Circle: Geometry, Patterns, and Applications

• Circle is a geometric object defined as the set of points equidistant from a center, playing a key role in classical constructions, discrete patterns, and algebraic generalizations.
• Discretized circle approximations and curvature equations facilitate the design of circle patterns in computational imaging and conformal geometry.
• Circle graphs and chord diagrams extend circle properties to graph recognition and combinatorial optimization, influencing recent advances in network analysis.

Searching arXiv for recent and foundational papers on circles across geometry, graph theory, and computational imaging. A circle in the Euclidean plane is the set of points at fixed distance from a center; with center $c=(x_0,y_0)$ and radius $r>0$, it admits the implicit equation $(x-x_0)^2+(y-y_0)^2=r^2$ and the parametric form $x(\theta)=x_0+r\cos\theta$, $y(\theta)=y_0+r\sin\theta$ for $\theta\in[0,2\pi)$ (Nguyen et al., 2022). In contemporary research, the same object also appears as the primitive of circle patterns with prescribed exterior intersection angles, as the chord model underlying circle graphs, and as a low-parameter representation for approximately spherical structures in medical imaging (Ba et al., 11 Dec 2025, Paul et al., 29 Dec 2025, Xiong et al., 2024).

1. Euclidean locus, parametrization, and classical constructions

In analytic form, the circle is simultaneously an implicit quadratic curve and a periodic parametrized contour. The parametric representation is particularly useful when the circle is discretized into ordered samples, for example $\theta_i=2\pi i/N$ and $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$, which makes the circle a natural periodic signal on a cycle graph (Xiong et al., 2024).

Classical straightedge-and-compass geometry gives a constructive description of the full family of circles through a prescribed point. For a fixed triangle $ABC$ with circumcircle $(O)$, a given point $r>0$0, and a chosen center $r>0$1, one defines $r>0$2 as the $r>0$3 rotation of $r>0$4 about $r>0$5 and constructs $r>0$6 so that $r>0$7 is a parallelogram. If $r>0$8 are the second intersections of $r>0$9 with $(x-x_0)^2+(y-y_0)^2=r^2$0, and $(x-x_0)^2+(y-y_0)^2=r^2$1 are the fourth vertices of the parallelograms $(x-x_0)^2+(y-y_0)^2=r^2$2, $(x-x_0)^2+(y-y_0)^2=r^2$3, and $(x-x_0)^2+(y-y_0)^2=r^2$4, then the circle through $(x-x_0)^2+(y-y_0)^2=r^2$5 passes through $(x-x_0)^2+(y-y_0)^2=r^2$6, has center $(x-x_0)^2+(y-y_0)^2=r^2$7, and radius $(x-x_0)^2+(y-y_0)^2=r^2$8 (Bradley, 2010). This identifies the entire family of circles through $(x-x_0)^2+(y-y_0)^2=r^2$9 with the set of admissible centers.

Several papers reframe the circle as a locus defined by nonstandard constraints. For a regular $x(\theta)=x_0+r\cos\theta$0-gon of circumradius $x(\theta)=x_0+r\cos\theta$1 and a point $x(\theta)=x_0+r\cos\theta$2, let $x(\theta)=x_0+r\cos\theta$3 be the distances from $x(\theta)=x_0+r\cos\theta$4 to the vertices and define $x(\theta)=x_0+r\cos\theta$5. For $x(\theta)=x_0+r\cos\theta$6, the locus $x(\theta)=x_0+r\cos\theta$7 is a circle centered at the polygon center when $x(\theta)=x_0+r\cos\theta$8, the single point at the center when $x(\theta)=x_0+r\cos\theta$9, and empty when $y(\theta)=y_0+r\sin\theta$0 (Meskhishvili, 2019). By contrast, in the upper half-plane model of hyperbolic geometry, the analog of the Apollonius circle for three collinear adjacent points is generically an algebraic curve of degree four; only special parameter choices reduce it to a semicircle, a half hyperbola, or a lemniscate-type curve (Ionaşcu, 2016). This suggests that the circular locus is robust under strong rotational symmetry, but not under arbitrary metric deformation.

2. Discrete circle patterns and conformal geometry

A planar circle pattern assigns to each vertex $y(\theta)=y_0+r\sin\theta$1 a Euclidean circle with radius $y(\theta)=y_0+r\sin\theta$2 and center $y(\theta)=y_0+r\sin\theta$3, while each edge $y(\theta)=y_0+r\sin\theta$4 carries a prescribed exterior intersection angle $y(\theta)=y_0+r\sin\theta$5. The corresponding center distance is

$y(\theta)=y_0+r\sin\theta$6

and for a triangulated domain the radii are determined by the discrete curvature equations $y(\theta)=y_0+r\sin\theta$7 after gluing Euclidean triangles with those side lengths (Ba et al., 11 Dec 2025). In this setting, the curvature at a vertex is computed from the sum of incident triangle angles, with

$y(\theta)=y_0+r\sin\theta$8

in the Euclidean polygonal formulation (Ba et al., 11 Dec 2025).

On surfaces of finite topological type, circle patterns are studied in both Euclidean and hyperbolic background geometries, including the case of obtuse exterior intersection angles. Under the local condition $y(\theta)=y_0+r\sin\theta$9 with $\theta\in[0,2\pi)$0 on each face, the triangle of centers is well defined, and the Chow–Luo combinatorial Ricci flows

$\theta\in[0,2\pi)$1

provide convergent constructions in the hyperbolic and Euclidean cases, respectively, when the corresponding global inequalities are satisfied (Ge et al., 2019). The same paper characterizes the image of the curvature map by explicit cut inequalities involving $\theta\in[0,2\pi)$2 and Euler characteristics of subcomplexes $\theta\in[0,2\pi)$3 (Ge et al., 2019).

The sphere case admits an obtuse-angle extension of the Koebe–Andreev–Thurston framework. For a triangulation of the sphere with angle assignment $\theta\in[0,2\pi)$4, conditions $\theta\in[0,2\pi)$5–$\theta\in[0,2\pi)$6 yield an irreducible circle pattern on the Riemann sphere, and the corresponding hyperbolic polyhedral statement gives a compact convex hyperbolic polyhedron with prescribed dihedral angles under conditions $\theta\in[0,2\pi)$7–$\theta\in[0,2\pi)$8, unique up to isometry (Zhou, 2020). A complementary computational viewpoint minimizes the least-squares curvature energy

$\theta\in[0,2\pi)$9

by gradient descent in the radii and then recovers centers by solving the sparse symmetric positive definite systems $\theta_i=2\pi i/N$0 and $\theta_i=2\pi i/N$1 associated with the discrete Dirichlet energy

$\theta_i=2\pi i/N$2

(Ba et al., 11 Dec 2025).

3. Circle graphs and combinatorial structure

A circle graph is the intersection graph of chords drawn in a circle. In chord-diagram language, a graph $\theta_i=2\pi i/N$3 is a circle graph when there exists a bijection from vertices to chords such that two vertices are adjacent exactly when the corresponding chord endpoints alternate around the circle (Paul et al., 29 Dec 2025). The same structure may be encoded by a double occurrence circular word, and for prime circle graphs the chord diagram is unique up to reversal (Paul et al., 29 Dec 2025).

Split decomposition organizes the recognition problem. A graph is circle if and only if every prime node of its split-tree is labeled by a circle graph (Paul et al., 29 Dec 2025). Earlier work achieved the first subquadratic recognition algorithm, with running time

$\theta_i=2\pi i/N$4

using incremental Lexicographic Breadth-First Search, split decomposition, and the data structure of consistent symmetric cycles for chord diagrams (Gioan et al., 2011). More recent work replaces the union-find component of the split decomposition by a PC-tree–based representation specialized to circle graphs, yielding the first linear-time recognition algorithm:

$\theta_i=2\pi i/N$5

(Paul et al., 29 Dec 2025).

Structural characterization is especially explicit within split graphs. In that class, circle-ness is equivalent to the absence of a family of minimal forbidden induced subgraphs including tent-with-center, odd $\theta_i=2\pi i/N$6-sun with center, even $\theta_i=2\pi i/N$7-sun, $\theta_i=2\pi i/N$8, $\theta_i=2\pi i/N$9, $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$0, $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$1, $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$2, $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$3, and $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$4 (Bonomo-Braberman et al., 2020). Recognition, however, does not imply easy optimization: Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are all W[1]-hard on circle graphs when parameterized by solution size, whereas Connected Acyclic Dominating Set is polynomial-time solvable (Bousquet et al., 2012).

4. Circle chains, closure phenomena, and special circles in triangle geometry

A closed chain of circles $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$5 with neighboring intersections $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$6 supports a family of adapted polygons $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$7 such that $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$8 are collinear. Writing $(x_i,y_i)=(x_0+r\cos\theta_i,\ y_0+r\sin\theta_i)$9 for the reversion map through $ABC$0, the global composition is

$ABC$1

If $ABC$2 fixes one point of $ABC$3, then $ABC$4 is the identity on $ABC$5, so the polygon closes for every starting point (Hungerbühler, 10 Feb 2025). The exact criterion is

$ABC$6

where $ABC$7 is the transfer angle between the tangents at the pivot $ABC$8 (Hungerbühler, 10 Feb 2025). For a chain of touching circles, $ABC$9; therefore an even chain closes in one round and an odd chain closes after two rounds (Hungerbühler, 10 Feb 2025).

The same framework yields fixed auxiliary circles from variable polygons. If $(O)$0 is the side line through $(O)$1, then for each pair $(O)$2 the point $(O)$3 lies on a fixed circle $(O)$4 through $(O)$5 and $(O)$6, independent of the starting point, and for any triple $(O)$7 the circles $(O)$8, $(O)$9, and $r>0$00 pass through a common point (Hungerbühler, 10 Feb 2025). Miquel’s six circles theorem and Steiner’s quadrilateral theorem arise as special cases of this closure formalism (Hungerbühler, 10 Feb 2025).

Triangle geometry supplies another distinguished family. An Omega circle is any circle through the Brocard point $r>0$01, and circles through $r>0$02 carry triangles indirectly similar to the reference triangle in the same sense that Hagge circles do for the orthocenter. The same paper shows that the three points where the medians intersect the orthocentroidal circle determine circles carrying triangles directly similar to triangles inscribed in the circumcircle (Bradley, 2010).

5. Circle-based representations in medical imaging

For ball-shaped medical objects such as glomeruli, nuclei, and eosinophils, circle representation offers a low-parameter alternative to box or polygon initialization. CircleSnake is an end-to-end contour-based instance segmentation framework in which the detection head predicts a center $r>0$03 and radius $r>0$04, the initial contour is a discretized circle with $r>0$05 ordered vertices, and refinement is performed by a circular graph convolutional network on the cycle graph with kernel size $r>0$06 and three deformation iterations (Xiong et al., 2024). Relative to octagon proposals, the method counts only $r>0$07 for forming the bounding circle once the center is localized, compared with $r>0$08 for octagon initialization (Xiong et al., 2024). The complete detection objective is

$r>0$09

with $r>0$10 and $r>0$11 (Xiong et al., 2024).

Empirically, CircleSnake reports superior segmentation accuracy and rotation consistency on glomeruli, nuclei, and eosinophils. On glomeruli, the segmentation configuration with DLA-34 attained $r>0$12, $r>0$13, $r>0$14, and Dice $r>0$15; on nuclei the Dice score was $r>0$16; and on eosinophils it was $r>0$17 (Xiong et al., 2024). The earlier glomeruli-only study already identified CircleSnake as the first end-to-end circle representation deep segmentation pipeline, reporting improvement from detection average precision $r>0$18 to $r>0$19 and Dice $r>0$20 to $r>0$21 over DeepSnake on that dataset (Nguyen et al., 2022).

Circle representation also changes ensemble design in detection. Weighted Circle Fusion matches detections by circle IoU, fuses parameters by the confidence-weighted means

$r>0$22

with $r>0$23, and filters the fused circles by $r>0$24 and $r>0$25 after matching at $r>0$26 (Yue et al., 2024). On murine whole-slide glomerular detection, Weighted Circle Fusion achieved $r>0$27 against $r>0$28 for the best individual model, and under $r>0$29 rotation it reached $r>0$30 (Yue et al., 2024).

6. Learning-oriented and algebraic generalizations

In graph learning, “circle” can denote local social structure rather than Euclidean geometry. Circle Feature Graphormer defines two pairwise features for link prediction: a modified swing score

$r>0$31

and a bridge score

$r>0$32

where $r>0$33 counts common neighbors bridging the exclusive neighborhoods of $r>0$34 and $r>0$35 (Lv et al., 2023). These terms enter Graphormer attention as additive biases, and on ogbl-citation2 the CFG$r>0$36 model reported test MRR $r>0$37 against $r>0$38 for SIEG, with the paper stating that CFG$r>0$39 adds the bridge feature and yields additional improvements (Lv et al., 2023).

A more literal generalization appears in arithmetic geometry. The $r>0$40-unit circle $r>0$41 is defined as the completion of the Carlitz torsion set $r>0$42 at a fixed branch of infinity, and the ambient space

$r>0$43

is a vector space over $r>0$44 of dimension $r>0$45 (Ward, 2018). In this setting, $r>0$46 has a center, curvature, additive and multiplicative Möbius transformations, a partition of the ambient space into translates $r>0$47, a Dirichlet approximation theorem, a reciprocity law, and an associated hyperbolic plane

$r>0$48

together with modular forms and Eisenstein series (Ward, 2018). This suggests that, even outside Euclidean geometry, “circle” remains a privileged structure for symmetry, completion, and automorphic action.