Pairs Of Circles Research Articles

Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the former. These diagonals and the central angle of the pair of peripherals, the shortest diagonals, are used to construct n rhombi. The rhombi define the shape of lenses tangential to them. To construct the arcs of the lenses, beams perpendicular to the sides of each rhombus are drawn. Four beams radiating from the top and bottom vertices of each rhombus intersect in the centers of a pair of coaxal circles. Thus, the vertical axis of each rhombus coincides with the radical axis of the pair. The intersection of the pair represents the corresponding lens. All n lenses form a tangential sequence along the central diagonal. Their cusps circumscribe the polygon and the lenses themselves. The area covered by the lenses converges to (2/3) πr2.

Orthogonal ring patterns in the plane

We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring patterns analogues of the Doyle spiral, Erf and z^alpha functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.

A new method for researching and constructing spherical bicentric polygons based on geometric mapping

A bicentric polygon is a special planar polygon that has both a circumcircle and an incircle. Many interesting properties of a planar bicentric polygon have been discovered. However, searching the geometric properties of a spherical bicentric polygon with elementary geometric method is very difficult. In this paper, we propose a special mapping for a pair of circles between a plane and a sphere. This mapping provides us a new method to research the spherical bicentric polygons. With this mapping, we can translate the Fuss relation, which describes the relation of the circumcircle and incircle of a bicentric polygon, from a planar bicentric polygon to a spherical polygon. Based on the mapping, we can derive the geometric properties of spherical bicentric polygons from the planar bicentric polygons, and construct a series of spherical bicentric n-polygons from a planar bicentric n-polygon.

On the Nature of Bondi–Metzner–Sachs Transformations

This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable ζ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of ζ. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of ζ. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold.

Poncelet's discovery of homology

Homology was among the concepts introduced in Jean Victor Poncelet's 1822 Traité des Propriétés Projectives des Figures. Homology is a projective transformation which has an axis, a line of fixed points. The Traité develops a straightedge construction of points under homology, essentially that found in work on perspective drawing and by Phillipe de la Hire, 1673. However, Poncelet's very distinct path to homology was through similitude, where the radical axis of a pair of circles became the axis of homology. We end with Poncelet's application of homology involving the focus of a conic section.

Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches

The Einstein field equations of general relativity constrain the local curvature at every point in spacetime, but say nothing about the global topology of the Universe. Cosmic microwave background anisotropies have proven to be the most powerful probe of non-trivial topology since, within ΛCDM, these anisotropies have well-characterized statistical properties, the signal is principally from a thin spherical shell centered on the observer (the last scattering surface), and space-based observations nearly cover the full sky. The most generic signature of cosmic topology in the microwave background is pairs of circles with matching temperature and polarization patterns. No such circle pairs have been seen above noise in the WMAP or Planck temperature data, implying that the shortest non-contractible loop around the Universe through our location is longer than 98.5% of the comoving diameter of the last scattering surface. We translate this generic constraint into limits on the parameters that characterize manifolds with each of the nine possible non-trivial orientable Euclidean topologies, and provide a code which computes these constraints. In all but the simplest cases, the shortest non-contractible loop in the space can avoid us, and be shorter than the diameter of the last scattering surface by a factor ranging from 2 to at least 6. This result implies that a broader range of manifolds is observationally allowed than widely appreciated. Probing these manifolds will require more subtle statistical signatures than matched circles, such as off-diagonal correlations of harmonic coefficients.

Modeling relational responding with artificial neural networks

Relational responding refers to behavior that conforms to a rule for com- paring stimuli. Lazareva et al. (2014) trained pigeons to choose either the smaller or the larger of two circles, using 1–3 pairs of circles for training and 17–19 new pairs for testing. The pigeons showed relational responding on some test pairs and systematic failures on others. We present a simple artificial neural network model that reproduces the animals’ behavior well, similarly to Lazareva et al.’s (2014) statistical model based on stimulus features and stimulus relationships. We analyze how the network model gener- alizes from training to test stimuli, and show that it can reconcile contrasting ideas about relational responding from the seminal works by Köhler (1929, 1918/1938, 1924), positing that animals can learn relational rules such as “choose the larger stimulus,” and Spence (1937), positing that relational re- sponding can be explained based on stimulus generalization.

Topological Classification and Determination of Non-Degenerate Intersections of Two Dupin Cyclides

The surface-surface intersection is a fundamental task in CAD/CAM. We present the classification and a full enumeration of the topology of all non-degenerate intersections of two Dupin cyclides. Inversion geometry is first used to transform the intersection of two Dupin cyclides to that of a cyclide and a torus. Then the topology of the intersection curve is reduced to the arrangements of the main circle of the torus and another pair of circles, which can be characterized by a simple algebraic sequence. Based on the classification and enumeration results, an efficient determination algorithm for the topology of two Dupin cyclides is also provided. • A classification of the non-degenerate intersections of two Dupin cyclides is given by a simple algebraic sequence. • A full enumeration of all non-degenerate intersections of two Dupin cyclides based on a simple algebraic sequence is provided. • An efficient and robust algorithm of determining the inter- section topology of two Dupin cyclides is provided.

The Boy We Killed

The Boy We Killed Adam Szetela (bio) My coach's ear looks like a smashed piece of cauliflower. They call this cauliflower ear. It's when the cartilage breaks. Unlike a broken nose, it can't be put back together again. Wrestlers treat cauliflower ear like the marines treat a USMC tattoo. It's a permanent badge of respect. I don't have cauliflower ear. I have a scar underneath my left eye. I got this scar when I lifted Nate above my head. When I slammed him back on the mat, his hip bone broke my face. His hip bone was so sharp because he was so thin. When you wrestle, the only thing as bad as getting pinned is not making weight. To make weight, wrestlers run in trash bags duct taped to their bodies. They sit in saunas with heavy sweatshirts, rain jackets, and scarfs wrapped around their throats. On the day of a meet, they might vomit until dark-green-to-yellowish-brown bile stains the toilet wall. When Mark didn't make weight, we held him down. As he screamed for help, someone shoved a dirty mop in his face. We used the mop to clean the mats. The mats were covered in bacteria that lived off our sweat. The bacteria caused skin infections. Impetigo. Scabies. Staph. The next day, Mark had tinea corporis on his face. Another name for tinea corporis is ring worm. Wrestlers and fighters get it. The pair of circles on Mark's face looked like a pair of balls below the dick of his nose. We called him Dick Face, Cock Mask, and Ball Breath. We took turns punching him in the stomach. If you caught a skin disease, you caught a beatdown. You also got your ass kicked if you were a fish. When a wrestler is about to get pinned, they try to keep their shoulders off the mat. They writhe. They flail. They flop around like a fish out of water. A fish is someone who gets pinned. Doug was the biggest fish on our team. We kicked him in the balls. We punched words into his chest like it was a broken typewriter. Fat fuck. Faggot. Dildo. We twisted his nipples until they were purple, black, and covered in blood. Until they hung from his skin like burnt pepperonis. Our coach had even less tolerance for fishes and kids who missed weight. During practice, they were used as dummies to demonstrate moves. It was one thing to get a bloody nose or a black eye from a teenager. It was another to get one from a grown man. Even on the bus ride home from meets, there were beatdowns. If someone got pinned, we would pin them against the window and stick our wet fingers in their ears. If they missed weight, we would break open their coolers and throw their food on the road. One time we grabbed a freshman by his legs and pretended to push him out the bus window. This kid screamed as the metal frame tore through the flesh on his ribs. [End Page 274] We didn't want him to hit the pavement. We just didn't want him to get pinned again. Before our last meet of the season, our coach gathered us in the center of the wrestling room. There was an incident at a high school a few towns over. One of their wrestlers was out for a jog to cut weight. One of his teammates came around the corner in a car. It was a prank gone wrong. The kid was run over an hour before weigh ins. His skin and bones were smushed into the pavement. He was dead. I don't remember how long the silence lasted before we broke into partners to drill takedowns. I don't remember who I was paired with. Maybe it was Nate or Mark. Maybe it was the freshman we pretended to push out the bus window. The freshman whose name I don't even remember. But I do remember that no one got a mop to the face that night. No one had their dinner thrown in...

Greater sensitivity in separation discrimination with closer spacing of separation levels tested.

We explored the effects of spacing in the levels of separation tested in a separation discrimination task. Participants indicated, for pairs of test circles, whether the separation between them was greater than a standard separation. A critical set of equally-spaced separation levels was tested in two conditions. In one condition additional separation levels were interleaved between the critical levels. In the other condition additional separation levels were not interleaved. Overall, the same average level and range of levels were tested in the two conditions, and the levels tested were equally spaced in both conditions. Critically, the levels tested were more closely spaced in one condition than the other. The sensitivity of the discrimination was greater in the condition with the more as opposed to less-closely spaced levels of separation. We suggest an explanation under which separation is assessed from the number of “separation fields” between the points at which the test stimuli register and under which the separation fields are smaller or more densely distributed when the levels of separation tested are more as opposed to less-closely spaced.

Using circles games to investigate the referential use of negation

ABSTRACT Studies on the spontaneous production of negation suggest that it can be modulated by pragmatic principles of successful communication such as informativity and relevance. The present study investigates whether negation production is additionally modulated by a more general principle of effort minimisation. In a series of circles games, subjects were presented with pairs of circles and asked to complete a sentence that would allow a listener to identify one of the two circles. Negation was only produced when an affirmative description for the circle at issue was harder, i.e. there was no simple intuitive way to describe the circle's pattern. The length of the concurrent descriptions did not strictly influence the production of negation. The results suggest that the use of negation becomes more frequent as the effort to produce it decreases with respect to a concurrent affirmation, even at the cost of greater informativity of affirmation.

Quaternionic Product of Circles and Cycles and Octonionic Product for Pairs of Circles

Quaternionic Product of Circles and Cycles and Octonionic Product for Pairs of Circles

A parallel algorithm for computing Voronoi diagram of a set of circles using touching disc and topology matching

We present a parallel algorithm for computing the Voronoi diagram of a set of circles, C. The algorithm starts with a parallel computation of the Voronoi cell of each circle ci∈C, using a subset of circles, Ci⊂C, and a reduced touching disc method. Unlike the actual touching disc method, a reduced touching disc method does not compute antipodal discs (ADs) between all pairs of circles. We employ the Delaunay graph, D, of the center points of circles in C to set an initial Ci. For computing an initial Voronoi cell for ci∈C, it is enough to compute ADs between ci and each of the circles cj∈Ci. Computation of Voronoi cells updates initially assumed Ci, and the algorithm iteratively computes those Voronoi cells until no further updating of Ci occurs. A parallel procedure corrects the Voronoi cells, iteratively, using a topology matching approach. A reduced touching disc method and a cell-wise computation of the Voronoi diagram allow input in non-general position. Results and performance of the algorithm show robustness and speed of the algorithm in handling a large set of circles.

An extension of Asgeirsson's mean value theorem for solutions of the ultra-hyperbolic equation in dimension four

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae.The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the condition for functions on line space to come from line integrals of functions in Euclidean 3-space, and hence it appears as a compatibility condition for tomographic data.The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae.John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.

On pairs of quadratic forms in five variables

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx. 22 (2010) 403–419].

Entanglement types for two-qubit states with real amplitudes

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states—those locally equivalent to the Bell states—form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled or so-called product states are \(\pi /4\) units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval \([0,\pi /4]\) with the point 0 corresponding to the maximally entangled states. The point \(\pi /4\) corresponds to the unentangled states which geometrically are described by a torus. Finally, for every \(0< d < \pi /4\) the point d corresponds to a disjoint pair of torus. Finally, we also show how this geometrical interpretation allows to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.

On Alternating Quasipositive Links

An effectively verifiable condition for quasipositivity of links is given. In particular, it is proven that if a quasipositive link can be represented by an alternating diagram satisfying the condition that no pair of Seifert circles is connected by a single crossing, then the diagram is positive and the link is strongly quasipositive.

Errors of Analysis in Distance Assessment: Effects of Vertical on Horizontal Components.

(a) Participants indicated, for pairs of circles whose locations varied on the horizontal and vertical axes of a frontal plane, whether the horizontal distance between the circles exceeded a target horizontal distance. The error rate depended on the vertical as well as the horizontal distance between the circles. (b) Participants indicated, for pairs of circles that were varying horizontal (or vertical) distances and a constant vertical (or horizontal) distance apart in a frontal plane, whether the horizontal (or vertical) distance between them matched a target horizontal (or vertical) distance. Incorrect "match" responses were more likely if the horizontal (or vertical) distance between the circles was less than as opposed to greater than the target distance. The results suggest that distance judgments for pairs of stimuli varying on the horizontal and vertical axes are based on the overall distance between the stimuli, with the relevant axis given more weight than the irrelevant axis in assessment of the distance. The results do not support the view that that such distance judgments are based on the relevant distance between the stimuli, with the relevant and irrelevant axes being erroneously interchanged on some iterations of the assessment process.

Method of O.Podgorny determination of the phase reflected wave corresponding wave phase from a point source

A method for constructing, using a computer, the envelope of a family of circles of variable radius, the centers of which are located on a given curve G and touch the circle C with the center at the origin, is given. In this case, the limits of the parameter t Î [a, b], which determines the position of the center of each family circle on the curve G., are taken into account. In solving this problem, in the general case, the question arises of ordering the set of calculated coordinates of points on the corresponding branches of the envelope, which is difficult to formalize. In this paper, it is proposed to simplify the process of approximate finding of points on the envelope by calculating the coordinates of the points of intersection of "neighboring" pairs of circles. Namely, from N circles of the family SN, their pairs SK and SK+1 (K = 1..N-1) are sequentially selected, and common points are calculated for them. Then it can be argued that the set of found intersection points of all circles will be located on the branches of the desired envelope. О. Podgorny suggested using the above geometric model for interpretation at the graphic level of the relationship between the phases of the incident wave on the curve Г and the reflected wave from the curve G. Let the origin point be the source of the incident wave. Then the phase of the reflected wave from curve G can be considered orthotomics. The possibility of using orthotomics as lines for which the reflected rays constitute, respectively, a bundle or congruence of normals, О. Podgorny also demonstrated for the case of parallel incident rays. Here, as a basis for constructive constructions, mappings from a line in a plane were considered. The direct and orthotomics in the plane are the fronts of the incident and reflected waves, respectively, since the incident and reflected rays are their normals. The paper presents a computer implementation of the graphical method for determining the individual phases of the reflected waves, which correspond to the phases of the wave with a point source. These studies are important in solving problems in the field of architectural and construction physics and solar technology, which require the further development of constructive methods for modeling the reflected streams of sound or light.

Two Separate Circles With Same-Radius: Projective Geometric Properties and Applicability in Camera Calibration

In the domain of computer vision, camera calibration is a key step in recovering the two-dimensional Euclidean structure. Circles are considered important image features similar to points, lines, and conics. In this paper, a novel linear calibration method is proposed using two separate same-radius (SSR) circles as the calibration pattern. We show that the distinct pair of dual circles encodes three lines, two of which are parallel to each other and perpendicular to the remaining line. When any two coplanar or parallel circles degenerate to SSR circles, a solution can be found to recover another pair of parallel lines based on the geometric properties of the SSR circles. Using the vanishing points obtained as the key helper for determining the imaged circular points and the orthogonal vanishing points, we deduce the constraints on the image of the absolute conic (IAC) and then employ it for complete camera calibration. Furthermore, a closed-form solution for the extrinsic parameters can be obtained based on the projective invariance of the conic dual to the circular points. Evaluations based on simulated and real data confirmed the effectiveness and feasibility of the proposed algorithms.