Complex velocity potential for an ellipse and a circle in translation and rotation

This paper investigates the analytical solutions of two-dimensional Neumann-type external boundary-value problem for an ellipse and a circle in complex analysis treatment. The transformation of harmonics between two planes is expanded in Laurent series. The circle theorem is extended so as to suit to the system of the ellipse and the circle. By means of Basset's process, the harmonic expressions of the system are found in the form of recurrence formulae which are suitable for numerical computations. The complex velocity potential of the system in translation and rotation is represented by the summation of the harmonic expressions.     

The Poisson integral and Green’s function for one strongly elliptic system of equations in a circle and an ellipse

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green’s function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

New properties of the Busemann ellipsea

A combination of the Busemann ellipse, the inscribed unit circle and a circle of radius √2 about the same centre is considered. For supersonic two-dimensional potential gas flows, it is shown that the inclinations of the velocity vector in motion along an arbitrary characteristic, the characteristic itself and the characteristic of the other family have values equal to, respectively. the difference between the areas of the elliptical and circular (R = 1) sectors, the difference between the areas of the elliptical and circular (R = √2) sectors, and the area of the elliptical sector, apart from unimportant multiplicative and additive constants. The straight sides of the sectors in question are the semiminor antis of the ellipse and the radius vector of the velocity. The obvious analogy with one of Ke:pler's laws is pointed out. The existence of a point of intersection of the ellipse and the second circle illustrates a well-known result of Khristianovich concerning the points of inflexion of characteristics with a monotone velocity distribution. It is shown how the combination of the ellipse and the inscribed circle illustrates the simplification of the compatibility conditions and the Darboux equation for trans- and hypersonic flows.     

ON THE CONVERGENCE OF CIRCLE PACKINGS TO THE QUASICONFORMAL MAP

Rodin and Sullivan （1987） proved Thurston＇s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.     

Diamond-free circle graphs are Helly circle

The diamond is the graph obtained from K₄ by deleting an edge. Circle graphs are the intersection graphs of chords in a circle. Such a circle model has the Helly property if every three pairwise intersecting chords intersect in a single point, and a graph is Helly circle if it has a circle model with the Helly property. We show that the Helly circle graphs are the diamond-free circle graphs, as conjectured by Durán. This characterization gives an efficient recognition algorithm for Helly circle graphs.     

Discrete linearized least-squares rational approximation on the unit circle

We already generalized the Rutishauser—Gragg—Harrod—Reichel algorithm for discrete least-squares polynomial approximation on the real axis to the rational case. In this paper, a new method for discrete least-squares linearized rational approximation on the unit circle is presented. It generalizes the algorithms of Reichel—Ammar—Gragg for discrete least-squares polynomial approximation on the unit circle to the rationale case. The algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel. Examples illustrate the numerical behavior of the algorithm.     

Triplets of mutually orthogonal circles associated with any ellipse

The author has shown (J. Geom. 94, 159–173, 2009) that, for any general point P on a given ellipse H, four concyclic notable points exist which determine a circle (denoted by Ω) orthogonal to Monge’s circle. Now, it is shown that a new set of notable concyclic points exists; such points determine a circle (denoted by Δ) orthogonal to both Monge’s circle and the circle Ω. Moreover, it is possible to introduce a new ellipse (denoted by H _Δ) concentric with the circle Δ, which is tangent to the ellipse H at P, shares the same circle Ω with the ellipse H and admits the circle Δ as its own Monge’s circle. Only elementary facts from trigonometry and analytic geometry are used.     

Locating a minisum circle in the plane

We consider the problem of locating a circle with respect to existing facilities in the plane such that the sum of weighted distances between the circle and the facilities is minimized, i.e., we approximate a set of given points by a circle regarding the sum of weighted distances. If the radius of the circle is a variable we show that there always exists an optimal circle passing through two of the existing facilities. For the case of a fixed radius we provide characterizations of optimal circles in special cases. Solution procedures are suggested.     

Queueing systems on a circle

Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey.We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.     

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.     

基于斜率的圆锥曲线新定义

为解决圆锥曲线对圆、椭圆、抛物线和双曲线统一的定义问题,可定义圆锥曲线是动点与二定点连线（或其中一连线为折线）斜率之积为定值的轨迹。此法不但较好地解决了圆锥曲线定义的不统一问题,而且数学推导也异常简单,有着明显的优点。此外,还论述了按此定义,用《几何画板》画各种圆锥曲线时,如何有效设置生成点的问题。     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

Holding a regular pyramid by a circle

Among other things, we prove the following (1) If a regular pyramid whose base is a regular polygon of circum-radius 1 has height at least 1, then the pyramid can be held by a circle, while for every ${0 < \varepsilon < 1}$ , there is a regular pyramid with height ${1 - \varepsilon}$ and base polygon of circum-radius 1 that cannot be held by any circle. (2) A regular pyramid of height h whose base is an equilateral triangle of circum-radius 1 can be held by a circle if and only if h > 0.479 . . . (which complements a theorem by Tanoue). (3) A regular pyramid with square base of unit circum-radius can be held by a circle if and only if its height is greater than 0.828 . . ..     

Partial circle geometries and antiregular quadrangles

A new and rather general definition of circle geometries is given. This definition is such that circle planes and chain spaces are circle geometries. Also the geometry of points and traces of an antiregular quadrangle is a partial circle geometry. Orthogonal quadrangles can then be characterised as those antiregular generalised quadrangles where in the associated partial circle geometry the Miquel condition is satisfied.     

关于圆和椭圆逼近显示的最优算法(英文)

本文给出了一个逼近显示圆的新算法。该算法是通过相交多边形而不是内接多边形逼近圆。由于构造相交多边形时其面积等于圆面积 ,因此新算法是最优逼近。同时还推广到椭圆     

On the linear representations of circle expanding maps

In this paper, it is proved that the two real analytic expanding endomorphisms of the unit circle are equivalent if and only if they have equal eigenvalues along corresponding cycles. A sufficient and necessary condition that a real analytic solution of Cvitanovic-Feigenbaum equation induces a real analytic expanding map is given. Supported by the NSFC.     

Number-theoretic interpretation and construction of a digital circle

This paper presents a new interpretation of a digital circle in terms of the distribution of square numbers in discrete intervals. The number-theoretic analysis that leads to many important properties of a digital circle succinctly captures the original perspectives of digital calculus and digital geometry for its visualization and characterization. To demonstrate the capability and efficacy of the proposed method, two simple algorithms for the construction of digital circles, based on simple number-theoretic concepts, have been reported. Both the algorithms require only a few primitive operations and are completely devoid of any floating-point computation. To speed up the computation, especially for circular arcs of high radii, a hybridized version of these two algorithms has been given. Experimental results have been furnished to elucidate the analytical power and algorithmic efficiency of the proposed approach. It has been also shown, how and why, for sufficiently high radius, the number-theoretic technique can expedite a circle construction algorithm.     

Rotating an interval and a circle

We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea.

     

A geometrical construction of inverse points with respect to an ellipse

In this note, we introduce the inverse point respect to an ellipse, which generalizes the classical inversion in a circle. Specifically, we show a geometrical construction of inverse points with respect to an ellipse. For this, we extend the classical method to construct the inverse of a point respect to a circle.