewise-linear iteration. With

$C_A,C_B$26

the recurrence becomes a quadratic-root formula, which is then transformed by

$C_A,C_B$27

After three steps one obtains the map

$C_A,C_B$28

and every orbit eventually enters a $C_A,C_B$29-periodic interval. The geometric $C_A,C_B$30-periodicity is therefore the dynamical shadow of eventual $C_A,C_B$31-periodicity for $C_A,C_B$32 (Ivanov et al., 2013).

A broader closure theory applies to closed chains $C_A,C_B$33 in which each neighboring pair intersects or is tangent at a pivot $C_A,C_B$34. If $C_A,C_B$35 denotes the reversion map from $C_A,C_B$36 to $C_A,C_B$37, then for any starting point $C_A,C_B$38 the polygonal chain $C_A,C_B$39 has side $C_A,C_B$40 passing through $C_A,C_B$41. The composite

$C_A,C_B$42

is the identity if and only if the transfer angles satisfy

$C_A,C_B$43

equivalently

$C_A,C_B$44

The same paper associates to every pair of indices $C_A,C_B$45 a circle $C_A,C_B$46 through $C_A,C_B$47 traced by the intersection of the lines $C_A,C_B$48 and $C_A,C_B$49 as the starting point varies, and proves that for every triple $C_A,C_B$50, the circles $C_A,C_B$51 pass through a common point $C_A,C_B$52. Miquel’s six circles theorem and Steiner’s quadrilateral theorem appear as special cases (Hungerbühler, 10 Feb 2025).

7. Specialized families and broader extensions

Within triangle geometry, several distinguished circle families are defined by incidence and tangency conditions. An Omega circle of a triangle $C_A,C_B$53 is any circle passing through the first Brocard point

$C_A,C_B$54

in areal coordinates. If an Omega circle $C_A,C_B$55 meets the lines $C_A,C_B$56 again at $C_A,C_B$57, respectively, then $C_A,C_B$58 is indirectly similar to $C_A,C_B$59. If $C_A,C_B$60 are the second intersections of $C_A,C_B$61 with the circles $C_A,C_B$62, $C_A,C_B$63, $C_A,C_B$64, then the lines $C_A,C_B$65, $C_A,C_B$66, and $C_A,C_B$67 are concurrent at a point $C_A,C_B$68. The same source also studies circles through the intersections of the medians with the orthocentroidal circle, producing directly similar triangles rather than indirectly similar ones (Bradley, 2010).

Ajima circles are defined relative to a triangle $C_A,C_B$69 and a circle $C_A,C_B$70 through $C_A,C_B$71 and $C_A,C_B$72. An Ajima circle $C_A,C_B$73 lies inside $C_A,C_B$74, is tangent to $C_A,C_B$75 and $C_A,C_B$76, and is externally tangent to $C_A,C_B$77. Writing

$C_A,C_B$78

and letting $C_A,C_B$79 be the angular measure of the arc $C_A,C_B$80 on $C_A,C_B$81, the radius $C_A,C_B$82 of $C_A,C_B$83 is

$C_A,C_B$84

In the semicircle case $C_A,C_B$85, so

$C_A,C_B$86

The paper also records that if the three Ajima circles $C_A,C_B$87 are erected, then their common external tangents along the sides have equal length $C_A,C_B$88, the six touch-points are concyclic on a circle centered at the incenter, and the three Gergonne cevians are radical axes for pairs of Ajima circles (Rabinowitz et al., 2023).

Arithmetic and non-Euclidean variants further broaden the concept. For a finite integer set $C_A,C_B$89, the existence of an integer circumscribed circle is equivalent to the condition that $C_A,C_B$90 cover no torus $C_A,C_B$91 for $C_A,C_B$92; equivalently, $C_A,C_B$93, where $C_A,C_B$94 denotes the integer circumscribed spectrum. If

$C_A,C_B$95

then the integer spectrum is

$C_A,C_B$96

while the rational spectrum has the form

$C_A,C_B$97

with $C_A,C_B$98 the product of all primes $C_A,C_B$99 such that $r_A,r_B$00 covers $r_A,r_B$01 (Karpenkov et al., 2024).

In constant-curvature geometry, generalized Apollonius circles coincide with equioptic curves of pairs of circles. If $r_A,r_B$02 have centers $r_A,r_B$03 and radii $r_A,r_B$04, then from an external point $r_A,r_B$05 the equal-angle condition for the tangent pairs is equivalent to

$r_A,r_B$06

Thus the equioptic curve of two geodesic circles is exactly the generalized Apollonius curve of their centers (Csima, 14 May 2026).

Finally, in self-dual symmetric $r_A,r_B$07-spaces, circles are characterized by transformation-theoretic and Riemannian properties. A diffeomorphism belongs to the big transformation group $r_A,r_B$08 if and only if it sends circles to circles. Moreover, if $r_A,r_B$09 is a circle, then there exists a maximal compact subgroup $r_A,r_B$10 such that

$r_A,r_B$11

is a diametrical unit-speed geodesic in the canonical $r_A,r_B$12-invariant metric; equivalently, it is a diagonal geodesic in a maximal totally geodesic flat torus. This identifies an invariantly defined “circle” with a geodesic after projective reparametrization (Salvai, 2019).

Taken together, these developments show that circles form not a single theory but a network of compatible theories: Euclidean incidence structures, separators and representers in combinatorics, dynamically constrained chains, arithmetic loci in $r_A,r_B$13, constant-curvature equioptic curves, and homogeneous-space trajectories preserved by large transformation groups.