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Metasurface has the ability to flexibly modulate the wavefront and detect the polarization states, thus receiving widespread research attention. The combination of polarization multiplexing techniques and focused beams with polarization information provides a new approach to compact polarization detection behavior. Here, an all-dielectric metasurface based on polarization multiplexing encoding technique is demonstrated and assigns the independent phase distributions to x- and y-polarized channels to separate different polarized components. Combining Stokes parameters and visualized polarization ellipses, realizing one-to-one mapping of the incident polarization states with far-field images. We obtain the Poincaré sphere and visualized polarization ellipses based on Stokes parameters to verify the polarization detection ability of the designed metasurface. The proposed scheme is expected to provide potential applications in fields such as full-Stokes polarization detection, high-resolution imaging, and terahertz communication. [ABSTRACT FROM AUTHOR]

Polycrystalline diffraction is a robust methodology employed to assess elastic strain within crystalline components. The Extended Caking (exCaking) method represents a progression of this methodology beyond the conventional azimuthal segmentation (Caking) method for the quantification of elastic strains using Debye–Scherrer 2D X-ray diffraction rings. The proposed method is based on the premise that each complete diffraction ring contains comprehensive information about the complete elastic strain variation in the plane normal to the incident beam, which allows for the introduction of a novel algorithm that analyses Debye–Scherrer rings with complete angular variation using ellipse geometry, ensuring accuracy even for small eccentricity values and offering greater accuracy overall. The console application of the exCaking method allows for the accurate analysis of polycrystalline X-ray diffraction data according to the up-to-date rules presented in the project repository. This study presents both numerical and empirical examinations and error analysis to substantiate the method's reliability and accuracy. A specific validation case study is also presented to analyze the distribution of residual elastic strains in terms of force balance in a Ti-6Al-4V titanium alloy bar plastically deformed by four-point bending. [ABSTRACT FROM AUTHOR]

Long live the vis-viva equation. There is sometimes more than one way of telling a story with the same ending, and this is particularly true in the field of applied mathematics where there are often different ways of obtaining the same result to a given problem, so that what might be lost or not appreciated in one approach can be found and appreciated in another. It is the purpose here to illustrate this by presenting a well-known example drawn from the field of orbital dynamics, namely the development of what is called the Vis-viva equation. This equation is simply an expression relating the square of the velocity of an orbiting object, for example a planet orbiting a sun, to orbit parameters and scientific constants. It is a standard workhorse equation that is used extensively today by orbit control specialists wishing to determine and affect velocities of spacecraft orbiting significantly larger masses, and it was developed from work carried out centuries ago by Gottfried Leibniz (1646 – 1716). The first story will simply acknowledge the equation and how it arose. The second story will present an alternative approach based on calculus, trigonometry, algebra and computations set against the backdrop of an ellipse geometry. This second story leads to the vis-viva equation in disguise, so to speak, and examples of the speeds of two planets, Earth and Mercury, orbiting the Sun will be discussed. Where relevant, the nomenclature of orbit dynamics will be acknowledged. The discussion will involve primarily a planet's orbit velocity of translation and detailed considerations will not be given to effects due to its speed of rotation. [ABSTRACT FROM AUTHOR]

We calculated virial coefficients for hard ellipses, stadia, rectangles, planar lenses and rhombi up to the eighth order employing Mayer-sampling Monte Carlo simulations. We improved and extended available data for ellipses, stadia, and rectangles and provide virial coefficients for planar lenses and rhombi. The influence of the aspect ratio ν on the virial coefficients is discussed, including hard discs as the limit of ellipses, stadia, and planar lenses and hard squares as the limit of rectangles and rhombi with aspect ratio $ \nu \to 1 $ ν → 1. In two dimensions, both in the limit $ \nu \to 0 $ ν → 0 and $ \nu \to \infty $ ν → ∞ , anisotropic figures approach hard needles. The dependence of reduced virial coefficients $ \tilde {B}_i=B_i/B_2^{i-1} $ B ~ i = B i / B 2 i − 1 on the shape parameter $ \alpha =B_2^\ast -1 $ α = B 2 ∗ − 1 , with $ B_2^\ast = B_2/V_\mathrm {P} $ B 2 ∗ = B 2 / V P and $ V_\mathrm {P} $ V P the particle area is analysed. The reduced virial coefficients $ \tilde {B}_i(\alpha) $ B ~ i (α) are approximated as functions of the shape parameter employing third-order polynomials, enabling an interpolation for all aspect ratios ν. [ABSTRACT FROM AUTHOR]

To align with the development trend of large aperture, high stiffness, and high expansion ratio for space antennas, we propose a scissor-like hoop truss deployable antenna mechanism with arbitrary curvature mesh support. Compared to classical AstroMesh-hoop structures, this configuration features parabolic support ribs under the mesh surface, eliminating the need for additional high columns. It has a lower stowed height and higher in-plane stiffness, which helps maintain high surface accuracy. Firstly, the Four-Parameter Method is proposed for the dimensional design of scissor-like chains with arbitrary curvature from the perspective of the deployable antenna support structure. This method allows for controlling the deployment configuration of the scissor-like support structure by adjusting the semi-major axis length and the focal length of the initial ellipse, the angles of the unit lines, and the x-coordinates of the fitting points. Then, a three-dimensional scissor-like hoop design with joint scale that satisfies the deployable conditions is performed. Subsequently, two major issues called over-deployment and bistability are discussed, which need to be considered when applying scissor-like elements (SLEs) to space-deployable antennas. The mechanisms behind these issues are analyzed from a kinematic perspective, and optimization methods are proposed. The effectiveness of the optimization methods is verified through dynamic software simulations and a prototype. The high in-plane stiffness of the configuration is verified by finite element analysis. • The Four-Parameter Method for the dimensional design of SLE's is proposed. • A scissor-like hoop truss mechanism with arbitrary curvature support is proposed. • Two major issues when applying SLE's to space-deployable reflectors are proposed.\ [ABSTRACT FROM AUTHOR]

A novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors, which is the perimeter of an ellipse whose focal points represent the degree‐point and its dual‐point of a pair of adjacent vertices in a graph G=(V,E)$$ G=\left(V,E\right) $$. We first analyze the chemical applicability of this elliptic Sombor index, and then establish its mathematical properties and determine the extremal values for this elliptic Sombor index among all (molecular) trees and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

This is a brief review of five-year studies of features of nontrivial three-dimensional topological structures of the electromagnetic field—optical strips—formed by vectors characterizing the sizes and orientation of the polarization ellipse of the light field with tracing of change in the orientations of these vectors along closed contours of various shapes, which are linked to lines of polarization singularity or are located immediately near them. The characteristics of polarization strips formed on arbitrarily oriented small circular contours around a single circular polarization singularity line of the electromagnetic field are described. The topology of polarization strips of light scattered by a dielectric sphere and a metallic spheroid with subwavelength sizes when elliptically polarized harmonic plane waves are incident on them is discussed. Optical polarization strips existing in the focal region of a parabolic mirror that tightly focus various laser beams incident on it is described. The topology of polarization ellipse strips in the electromagnetic field of laser radiation propagating in a planar metamaterial is discussed. [ABSTRACT FROM AUTHOR]

One nappe of a right circular cone, cut by a transverse plane, splits the cone into an infinite frustum and a cone with an elliptical section of finite volume. There is a standard way of computing this finite volume, which involves finding the parameters of the so-called shadow ellipse, the characteristics of the oblique ellipse (the cut) and, finally, the projection of the vertex of the cone onto the oblique ellipse. This paper shows that it is possible to compute that volume just by using the information of the shadow ellipse and the height of the cone. Indeed, the finite slant cone has the same volume of an elliptic right cone, with the base being the shadow ellipse of the cut portion and with the height being the distance between the vertex of the cone and the intersection of the height of the original cone with the cutting plane. This is proved by introducing a volume-preserving shear transformation of the elliptical slant cone to a right cone, so that the standard volume formula for a cone can be straightforwardly applied. This implies a simplification in the procedure for computing the volume, since the oblique ellipse—i.e., the difficult part—can be neglected because only the shadow ellipse needs to be determined. [ABSTRACT FROM AUTHOR]

Edge detection is one of the most fundamental and critical operations in image analysis and computer vision. This paper proposes an adaptive edge detection method that combines multi-scale closest neighbor operator with grid partition technique (MSCNOGP). The multi-scale closest neighbor operator can be used to remove both noisy data and small area textures, while the grid partition technique can improve the precision of the edges. By utilizing the concepts of both twin edge pixel and grid divergence, the resulting edges can be further improved. Compared with prevailing traditional methods, the MSCNOGP method achieves both the best precision and almost the best visual effect, where both line segment fitting and ellipse fitting are applied for testing different edge detection methods. The performance on the F-measure score of the MSCNOGP method is much better than those of prevailing traditional methods. [ABSTRACT FROM AUTHOR]

The off-axis vortex Gaussian beam (OavGB) in strongly nonlocal nonlinear media (SNNM) with oblique incidence showed novel transmission characteristics. When the off-axis vortex parameters were not equalled to the intensity distribution parameters, the analytical expression for the propagation of the OavGB in SNNM was obtained by using the ABCD matrix method. The expression for the second-order moment beamwidth and the trajectory equation for the center of the beam's mass were obtained by analytical method. According to numerical simulations, the projection of the mass's center on the cross-section was found to be an ellipse, a line segment, or a circle, with the direction of rotation of the projected trajectory around the z-axis depending on the direction of the orbital angular momentum, and the direction of rotation of the beam depending on the sign in front of the imaginary part of the vortex point. The direction of the OavGB's rotation and the direction of revolution of the mass's center may be the same or opposite, But the rotation period of the beam and the revolution period of the mass's center were equated. The projected trajectory of the mass's center of the beam on the cross-section can be controlled by the magnitude of the parameters a, b, c, d, ζ , f. These laws can be used to control beam transmission paths and information encoding. [ABSTRACT FROM AUTHOR]

In this paper, according to the actual size of gas turbine blade interior space, eight vortex cooling models with different cross-sectional shapes of coolant chambers are established. Rectangular, Circle, Trapezoidal up, Trapezoidal down, Ellipse up, Ellipse down, Sine up and Sine down are the eight coolant chamber shapes. Functions of variable coolant chamber structures on vortex cooling flow and heat exchange characteristics are numerically analyzed by verifying the Standard k - ω turbulence model and conducting grid independence analysis. Research results show that the heat exchange ability of Sine up coolant chamber is the best compared with the other coolant chambers. The cool air enters the vortex chamber at a faster speed and impacts on the target surface to form a high-pressure area and a low-temperature swirl. The more the cool air moves downstream, the greater the turbulent kinetic energy becomes, which is more conducive to the heat transfer. Therefore, Sine up coolant chambers have the highest thermal performance factor. Due to the low drag coefficient and high target surface average Nusselt number of Ellipse up, its thermal exchange performance is second only to that of Sine up. Circle coolant chamber will not bring favorable factors to engineering application because of the slowest flow speed and the worst heat exchange capacity. For Trapezoidal up and Trapezoidal down, the flow velocity, pressure distribution, turbulent kinetic energy intensity and heat transfer uniformity are almost the same as Rectangle coolant chamber. The models of Ellipse down and Sine down coolant chambers exhibit less turbulent kinetic energy strength to heat transfer and higher temperature in the vortex chamber, and it is not suggested for cooling system optimal design. [ABSTRACT FROM AUTHOR]

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's endpoints and eccentricity 3 / 2 . Using an optimization approach, we prove that the convex sets bounded by these ellipses intersect, answering a Tverberg-type question of Andy Fingerhut from 1995. [ABSTRACT FROM AUTHOR]

In an effort to illuminate the general principles governing the critical behavior of model fluids, we investigate in this study how the shape and the (attractive) interaction range of the molecule affect the gas–liquid equilibrium and the critical behavior of the system. A combination of Monte Carlo simulations and analytical theory is employed to compute critical properties, i.e., temperature and density, of a system of hard-core ellipses with an attractive square-well potential in two-dimensional space. The critical temperature is found to decrease monotonically as the asphericity of the molecule is increased. This trend can be successfully explained in terms of the strength of the effective attraction acting between molecules measured, for instance, by the second virial coefficient. The critical density shows a complex dependence on both the range of attraction and the asphericity of the molecule. We find that the properties of particle clusters formed in near-critical states reproduce some of the most important features of the critical density, including multiple minima and maxima. It is shown that a model based on the extent of the overlap between attractive shells surrounding the ellipses captures the variation of the size of the clusters. Based on the obtained results, we discuss implications of varying the shape of the attraction potential for critical density. [ABSTRACT FROM AUTHOR]

In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on the Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being an annulus or a confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the conditions that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding. [ABSTRACT FROM AUTHOR]

Analogous to the Poincaré sphere, a hybrid order Poincaré sphere is used to represent the ellipse field singularities (C-points). We study the tight focusing properties of generic bright and dark hybrid order Poincaré sphere beams in the presence of primary coma. The role of the polarization singularity index and handedness of the polarization of the hybrid order Poincaré sphere beams on the focused structure has been discussed. Results have been presented for the total intensity, component intensities, and component phase distributions for left- and right-handed bright and dark star and lemon types singularities. The presence of primary coma distorted the focal plane intensity distributions for both positive and negative index generic C-points. Coma is known to disturb the circular symmetry of the focal plane intensity distribution. Similarly in tight focusing polarization is known to disturb the symmetry. Therefore, a beam with structured and inhomogeneous polarization distribution tightly focused under the influence of coma is a fit case to study. It is found that the presence of primary coma aberration in the focusing system produces a positional shift of the high-intensity peaks and a reduction of the intensity on one side of the center. As the strength of the primary coma increases, the focal plane intensity distributions shift more and more toward the right from the initial position. Unlike the scalar vortex case, in the case of hybrid order Poincaré sphere beams, the focal plane intensity distribution undergoes rotation, as the helicity of the hybrid order Poincaré sphere beams is inverted, in addition to shifting. All the component phase distributions are found to be embedded with phase vortices of charge ± 1 . [ABSTRACT FROM AUTHOR]

Given three non-parallel segments OP1, OP2, OP3 in a plane ω, we consider the ellipses εP1,P2, εP2,P3, εP3,P1 having as conjugate semi-diameters the pairs (OP1,OP2), (OP2,OP3) and (OP3,OP1), respectively. We find the necessary and sufficient conditions for (i) the existence of a common point P ∈ εP1,P2 ∩ εP2,P3 ∩ εP3,P1 and (ii) the existence of a pair of parallel and distinct lines tangent to the three ellipses. In this later case, we solve the problem by introducing the definition of cylindrical Pohlke's projection. [ABSTRACT FROM AUTHOR]

We report the results of the stellar occultation by (UII) Umbriel on 2020 September 21. The shadow crossed the USA and Canada, and 19 positive chords were obtained. A limb parameter accounted for putative topographic features in the limb fittings. Ellipse fittings were not robust – only upper limits were derived for the true size/shape of a putative Umbriel ellipsoid. The adopted spherical solution gives radius = 582.4 ± 0.8 km, smaller/close to 584.7 ± 2.8 km from Voyager II. The apparent ellipse fit results in a true semi-major axis of 584.9 ± 3.8 km, semi-minor axis of 582.3 ± 0.6 km, and true oblateness of 0.004 ± 0.008 for a putative ellipsoid. The geometric albedo was p V = 0.26 ± 0.01. The density was ρ = 1.54 ± 0.04 g cm−3. The surface gravity was 0.251 ± 0.006 m s−2 and the escape velocity was 0.541 ± 0.006 km s−1. Upper limits of 13 and 72 nbar (at 1σ and 3σ levels, respectively) were obtained for the surface pressure of a putative isothermal CO2 atmosphere at T = 70 K. A milliarcsecond precision position was derived: α = 02h30m28 |${_{.}^{\rm s}}$| 84556 ± 0.1 mas, δ = 14o19′36″.5836 ± 0.2 mas. A large limb parameter of 4.2 km was obtained, in striking agreement with opposite Southern hemisphere measurements by Voyager II in 1986. Occultation and Voyager results indicate that the same strong topography variation in the surface of Umbriel is present on both hemispheres. [ABSTRACT FROM AUTHOR]

In this work, for the first time, an exact solution of the contact problem of interaction with a multilayer base of two semi-infinite stamps, the ends of which are parallel to each other, is constructed. The stamps are assumed to be absolutely rigid, and the distance between them can have any finite value. The task is an important stage in the algorithm for constructing models of a new type of crack in materials of different rheologies. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been little studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The problem considered in this article can be considered as the result of the formation of a new type of crack with absolutely rigid banks and a deformable lower boundary. Thanks to it, after the solution, it becomes possible to switch to deformable stamps and a crack of a new type in the rheological medium. The solution of this problem turned out to be possible due to the construction of exact solutions of the Wiener–Hopf integral equations on a finite segment. This paper shows how the solution of one of the previously unsolved problems allows us to investigate and solve exactly other problems and to identify previously unknown properties and resonances. As a result of constructing an exact solution to the problem, the fact that the solution of dynamic contact problems for stamp systems is not unique was confirmed and a dispersion equation for finding resonant frequencies was constructed. [ABSTRACT FROM AUTHOR]

Currently, there are two main types of anti-floating methods for underground structures; one is the passive anti-floating method represented by anti-draft piles, the other is the active anti-floating method which focuses on interceptor-discharge pressure-reducing (IDPR). In the design of an IDPR anti-floating system, the relief well system situated within the cut-off wall serves as the primary drainage channel. The determination of the seepage field distribution within the multi-well system is vital for the overall design. For the seepage field analysis of the IDPR anti-floating multi-well system, currently numerical analysis is usually used, and there is a lack of simplified analysis methods. The simplified analysis methods already available are based on the uniform distribution of wells in circular pits, while the conversion of non-circular pits into circular pits produce large errors, which are not conducive to promoting the use of the method. To address this, we propose a simplified calculation approach suitable for multi-well systems (arbitrary layout) within elliptical pits. The analytical solution of non-uniformly distributed wells in circular pits is deduced through the principle of superposition. Then, the ellipse is mapped into a circle by using conformal mapping. The resistance coefficient method is adopted, and the internal and external seepage fields are connected in series to obtain the total flow rate, as well as the distribution of the seepage field. This is based on the consideration of the permeability of the waterproof curtains and the bypassing seepage. According to the verification of the calculation example, the results of the simplified algorithm are similar to the results of the finite element method, which proves the accuracy of the method; at the same time, when applied to the actual engineering, the obtained calculation results coincide with the measured data, which proves the practicability and reliability of the method. The simplified method can provide an effective way to design an IDPR anti-floating system. [ABSTRACT FROM AUTHOR]

In this research article, a novel double-split elliptical split ring resonator (DS-ESRR) is proposed to achieve frequency-notching behavior of ultrawideband filtenna, where the semimajor and minor axes of the ellipse are taken as a bivariate random variable and expressed in Ramanujan's correction coefficient so that more degree of freedom is available for choosing degenerated impedance and thus for variable frequency-notching applications. To verify this hypothetical method, finite sets of variables for DS-ESRR are presented in this proposed work, and a mathematical expression is formulated to estimate the resonant frequency of the DS-ESRR, such that for practical applications, frequency-notching parameters can be easily estimated accurately rather than previously used SRR like circular or square-shaped geometry. DS-ESRR is deployed at the back of a CPW-fed ultrawideband antenna for notching filter application, and the computed data is compared with the eigen mode simulation results which reveal good agreement with each other. [ABSTRACT FROM AUTHOR]

We perform mathematical modeling of the shape of dental arches. It is assumed that dental arches can be modeled by second-order curves. As a result of the analysis of the data of measurements performed for the diagnostic models of dentures, we compute both the coefficients of the quadratic form of the indicated second-order curves and the invariant of this form. Since the invariant of the second-order curve appears to be greater than zero, we make conclusion that the shapes of dental arches are described by the equation of ellipse. The parameters of the canonical equation of ellipse used to model the shape of dental arches are determined. We also develop a procedure for specifying the parameters of the model with an aim to use them in the processes of orthopedic and orthodontic treatments. [ABSTRACT FROM AUTHOR]

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors. [ABSTRACT FROM AUTHOR]

A dataset of 85 Lagrangian surface drifter trajectories covering the central North Sea area and the Skagerrak from 2017 - 2021 of 17 deployments is presented. The data have been quality controlled, uniformly structured and assimilated in a standard NetCDF format. Using appropriate methods presented in detail here, surface currents were calculated from the drifter position data and gridded surface current maps with 0.125° spatial resolution were derived. The maps present for the first time mean currents of the south-eastern North Sea and the Skagerrak from Lagrangian observations. Tidal energy spectra were analyzed separately for the southern and northern areas of the North Sea, and tidal ellipses were calculated to determine the tidal impact on surface currents. Significant differences between the shallow shelf and the deeper areas of the North Sea are evident. While the shallow nearshore areas are dominated by tidal currents, deeper areas such as the Skagerrak register a high mean residual circulation driven by high density gradients. Measurements using Eulerian approaches and remote sensing methods are restricted in temporal and spatial coverage, in particular, to capture submesoscale dynamics. For this reason, Lagrangian measurements, to a large extent, provide new insights in the complex submesoscale dynamics of the North Sea. Exemplarily, the Skagerrak region is used to illustrate the ability of reconstructing mesoscale and submesoscale current patterns using drifter observations. This unique dataset covering the entire south-eastern North Sea and the Skagerrak offers further analysis possibilities and can be used for the investigation of various hydrodynamic and environmental issues, e.g., the analysis of submesoscale current dynamics at ocean fronts, the determination of the kinetic eddy energy and the propagation of pollutants in the North Sea. [ABSTRACT FROM AUTHOR]

In this article, we establish new fixed ellipse theorems that will provide a geometrical perspective to the classical metric fixed point theory. Also, we provide novel fixed ellipse consequences as the expanding of some existing and known fixed circle theorems in literature to ellipse case, and give non‐trivial illuminating examples, which show the impressiveness of our results. On the other hand, using the given fixed ellipse, we give some solutions for the open question which is posed by Rhoades with respect to the presence of contractive mappings that permissive discontinuity at the fixed point. Furthermore, we give an application to usage respecting activation functions handled at neural networks via the discontinuity and the obtained results of fixed ellipse. [ABSTRACT FROM AUTHOR]

Elongated ellipsoidal liquid crystal microdroplet reorientation dynamics are discussed in this paper for biosensor applications. To investigate the effect of elongated droplets on nematic liquid crystal droplet biosensors, we simulated a model of a liquid crystal droplet using ellipse geometry. Director reorientation is examined in relation to the elongated droplet shape. In addition, we examined aspect ratio as a factor affecting biosensor response time in relation to surface viscosity and anchoring energy. Finally, the findings suggest that the aspect ratio should be taken into account when designing biosensors. These results can be used to develop more effective biosensors for a variety of applications. This model then predicts the director reorientation angle, which is dependent on the anchoring energy and surface viscosity. This model further suggests that both surface viscosity and homeotropic anchoring energy play an important role when it comes to the director reorientation angle. We developed and applied a nonlinear unsteady-state mathematical model utilizing torque balance and Frank free energy according to the Leslie–Ericksen continuum theory for simulating elongated nematic liquid crystal biosensor droplets with aqueous interfaces. Using the Euler–Lagrange equation, a transient liquid crystal–aqueous interface realignment is modeled by changing the easy axis when surfactant molecules are added to the interface. The realignment at the surface of the droplet is assumed to be driven by the effect of the surfactant, which causes an anchoring transition. According to the results, the response time of the biosensor depends on the aspect ratio. Therefore, the elongation has the potential to control biosensing response time. The result of our study provides a better understanding of director reorientation in elongated liquid crystal droplets in biosensing applications through the numerical results which are presented in this paper. [ABSTRACT FROM AUTHOR]

In computer vision, camera pose estimation from correspondences between 3D geometric entities and their projections into the image has been a widely investigated problem. Although most state-of-the-art methods exploit low-level primitives such as points or lines, the emergence of very effective CNN-based object detectors in the recent years has paved the way to the use of higher-level features carrying semantically meaningful information. Pioneering works in that direction have shown that modelling 3D objects by ellipsoids and 2D detections by ellipses offers a convenient manner to link 2D and 3D data. However, the mathematical formalism most often used in the related litterature does not enable to easily distinguish ellipsoids and ellipses from other quadrics and conics, leading to a loss of specificity potentially detrimental in some developments. Moreover, the linearization process of the projection equation creates an over-representation of the camera parameters, also possibly causing an efficiency loss. In this paper, we therefore introduce an ellipsoid-specific theoretical framework and demonstrate its beneficial properties in the context of pose estimation. More precisely, we first show that the proposed formalism enables to reduce the pose estimation problem to a position or orientation-only estimation problem in which the remaining unknowns can be derived in closed-form. Then, we demonstrate that it can be further reduced to a 1 Degree-of-Freedom (1DoF) problem and provide the analytical derivations of the pose as a function of that unique scalar unknown. We illustrate our theoretical considerations by visual examples and include a discussion on the practical aspects. Finally, we release this paper along with the corresponding source code in order to contribute towards more efficient resolutions of ellipsoid-related pose estimation problems. The source code is available here: https://gitlab.inria.fr/vgaudill/p1e. [ABSTRACT FROM AUTHOR]

WC–Co cemented carbides build one of the important classes of metal matrix composites. We show in this paper that the use of machine vision methods makes it possible to obtain sufficiently informative statistical data on the topology of the interfaces between tungsten carbide grains (WC) and a cobalt matrix (Co). For the first time, the outlines of the regions of the cobalt binder were chosen as a tool for describing the structure of cemented carbides. Numerical processing of micrographs of cross sections of three WC–Co alloys, which differ in the average grain size, was carried out. The distribution density of the angles in the contours of cobalt "lakes" is bimodal. The peaks close to 110° (so-called outcoming angles) correspond to the contacts between the cobalt binder and the WC/WC grain boundaries. The peaks close to 240° (or incoming angles) correspond to the WC "capes" contacting the cobalt "lakes" and are determined by the angles between facets of WC crystallites. The distribution density of the linear dimensions of the regions of the cobalt binder, approximated with ellipses, were also obtained. The distribution density exponentially decreases with the lengths of the semi-axes of the ellipsoid, approximating the area of the cobalt binder. The possible connection between the obtained data on the shape of cobalt areas and the crack trajectories in cemented carbides is discussed. [ABSTRACT FROM AUTHOR]

In this paper deals with the axisymmetric temperature distribution of an elliptical cylinder subjected to the interior heat supply within which sources are generated in step with a linear function of the temperature. The solution of the heat conduction equation and the corresponding initial and boundary conditions are obtained in an analytical kind by using the Green’s function methodology. The outcomes are obtained in a series in terms of Mathieu functions. The solution is more verified by comparing them to the limiting case of a circle as a special kind of ellipse. Finally, the governing equation for thermally induced deflection results and its associated stresses are found throughout the bending of a simply supported cylinder. The numerical results obtained using these computational tools and illustrated graphically. [ABSTRACT FROM AUTHOR]

Parking lots have wide variety of shapes because of surrounding environment and the objects inside the parking lot, such as trees, manholes, etc. In the case of paving the parking lot, as much area as possible should be covered by the construction vehicle to reduce the need for manual workforce. Thus, the coverage path planning (CPP) problem is formulated. The CPP of the parking lots is a complex problem with constraints regarding various issues, such as dimensions of the construction vehicle and data processing time and resources. A strategy based on convolutional neural networks (CNNs) for the fast estimation of the CPP's average track length, standard deviation of track lengths, and number of tracks was suggested in this article. Two datasets of different complexity were generated to analyze the suggested approach. The first case represented a simple case with a working polygon constructed out of several rectangles with applied shear and rotation transformations. The second case represented a complex geometry generated out of rectangles and ellipses, narrow construction area, and obstacles. The results were compared with the linear regression models, with the area of the working polygon as an input. For both generated datasets, the strategy to use an approximator to estimate outcomes led to more accurate results compared to the respective linear regression models. The suggested approach enables us to have rough estimates of a large number of geometries in a short period of time and organize the working process, for example, planning construction time and price, choosing the best decomposition of the working polygon, etc. [ABSTRACT FROM AUTHOR]

The spatial circular restricted three-body problem in the nonresonant case is investigated. Namely, we apply Gaussian averaging to obtain averaged equations of motion in terms of osculating elements and then investigate them. A Keplerian ellipse with a focus at the main body (the Sun) is taken as an unperturbed orbit, assuming the semi-major axis of the ellipse to be shorter than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula, we derive a twice-averaged disturbing function in the form of an explicit analytical series with coefficients that are expressed in terms of Gauss and Clausen hypergeometric functions. An investigation of the averaged disturbing function along its curves of nonanalyticity shows that the series are asymptotic in the sense of Poincaré. For a reduced system with one degree of freedom, the phase portraits of oscillations in the plane of Keplerian elements , are constructed in the second and fourth approximations. It is shown that the topology of the phase portrait in the fourth approximation is more complex than in the first and second approximations, provided that the constant of the Lidov–Kozai integral belongs to the interval (0, 0.382). [ABSTRACT FROM AUTHOR]

Copyright of Journal of China University of Petroleum is the property of China University of Petroleum and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case. [ABSTRACT FROM AUTHOR]

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2õ (n²/3 ) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R³, a maximum subset of pointswith diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we showthat, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2n1-ε, unless the Exponential Time Hypothesis fails. [ABSTRACT FROM AUTHOR]

We examine the fluid phase behavior of binary mixtures of hard superellipses using the scaled particle theory. The superellipse is a general two-dimensional convex object that can be tuned between the elliptical and rectangular shapes continuously at a given aspect ratio. We find that the shape of the particle affects strongly the stability of isotropic, nematic, and tetratic phases in the mixture even if the side lengths of both species are fixed. While the isotropic–isotropic demixing transition can be ruled out using the scaled particle theory, the first order isotropic–nematic and the nematic–nematic demixing transition can be stabilized with strong fractionation between the components. It is observed that the demixing tendency is strongest in small rectangle–large ellipse mixtures. Interestingly, it is possible to stabilize the tetratic order at lower densities in the mixture of hard squares and rectangles where the long rectangles form a nematic phase, while the squares stay in the tetratic order. [ABSTRACT FROM AUTHOR]

In this article, we classify the developable surfaces in three-dimensional Euclidean space R 3 that are foliated by general ellipses. We show that the surface has constant Gaussian curvature (CGC) and is foliated by general ellipses if and only if the surface is developable, i.e., the Gaussian curvature G vanishes everywhere. We characterize all developable surfaces foliated by general ellipses. Some of these surfaces are conical surfaces, and the others are surfaces generated by some special base curves. [ABSTRACT FROM AUTHOR]

A magnetorheological brake (MRB) works on the shearing effect of MR fluid in the magnetic field to develop resistance between the stator and the rotor, which are relative moving surfaces. Researchers achieved high torque volume ratios in magnetorheological fluid braking systems by adding multiple layers and using multipole approaches. Even though several designs already exist, there is a need for innovative ideas and advanced structures of MRBs for practical engineering applications. The number of magnetic poles, pole geometry and placement have a significant effect on the performance of multipole MR brakes. This research focused on the effect of the pole head geometry in a multipole dual disc compact MR brake to understand the relationship between the geometry and its magnetic flux flow. The magnetic field generated by the electromagnetic coil's orthogonal travel through pole heads, rotor and MR fluid layers results in friction to fulfill the action of braking that achieves maximum brake torque. This flow depends on the magnetic pole head area and corresponding core dimensions. This study considered polygon, square, hexagon, trapezoid and ellipse pole head profiles in place of the conventional circular pole and the best possible core dimensions were used for each geometry. To study the brake's performance, COMSOL Multiphysics software (version 5.5) commonly used for 3D electromagnetic modeling was used to simulate and examine the magnetic flux intensity for various pole head configurations. Results indicate an improvement compared to previous works, considering the same dimensional limitations and the flux leakage between the adjacent poles being much less. The hexagonal and elliptical pole heads had better magnetic field area, magnetic intensity distribution and higher magnetic flux lines piercing through the MR fluid perpendicularly than the circular and square pole heads. An increment of 3.88% for braking torque performance was obtained, and the braking torque was significantly enhanced. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin Institute of Technology. Social Sciences Edition / Haerbin Gongye Daxue Xuebao. Shehui Kexue Ban is the property of Harbin Institute of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The face gear tooth surface is a high-order variable curvature surface, and the curvature of the surface is complicated, so it is difficult to describe the characteristics of the face gear tooth surface by a specific equation. In this paper, the discrete curvature relation of the surface is used instead of an analytical equation to describe a spatial meshing surface, and the traditional meshing theory is neutralized to analyze the characteristics of the face gear tooth surface. Firstly, according to the structural characteristics of the face gear, the sampling method of the face gear tooth surface is analyzed, and the mathematical model of fitting the tooth surface contact point is established. Then, the discrete asymptotic surface development analysis method is studied, and the ellipsoid contact analysis method of the face gear pair is established by simplifying the conjugate surface and its region. Finally, the contact analysis method of the discrete tooth surface is studied, and the instantaneous contact ellipse of the face gear tooth surface is calculated, which formed a new numerical meshing method of space curved surface. [ABSTRACT FROM AUTHOR]

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function being either analytic on and within an ellipse and |$\lambda \leqslant 0$| or differentiable and |$\lambda \leqslant 1$|⁠ , where |$\lambda $| is the parameter in Gegenbauer projections. If the underlying function is analytic and |$\lambda>0$| or differentiable and |$\lambda>1$|⁠ , then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of |$n^{\lambda }$| and |$n^{\lambda -1}$|⁠ , respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all |$\lambda>-1/2$|⁠. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods. [ABSTRACT FROM AUTHOR]

Minkowski tensors contain information about shape and orientation of the underlying convex body. We make this precise by showing that reconstructing a centered ellipse in two-dimensional Euclidean space from its rank-2 surface tensor is a well-posed inverse problem. It turns out that this result can be restated equivalently with other geometric tomography data derived from the support function of the ellipse, such as the first three non-trivial Fourier coefficients. We present explicit reconstruction algorithms for all three types of input. The relevance of these findings is illustrated in an application to stationary particle processes. We define and discuss two shape ellipses, each containing information about the mean shape and orientation of the typical particle. [ABSTRACT FROM AUTHOR]

We know that circular shapes we encounter in daily life may appear to be elliptical from some viewing points. It is reasonable to expect that an ellipse may appear to be circular from certain viewpoints. In this paper, we investigate the locus of viewpoints from which an ellipse appears circular. It will be shown that the locus of viewpoints is a hyperbola passing through the two foci of the ellipse. Conversely, the locus of viewpoints from which a hyperbola looks circular is an ellipse passing through the two foci of the hyperbola. Further, the locus of viewpoints from which a parabola looks circular is itself a parabola, passing through the focus of the original parabola. There is a simple duality between the object to be observed and the observer. [ABSTRACT FROM AUTHOR]

Fracture characteristics have a significant effect on the mechanical and hydraulic properties of rock mass. In this paper, before modeling a discrete fracture network, the fracture is simplified as a non-similar ellipse. First, the multifactor coupling stereological relationship of the sampling trace length distribution is established, which proves that the trace length distribution is independent of the fracture occurrence distribution. Using the Volterra integral equation method, the stereological formula of the trace length is inversely solved to obtain the distribution expressions of the major axis and axial ratio of the elliptical fractures. The analytical solutions of the probability density function (PDF) of the characteristic size of the elliptical fractures are derived for cases in which the trace length follows a uniform distribution, fractal distribution, and polynomial distribution. Second, for cases in which the trace length conforms to a negative exponential distribution, gamma distribution, chi-square distribution, and lognormal distribution, the statistical eigenvalues of the major axis and axial ratio of the elliptical fractures are deduced. Finally, a Monte Carlo statistical simulation is performed using the Rock Mass Joint Network Simulation (RJNS3D) toolkit to verify the applicability of the derivation process and the correctness of fitting a polynomial function to the trace length distribution to solve the PDF of the characteristic size of the elliptical fractures. The proposed method can better predict the distribution of the required parameters, such as the major axis and axial ratio of the elliptical fractures, according to the typical two-dimensional data in the sampling windows. This method can be further applied to reconstruct fracture networks in practical engineering tasks and lay a foundation for the analysis of rock mass strength and deformation, representative elementary volume (REV), seepage, and surrounding rock stability. Highlights: A non-similar elliptical model is developed to simulate a fracture network in rock mass. The multifactor coupling stereological relationship of the sampling trace length distribution is established. The correct analytical solutions of the probability density function of the characteristic size of elliptical fractures are derived. Trace sampling in the sampling window is simulated by the Monte Carlo method. [ABSTRACT FROM AUTHOR]

Muffler with circular section and inner symmetric acoustical elements has been habitually used in industry. In order to improve the acoustical performance, an advanced eccentric muffler having ellipse cross-section, multiple chambers, and multi-segment perforated/non-perforated tubes is presented. Concerning about high order wave effect to the muffler, a finite element method (FEM) run on COMSOL package is adopted. In addition, the influence of sound transmission loss with respect to various geometric parameters has been explored. In order to simplify the optimization process, a simplified objective function established by an Artificial Neural Network (ANN) and FEM has been implemented and connected to a Genetic Algorithm (GA). Two case studies of muffler optimization for two kinds of design parameter sets (one of Lin and Lout and the other of Lin1 and Lout 1) are introduced. Two targeted frequencies (1000 Hz and 2500 Hz) are selected and applied in the two cases' muffler optimization. Simulated results reveal that the acoustical influence relating to various geometric parameters is huge. The sound attenuation will increase in the low frequency region if the curvature ratio δ (ratio of long axis to short axis in ellipse) or the X (horizontal length of the muffler's body) increases. Consequently, the study demonstrates a methodology in the optimization of a four-chamber ellipse and eccentric muffler internally inserted with multi-segment perforated/non-perforated tubes. [ABSTRACT FROM AUTHOR]

We propose a novel minimal solver for sphere fitting via its 2D central projection, i.e., a special ellipse. The input of the presented algorithm consists of contour points detected in a camera image. General ellipse fitting problems require five contour points. However, taking advantage of the isotropic spherical target, three points are enough to define the tangent cone parameters of the sphere. This yields the sought ellipse parameters. Similarly, the sphere center can be estimated from the cone if the radius is known. These proposed geometric methods are rapid, numerically stable, and easy to implement. Experimental results—on synthetic, photorealistic, and real images—showcase the superiority of the proposed solutions to the state-of-the-art methods. A real-world LiDAR-camera calibration application justifies the utility of the sphere-based approach resulting in an error below a few centimeters. [ABSTRACT FROM AUTHOR]

A function from the domain (x-set) to the codomain (y-set) connects each x element to precisely one y element. Since each x-point originating from the domain corresponds to two y-points on the graph of a closed curve (i.e., circle, ellipse, superellipse, or ovoid) in a rectangular (Cartesian) diagram, it does not fulfil the function's requirements. This non-function phenomenon obstructs the nonlinear regression application for fitting observed data resembling a closed curve; thus, it requires transforming the rectangular coordinate system into a polar coordinate system. This study discusses nonlinear regression to fit the circumference of a tree stem's cross-section and its sapwood–heartwood transition by transforming rectangular coordinates (x, y) of the observed data points' positions into polar coordinates (r, θ). Following a polar coordinate model, circular curve fitting fits a log's cross-sectional shape and sapwood–heartwood transition. Ellipse models result in better goodness of fit than circular ones, while the rotated ellipse is the best-fit one. Deviation from the circular shape indicates environmental effects on vascular cambium differentiation. Foresters have good choices: (1) continuing using the circular model as the simplest one or (2) changing to the rotated ellipse model because it gives the best fit to estimate a tree stem's cross-sectional shape; therefore, it is more reliable to determine basal area, tree volume, and tree trunk biomass. Computer modelling transforms the best-fit model's formulas of the rotated ellipse using Python scripts provided by Wolfram engine libraries. [ABSTRACT FROM AUTHOR]

In this paper, we prove a common fixed-point theorem for four self-mappings with a function family on -metric spaces. In addition, we investigate some geometric properties of the fixed-point set of a given self-mapping. In this context, we obtain a fixed-disc (resp. fixed-circle), fixed-ellipse, fixed-hyperbola, fixed-Cassini curve and fixed-Apollonious circle theorems on -metric spaces. [ABSTRACT FROM AUTHOR]

The calculation of moments of area is one of the most fundamental aspects of engineering mechanics for calculating the properties of beams or for the determination of invariants in different kind of geometries. While a variety of shapes, such as circles, rectangles, ellipses, or their combinations, can be described symbolically, such symbolic expressions are missing for freeform cross-sections. In particular, periodic B-spline cross-sections are suitable for an alternative beam cross-section, e.g., for the representation of topology optimization results. In this work, therefore, a symbolic description of the moments of area of various parametric representations of such B-splines is computed. The expressions found are then compared with alternative computational methods and checked for validity. The resulting equations show a simple method that can be used for the fast conceptual computation of such moments of area of periodic B-splines. [ABSTRACT FROM AUTHOR]

Just before making landfall in Puerto Rico, Hurricane Maria (2017) underwent a concentric eyewall cycle in which the outer convective ring appeared robust while the inner ring first distorted into an ellipse and then disintegrated. The present work offers further support for the simple interpretation of this event in terms of the non-divergent barotropic model, which serves as the basis for a linear stability analysis and for non-linear numerical simulations. For the linear stability analysis the model's axisymmetric basic state vorticity distribution is piece-wise uniform in five regions: the eye, the inner eyewall, the moat, the outer eyewall, and the far field. The stability of such structures is investigated by solving a simple eigenvalue/eigenvector problem and, in the case of instability, the non-linear evolution into a more stable structure is simulated using the non-linear barotropic model. Three types of instability and vorticity rearrangement are identified: (1) instability across the outer ring of enhanced vorticity; (2) instability across the low vorticity moat; and (3) instability across the inner ring of enhanced vorticity. The first and third types of instability occur when the rings of enhanced vorticity are sufficiently narrow, with non-linear mixing resulting in broader and weaker vorticity rings. The second type of instability, most relevant to Hurricane Maria, occurs when the radial extent of the moat is sufficiently narrow that unstable interactions occur between the outer edge of the primary eyewall and the inner edge of the secondary eyewall. The non-linear dynamics of this type of instability distort the inner eyewall into an ellipse that splits and later recombines, resulting in a vorticity tripole. This type of instability may occur near the end of a concentric eyewall cycle. [ABSTRACT FROM AUTHOR]

Two‐dimensional (2D) images of light beams reflected off the objects in space impinge on the retinal photoreceptors of our two laterally separated eyes. Nevertheless, we experience our visual percept as a single 3D entity—our visual world that we tend to identify with the physical world. However, experiments point to different geometries in these two worlds. Using the binocular system with asymmetric eyes (AEs), this article studies the global geometric aspects of visual space in the Riemannian geometry framework. The constant‐depth curves in the horizontal field of binocular fixations consist of families of arcs of ellipses or hyperbolas depending on the AE parameters and the eyes' fixation point. For a single set of AE parameters, there is a unique symmetric fixation at the abathic distance such that the constant‐depth conics are straight frontal lines. Critically, the distribution of these constant‐depth lines is independent of such fixations. In these cases, a two‐parameter family of the Riemann metrics is proposed to account for the retinotopy of the brain's visual pathways and simulated constant‐depth lines. The obtained geodesics for a subset of metric parameters include incomplete geodesics that give finite distances to the horizon. The Gaussian curvature of the phenomenal horizontal field is analyzed for all the metric parameters. The sign of the curvature can be inferred from the global behavior of the constant‐depth ellipses and hyperbolas only when for the metric parameters for which the constant‐depth frontal lines at the abathic distance fixations are geodesics. [ABSTRACT FROM AUTHOR]