搜索结果：Circle

In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside any one of the three given circles, a circle of the similar radius and concentric with its own corresponding original circle is drawn.The tangent circle to these three similar circles is obtained. Then the inverted circles of the three similar circles and the tangent circle regarding an obtainable point and a computable power of inversion (reflection) constant are obtained. These circles (three inverted circles and an inverted tangent circle)will be tangent together.Just,we obtain another reflection point and power of inversion so that those three reflected circles (inversions of three similar circles) can be reflections of three original circles, respectively. In such a case,the reflected circle tangent to three reflected circles regarding same new inversion system will be tangent to the three original ones. This circle is our desirable circle. A drawing algorithm is also given for drawing desirable circle by straightedge and compass. A survey of

Thurston introduced the notion of a universal circle associated to a taut foliation of a $3$-manifold as a way of organizing the ideal circle boundaries of its leaves into a single circle action. Calegari--Dunfield proved that every taut foliation of an atoroidal $3$-manifold $M$ has a universal circle, but the uniqueness (or lack-thereof) of this structure remains rather mysterious. In this paper, we consider the foliations associated to an Anosov flow $\varphi$ on $M$, showing that several constructions of a universal circle in the literature are typically distinct. Moreover, the underlying action of the Calegari--Dunfield leftmost universal circle is generally not even conjugate to the universal circle arising from the boundary of the flow space of $\varphi$. Our primary tool is a way to use the flow space of $\varphi$ to parameterize the circle bundle at infinity of $\varphi$'s invariant foliations.

This paper presents a new Lemoine-type circle defined by a six-point configuration satisfying a cocyclicity criterion. We prove the result, establish a converse theorem, and relate the new circle to previously known Lemoine circles, in particular the one introduced by Q.T. Bui. We show that the new circle does not belong to the family of Tucker circles.

In this paper, we introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show that, under appropriate conditions, one can meaningfully and stably identify a discrete approximate circle bundle with an isomorphism class of true circle bundles. We also describe two cohomology invariants which uniquely determine the isomorphism class of a circle bundle, and provide algorithms to compute them given a discrete approximate representative. Finally, we propose a novel methodology for coordinatization and dimensionality reduction of circle bundle data. To illustrate the practical utility and viability of our algorithms, we present applications to both real and synthetic datasets from computer vision (e.g., modeling optical flow). The paper is accompanied by an open-source software package, with full documentation and tutorials, enabling reproducible implementation of the proposed algorithms and experiments, including those used to generate the figures in this paper.

Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $πr^2$, that is, the area of the circle. The present research is to study how often $C(r)$ will return a prime number of lattice points for $r \leq n$. The Prime Number Theorem predicts that the number of primes less than or equal to $n$ is asymptotic to $\frac{n}{\log n}$. We find that the number of Gauss Circle Primes for $r \leq n$ is also of order $\frac{n}{\log n}$ for $n \leq 2 \times 10^6$. We include a heuristic argument that the Gauss Circle Primes can be approximated by $\frac{n}{\log n}$.

We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle of the centers of two circles that coincides with their equioptic curves, as in Euclidean geometry.

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

We are interested in arrangements of circles and the regions surrounded by them. {\it Poincaré-Reeb graphs} have been fundamental and strong tools in studying shapes of regions surrounded by real algebraic curves, since around 2020. They are natural graphs the regions naturally collapse to and were first formulated by Sorea with several researchers. Studying shapes of such regions is one of fundamental studies in real algebraic geometry and combinatorics for example. This is surprisingly new and recently developing. Our study introduces labels on vertices and edges of such graphs encoding information of the circles where we concentrate on regions surrounded by circles. The author studied local changes of Poincaré-Reeb graphs by addition of circles under certain rules before and we discuss changes of new types. The author has started related studies motivated by singularity theory of real algebraic maps and found first that our regions are the images of natural real algebraic maps, generalizing natural projections of spheres.

We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The variables in these equations are associated to corners of triangles in the complex, with equations for interior vertices, edges, faces, and generators of first homology. The vertex equations are generalisations of the Descartes circle theorem, of higher degree but more symmetric than those previously found by the authors. We also provide some connections between the spinorial approach of previous work of the authors, and classical Euclidean geometry.

Recently, the use of circle representation has emerged as a method to improve the identification of spherical objects (such as glomeruli, cells, and nuclei) in medical imaging studies. In traditional bounding box-based object detection, combining results from multiple models improves accuracy, especially when real-time processing isn't crucial. Unfortunately, this widely adopted strategy is not readily available for combining circle representations. In this paper, we propose Weighted Circle Fusion (WCF), a simple approach for merging predictions from various circle detection models. Our method leverages confidence scores associated with each proposed bounding circle to generate averaged circles. We evaluate our method on a proprietary dataset for glomerular detection in whole slide imaging (WSI) and find a performance gain of 5% compared to existing ensemble methods. Additionally, we assess the efficiency of two annotation methods, fully manual annotation and a human-in-the-loop (HITL) approach, in labeling 200,000 glomeruli. The HITL approach, which integrates machine learning detection with human verification, demonstrated remarkable improvements in annotation efficiency. The Wei

The paper reports a generalized expression for filling the congruent circles (of radius r) in a circle (of radius R). First, a generalized expression for the biggest circle (r) inscribed in the nth part of the bigger circle (R) was developed. Further, it was extended as n such circles (r) touching each other and the bigger circle (R). To fill the bigger circle (R), the exercise was further repeated by considering the bigger circle radius as R-2r, R-4r and so on. In the process, a generalized expression was deduced for the total no. of such circles (r) which could be inscribed in this way of filling the bigger circle (R). The approach does not claim the closest packing always though it could be helpful for practical purposes.

We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$ induced by the rational chords is a strongly universal element of the family of circle graphs, and that it is invariant under local complementation. The only other known connected graphs that have the latter property are $K_2$ and the Rado graph.

In this paper, we introduce two local graph features for missing link prediction tasks on ogbl-citation2. We define the features as Circle Features, which are borrowed from the concept of circle of friends. We propose the detailed computing formulas for the above features. Firstly, we define the first circle feature as modified swing for common graph, which comes from bipartite graph. Secondly, we define the second circle feature as bridge, which indicates the importance of two nodes for different circle of friends. In addition, we firstly propose the above features as bias to enhance graph transformer neural network, such that graph self-attention mechanism can be improved. We implement a Circled Feature aware Graph transformer (CFG) model based on SIEG network, which utilizes a double tower structure to capture both global and local structure features. Experimental results show that CFG achieves the state-of-the-art performance on dataset ogbl-citation2.

The Valeriepieris circle is the smallest circle that can be draw on the globe containing half of the world's population. The Valeriepieris (VP) circle acts as a spatial median, effectively splitting spatial data into two halves in a unique way. In this paper the idea of the VP circle is generalised and a fast algorithm and corresponding software package to compute it are described. The VP circle is compared to other measures of centre and dispersion for population distributions and is shown to reflect expected differences between countries and changes over time. By studying the VP circle as a function of the included population fraction a new way of representing population distributions is constructed, as well as a mathematical model of its expected behaviour. Finally a measure of population `centralisation' is constructed which measures the tendency of a territory to be dominated by a single population centre or to have a more even distribution of population.

Given a graph $G$ with a fixed vertex order $\prec$, one obtains a circle graph $H$ whose vertices are the edges of $G$ and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in $\prec$. Therefore, the problem of determining whether $G$ has a $k$-page book embedding with spine order $\prec$ is equivalent to deciding whether $H$ can be colored with $k$ colors. Finding a $k$-coloring for a circle graph is known to be NP-complete for $k \geq 4$ and trivial for $k \leq 2$. For $k = 3$, Unger (1992) claims an efficient algorithm that finds a 3-coloring in $O(n \log n)$ time, if it exists. Given a circle graph $H$, Unger's algorithm (1) constructs a 3-\textsc{Sat} formula $Φ$ that is satisfiable if and only if $H$ admits a 3-coloring and (2) solves $Φ$ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details. In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unge

Apollonius of Perga, showed that for two given points $A,B$ in the Euclidean plane and a positive real number $k eq 1$, 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 $Γ_1,Γ_2$ and we show that geometric locus of the points $X$ with the ratio of the power with respect to the circles $Γ_1,Γ_2$ is constant, is also a circle. Using this we generalize the definition of Apollonius Circle, and generalize some results about Apollonius Circle.

A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequence of a more general result, describing the hyperideal circle patterns in the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding $\C P^1$-structures). This more general statement is obtained as a consequence of a theorem of Otal \cite{otal,bonahon-otal} on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.

We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find a nice relation involving the radii of the inner and outer Apollonius circles of the three circles in the triad.

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,π)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\infty$-convergence on compact subsets.