Polar Coordinates Research Articles

A simple model of a gravitational lens from geometric optics

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n(r)2=1+C2/r2, where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images.

Elastic collisions on a simulated circular air track

Elastic collisions of gliders on linear air tracks are often used to explore conservation of energy and momentum. If one is interested in the glider behavior over a long time span, the analysis involves repeated collisions and is complicated by reflections from the track end stops. Here we analyze elastic collisions on a novel circular air track; since such a track lacks end stops, the mathematical analysis of repeated collisions is amenable to our students. Our analysis uncovers a variety of interesting behaviors which depend on the ratio of the glider masses. We examine periodic sequences where the gliders return to their initial conditions and progressions where (when plotted in polar coordinates) the collision positions take on the locus of a spiral. One set of initial conditions produces an “angle trap” where one glider remains within a certain angular range. We also explore making one glider's mass hypothetically negative, which results in a novel “chasing” motion. Our results were obtained using a 3D interactive simulation (created using C++ within the Unreal engine), which we make available as supplementary material.

A model of the zone-of-interests test

Abstract This article offers the first comprehensive economic analysis of the zone-of-interests test from administrative law. Applying concepts from polar coordinates and inner products, we construct a model of the zone-of-interests test that provides answers to many of the puzzling questions stemming from the doctrine. The results suggest that the zone-of-interests test can be understood as Congress’s attempt to cabin judicial review to ensure that litigation based on administrative rule challenges would lead to outcomes that are timely and consistent with Congress’s intent behind the subject statute. Our model also illustrates a relationship between the zone-of-interests doctrine and the standing doctrine. In extensions, we also consider how the doctrine can be rationalized as safeguarding against substantive court errors and economizing on judicial resources, and why certain statutes may include citizen-suit provisions while others do not. (JEL K23, K41)

On the non-flatness nature of noncommutative Minkowski spacetime and the singular behavior of probes

It is more than a century-old concept that the Minkowski spacetime is flat. From the pure geometric point of view, we explicitly address the issue of whether a noncommutative Minkowski spacetime is flat or not. In the framework of the twisted-diffeomorphism approach to noncommutative gravity with canonical type noncommutative (NC) coordinate structure, one important result that we get is that the NC Minkowski spacetime parametrized either with spherical polar coordinates or with parabolic coordinates has nontrivial NC corrections to Riemann curvature tensor, Ricci tensor and curvature scalars. Another crucial result is that the curvature scalars have singular behavior at certain points, and these singularities are not coordinate singularities. The nature of these singularities clearly points towards the idea of high-energetic probes turning into black holes. The absence of any such noncommutative corrections, and thus any such singularities, in the cases of Cartesian coordinates and cylindrical coordinates leads to the conclusion that high-energetic probes do not turn into black holes when the canonical NC structure is considered for these coordinates.

The Riemann Hypothesis and Polar Coordinates

The Riemann Hypothesis and Polar Coordinates

A Gauss kernel non-local stress-driven plate theory

This study presents a novel stress-driven formulation, incorporating Kirchhoff’s plate assumptions and accounting for the non-local size-dependent feature in both plate geometric axes, which is practical, especially near the origin of the axis in the polar coordinates. The presented formulation does not impose any additional constraints on the elastic radial curvatures, which are known as constitutive boundary conditions. Therefore, taking into account the non-local behavior in both radial and circumferential directions does not lead to an overconstrained problem. The study focuses on the analysis of a circular plate with an outer radius (R), thickness (h), and the length scale parameter (L). The governing differential equations are derived in Cartesian coordinates and then transformed to the polar coordinates to analyze circular plates. When R tends to infinity, the governing equation is solved analytically. Using the Galerkin technique, a finite element solution is presented for the analysis of a bounded circular plate with a clamped boundary condition at the outer radius R. The study discusses the size effects and the gradient properties of functionally graded plates in stress-driven non-local theory. It reveals that while increasing the non-local parameter results in a reduction of transverse displacement, the moments remain unaffected.

Inverse problem for 2d Laplace equation in polar coordinate system

The article presents the results of study on finding the thermophysical characteristics of a heterogeneous material based on the heat conductivity equation in polar coordinates, carried out within the framework of the grant project of the Ministry of Science and Higher Education of the Republic of Kazakhstan AP19677594 «Development of methods for finding all nonlinear moisture-heat-conducting characteristics of nonhomogeneous medium, realization of methods by creating machine learning program». The temperature state of a body of a multilayer system of bodies can be characterized using a temperature field, which is understood as a set of instantaneous temperature values at all points of the studied space. It occurs when two bodies with different temperatures come into contact or within one body with areas with different temperatures.

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Sampling Theorems Associated with Offset Linear Canonical Transform by Polar Coordinates

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important signal analysis tool in many fields of signal processing and optics. This paper investigates two sampling theorems for interpolating bandlimited and highest frequency bandlimited functions in the OLCT and offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark’s interpolation formulas, we derive the sampling theorems for bandlimited functions in the OLCT and OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity.

Proportional-derivative control and motion stability analysis of a 16-pole legs rotor-active magnetic bearings system

This paper analyzes the motion stability of a 16-pole rotor-active magnetic bearings (rotor-AMB) system and investigates the complex vibrations under a proportional-derivative (PD) controller. First, electromagnetic theory and Newton’s second law are applied to derive the two-degree-of-freedom differential governing equations for the 16-pole rotor-AMB system, incorporating the PD control terms. The resulting differential equations include parametrically excited, quadratic nonlinear, and cubic nonlinear terms. Subsequently, the multiple time scales perturbation analysis method is performed on the obtained governing equations, yielding four-dimensional averaged equations in both Cartesian and polar coordinates. Finally, numerical simulations including the amplitude–frequency response characteristics, motion trajectories, energy–amplitude relationships, as well as bifurcation and chaotic motion of the system are studied. The results indicate that the PD controller affects the softening and hardening spring characteristics of the system and has significant control effects on the system’s amplitude, energy, and stability. Additionally, increasing the differential gain coefficient [Formula: see text] can change the system’s motion from chaotic to periodic.

Irregular array optimization for beamforming with a polar coordinate-based partition coding approach

Abstract An innovative irregular array configuration optimization method for enhancing beamforming is introduced. This study presents partition coding to optimize sensor positioning and quantity of a non-uniform concentric circular array. This novel approach transcends traditional techniques by integrating structural partitioning and performance optimization to quantify the array’s geometry-performance correlation. Sensor candidate positions are mapped in polar coordinates, with each configuration translated into a sensor position matrix form. A significant innovation lies in the adaptation of the partition coding genetic algorithm to enhance the encoding of candidate positions and to refine crossover and mutation operations, underpinned by an elite retention strategy for selecting the optimal array configuration. Both simulation and experimental results substantiate the method’s effectiveness, achieving high-resolution acoustic mapping with commendably low computational complexity.

An overset-grid finite-difference algorithm for seismic wavefield propagations modelling in the polar coordinate system with a complex free-surface topography

Summary Nowadays, finite-difference method has been widely applied to simulate seismic wavefield propagation in a local seismic model with a complex topography by utilizing curvilinear grids and traction image method in Cartesian coordinates system. For a global seismic model in polar coordinate system, it is still a challenge for the conventional finite-difference method to deal with the grid singularity at the center and the topographic free surface at the top. In this study, we develop a finite-difference method in polar coordinates for seismic wavefield propagation in the 2D global model. In the proposed finite-difference method, the overset-grid algorithm is used to handle the grid singularity at the center, and the curvilinear grid technique as well as the traction image method are applied to implement free-surface boundary condition on the complex topography. The proposed finite-difference method is validated in flat and topographic free-surface models by comparing synthetic waveforms with reference solutions. The seismic wavefield propagation in a realistic Mars profile is computed by the proposed finite-difference method. The proposed finite-difference method is an efficient and accurate method for seismic wave modeling in the polar coordinate system with a complex free-surface topography.

Human–exoskeleton interaction portrait

Human–robot physical interaction contains crucial information for optimizing user experience, enhancing robot performance, and objectively assessing user adaptation. This study introduces a new method to evaluate human–robot interaction and co-adaptation in lower limb exoskeletons by analyzing muscle activity and interaction torque as a two-dimensional random variable. We introduce the interaction portrait (IP), which visualizes this variable’s distribution in polar coordinates. We applied IP to compare a recently developed hybrid torque controller (HTC) based on kinematic state feedback and a novel adaptive model-based torque controller (AMTC) with online learning, proposed herein, against a time-based controller (TBC) during treadmill walking at varying speeds. Compared to TBC, both HTC and AMTC significantly lower users’ normalized oxygen uptake, suggesting enhanced user-exoskeleton coordination. IP analysis reveals that this improvement stems from two distinct co-adaptation strategies, unidentifiable by traditional muscle activity or interaction torque analyses alone. HTC encourages users to yield control to the exoskeleton, decreasing overall muscular effort but increasing interaction torque, as the exoskeleton compensates for user dynamics. Conversely, AMTC promotes user engagement through increased muscular effort and reduces interaction torques, aligning it more closely with rehabilitation and gait training applications. IP phase evolution provides insight into each user’s interaction strategy formation, showcasing IP analysis’s potential in comparing and designing novel controllers to optimize human–robot interaction in wearable robots.

An Approach to Near-Optimal Continuous-Thrust Solution for Plane Constellation Deployment

Paving the way in satellite constellation deployment, this article presents the Comprehensive Program (GCP) for planning constellation deployment missions by introducing a near-optimal solution for continuous-thrust maneuvers. The GCP establishes time constraints, enabling appropriate time scheduling and eliminating reconfiguration phases. It considers various cases of satellite departure and arrival sequences, collision avoidance constraints, and time limitations. A near-optimal solution for the continuous-thrust trajectory of each satellite is introduced, relying on the Fourier series approximation of the thrust pointing angle. The Fourier series with constant coefficients replaces the thrust angle in the satellite’s orbital motion equations in polar coordinates. The primary advantage of this near-optimal approach lies in its simplicity in accommodating space perturbations, posing no challenges in problem-solving. Additionally, the approach is beneficial due to its straightforward implementation and simple mathematical computations. This approach does not require pre-determining the revolution number of the transfer trajectory, unlike shape-based methods. The proposed method’s flexibility allows it to adapt and optimize the trajectory without prior assumptions about the number of revolutions required for the mission. Analyses demonstrate the proposed algorithm’s versatility and precision across various scenarios involving different orbital eccentricities, thrust forces, and thrust pointing angles. The constant coefficients of the Fourier series are determined by defining a set of objective functions and minimizing them using the Multi-Objective Particle Swarm Optimization algorithm (MOPSO). The first and second objective functions ensure that the transfer orbit reaches the desired deployment point, while the third and fourth ones guarantee the tangential conditions between the transfer orbit and the final orbit. An additional objective function minimizes the trajectory time. As a perturbation of interest, the effect of Earth’s oblateness is considered.

Circular phase shift migration based photoacoustic computational tomography

Abstract In the recent several decades, photoacoustic technique has proved to be capable of noninvasive biomedical imaging with scalable spatial resolution and high penetration depth. To achieve faithful image reconstruction, various method has been proposed, such as back projection, phase shift migration, model-based method, etc. These methods have simple implementations in homogeneous media, whereas in the cases of heterogeneous media (e.g., layered media), complicated modifications with respect to wave propagation at layer interface are always necessary. Here, we propose a frequency domain algorithm extension of phase shift migration in polar coordinates, which is obtained by making second order approximation to the optoacoustic point spread function in cylindrical coordinates and thereafter step by step wavenumber-frequency domain wave field extrapolation along the axial direction. Theoretical simulation and phantom experimental results demonstrate that the proposed circular phase shift migration method can well solve the layered heterogeneous reconstructive problem without modifying the wave propagation model. The acoustic diffraction limited acoustic resolution (up to 225 μm) and high imaging SNR (at least 16 dB) are obtained.

Preparation for CSST: Star-galaxy Classification using a Rotationally Invariant Supervised Machine Learning Method

Most existing star-galaxy classifiers depend on the reduced information from catalogs, necessitating careful data processing and feature extraction. In this study, we employ a supervised machine learning method (GoogLeNet) to automatically classify stars and galaxies in the COSMOS field. Unlike traditional machine learning methods, we introduce several preprocessing techniques, including noise reduction and the unwrapping of denoised images in polar coordinates, applied to our carefully selected samples of stars and galaxies. By dividing the selected samples into training and validation sets in an 8:2 ratio, we evaluate the performance of the GoogLeNet model in distinguishing between stars and galaxies. The results indicate that the GoogLeNet model is highly effective, achieving accuracies of 99.6% and 99.9% for stars and galaxies, respectively. Furthermore, by comparing the results with and without preprocessing, we find that preprocessing can significantly improve classification accuracy (by approximately 2.0% to 6.0%) when the images are rotated. In preparation for the future launch of the China Space Station Telescope (CSST), we also evaluate the performance of the GoogLeNet model on the CSST simulation data. These results demonstrate a high level of accuracy (approximately 99.8%), indicating that this model can be effectively utilized for future observations with the CSST.

Characterization of arteriosclerosis based on computer-aided measurements of intra-arterial thickness.

Our purpose is to develop a computer vision approach to quantify intra-arterial thickness on digital pathology images of kidney biopsies as a computational biomarker of arteriosclerosis. The severity of the arteriosclerosis was scored (0 to 3) in 753 arteries from 33 trichrome-stained whole slide images (WSIs) of kidney biopsies, and the outer contours of the media, intima, and lumen were manually delineated by a renal pathologist. We then developed a multi-class deep learning (DL) framework for segmenting the different intra-arterial compartments (training dataset: 648 arteries from 24 WSIs; testing dataset: 105 arteries from 9 WSIs). Subsequently, we employed radial sampling and made measurements of media and intima thickness as a function of spatially encoded polar coordinates throughout the artery. Pathomic features were extracted from the measurements to collectively describe the arterial wall characteristics. The technique was first validated through numerical analysis of simulated arteries, with systematic deformations applied to study their effect on arterial thickness measurements. We then compared these computationally derived measurements with the pathologists' grading of arteriosclerosis. Numerical validation shows that our measurement technique adeptly captured the decreasing smoothness in the intima and media thickness as the deformation increases in the simulated arteries. Intra-arterial DL segmentations of media, intima, and lumen achieved Dice scores of 0.84, 0.78, and 0.86, respectively. Several significant associations were identified between arteriosclerosis grade and pathomic features using our technique (e.g., intima-media ratio average [ , ]) through Kendall's tau analysis. We developed a computer vision approach to computationally characterize intra-arterial morphology on digital pathology images and demonstrate its feasibility as a potential computational biomarker of arteriosclerosis.

A Detailed Analysis of the Dynamic Behavior of a MEMS Vibrating Internal Ring Gyroscope.

This paper presents the development of an analytical model of an internal vibrating ring gyroscope in a Microelectromechanical System (MEMS). The internal ring structure consists of eight semicircular beams that are attached to the externally placed anchors. This research work analyzes the vibrating ring gyroscope's in-plane displacement behavior and the resulting elliptical vibrational modes. The elliptical vibrational modes appear as pairs with the same resonance frequency due to the symmetric structure of the design. The analysis commences by conceptualizing the ring as a geometric structure with a circular shape possessing specific dimensions such as thickness, height, and radius. We construct a linear model that characterizes the vibrational dynamics of the internal vibrating ring. The analysis develops a comprehensive mathematical formulation for the radial and tangential displacements in local polar coordinates by considering the inextensional displacement of the ring structure. By utilizing the derived motion equations, we highlight the underlying relationships driving the vibrational characteristics of the MEMS' vibrating ring gyroscope. These dynamic vibrational relationships are essential in enabling the vibrating ring gyroscope's future utilization in accurate navigation and motion sensing technologies.

On an isotropic porous solid cylinder: the analytical solution and sensitivity analysis of the pressure

Within this work, we perform a sensitivity analysis to determine the influence of the material input parameters on the pressure in an isotropic porous solid cylinder. We provide a step-by-step guide to obtain the analytical solution for a porous isotropic elastic cylinder in terms of the pressure, stresses, and elastic displacement. We obtain the solution by performing a Laplace transform on the governing equations, which are those of Biot’s poroelasticity in cylindrical polar coordinates. We enforce radial boundary conditions and obtain the solution in the Laplace transformed domain before reverting back to the time domain. The sensitivity analysis is then carried out, considering only the derived pressure solution. This analysis finds that the time t, Biot’s modulus M, and Poisson’s ratio v have the highest influence on the pressure whereas the initial value of pressure P0 plays a very little role.

DFT Modeling of Coordination Polymerization of Polar Olefin Monomers by Molecular Metal Complexes

Introducing polar functional groups into polyolefin chains through polar olefin monomer coordination (co)polymerization can directly and significantly improve the surface properties of polymer materials and expand their application range. Therefore, the related research has always received considerable attention from both academia and industry. Many experimental studies have been reported in this field, and molecular metal complexes have shown high catalytic activity and selectivity in polar olefin monomer polymerizations. Although considerable DFT calculations have also been conducted for better understanding of the (co)polymerization performance, the factors governing the activity, selectivity, and molecular weight of resulting polymers are still ambiguous. This review mainly focuses on the DFT studies of polar olefin monomer coordination (co)polymerization catalyzed by molecular metal complexes in recent years, discussing the chain initiation and propagation, the origin of polymerization activity and selectivity, and the specific role of additives in the (co)polymerization reactions.