Статті в журналах: "Spherical coordinate"

1

Lei, Kin, Dongxu Qi, and Xiaolin Tian. "A New Coordinate System for Constructing Spherical Grid Systems." Applied Sciences 10, no. 2 (January 16, 2020): 655. http://dx.doi.org/10.3390/app10020655.

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In astronomy, physics, climate modeling, geoscience, planetary science, and many other disciplines, the mass of data often comes from spherical sampling. Therefore, establishing an efficient and distortion-free representation of spherical data is essential. This paper introduces a novel spherical (global) coordinate system that is free of singularity. Contrary to classical coordinates, such as Cartesian or spherical polar systems, the proposed coordinate system is naturally defined on the spherical surface. The basic idea of this coordinate system originated from the classical planar barycentric coordinates that describe the positions of points on a plane concerning the vertices of a given planar triangle; analogously, spherical area coordinates (SACs) describe the positions of points on a sphere concerning the vertices of a given spherical triangle. In particular, the global coordinate system is obtained by decomposing the globe into several identical triangular regions, constructing local coordinates for each region, and then combining them. Once the SACs have been established, the coordinate isolines form a new class of global grid systems. This kind of grid system has some useful properties: the grid cells exhaustively cover the globe without overlapping and have the same shape, and the grid system has a congruent hierarchical structure and simple relationship with traditional coordinates. These beneficial characteristics are suitable for organizing, representing, and analyzing spatial data.

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2

POVSTENKO, Yuriy. "Solutions to Diffusion-Wave Equation in a Body with a Spherical Cavity under Dirichlet Boundary Condition." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 1, no. 1 (June 27, 2011): 3–16. http://dx.doi.org/10.11121/ijocta.01.2011.0035.

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Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium with a spherical cavity. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate Ï•, the Legendre transform with respect to the spatial coordinate μ, and the Weber transform of the order n+1/2 with respect to the radial coordinate r. In the central symmetric case with one spatial coordinate r the obtained resultscoincide with those studied earlier.

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3

Swaszek, P. "Uniform Spherical Coordinate Quantization of Spherically Symmetric Sources." IEEE Transactions on Communications 33, no. 6 (June 1985): 518–21. http://dx.doi.org/10.1109/tcom.1985.1096333.

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4

Kim, Jinwook, Sung-Hee Lee, and Frank C. Park. "Kinematic and dynamic modeling of spherical joints using exponential coordinates." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 10 (November 24, 2013): 1777–85. http://dx.doi.org/10.1177/0954406213511365.

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Traditional Euler angle-based methods for the kinematic and dynamic modeling of spherical joints involve highly complicated formulas that are numerically sensitive, with complex bookkeeping near local coordinate singularities. In this regard, exponential coordinates are known to possess several advantages over Euler angle representations. This paper presents several new exponential coordinate-based formulas and computational procedures that are particularly useful in the modeling of mechanisms containing spherical joints. Computationally robust procedures are derived for evaluating the forward and inverse formulas for the angular velocity and angular acceleration in terms of exponential coordinates. We show that these formulas simplify the parametrization of joint range limits for spherical joints, and lead to more compact equations in the forward and inverse dynamic analysis of mechanisms containing spherical joints.

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5

Dil, Emre, and Talha Zafer. "Transformation Groups for a Schwarzschild-Type Geometry in f(R) Gravity." Journal of Gravity 2016 (November 2, 2016): 1–8. http://dx.doi.org/10.1155/2016/7636493.

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We know that the Lorentz transformations are special relativistic coordinate transformations between inertial frames. What happens if we would like to find the coordinate transformations between noninertial reference frames? Noninertial frames are known to be accelerated frames with respect to an inertial frame. Therefore these should be considered in the framework of general relativity or its modified versions. We assume that the inertial frames are flat space-times and noninertial frames are curved space-times; then we investigate the deformation and coordinate transformation groups between a flat space-time and a curved space-time which is curved by a Schwarzschild-type black hole, in the framework of f(R) gravity. We firstly study the deformation transformation groups by relating the metrics of the flat and curved space-times in spherical coordinates; after the deformation transformations we concentrate on the coordinate transformations. Later on, we investigate the same deformation and coordinate transformations in Cartesian coordinates. Finally we obtain two different sets of transformation groups for the spherical and Cartesian coordinates.

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6

Vodolazov, Rodion, Valery Chapursky, and Andrey Filatov. "To the question of coordinate system choice within composition of the spatial generalized MIMO radar ambiguity function." ITM Web of Conferences 30 (2019): 15014. http://dx.doi.org/10.1051/itmconf/20193015014.

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The issue of the choice of spatial coordinate system which is suitable to physical sense of the radar concept is explored on the basis of the derivation of spatial generalized ambiguity function for MIMO antenna system and of its cross-section analysis. The comparison results of generalized ambiguity function cross-sections for two dimensional target coordinates in Cartesian, spherical and modified spherical coordinate systems are given for orthogonal LFM waveforms radiated by the elements of linear MIMO antenna system transmit. This proves the desirable choice between coordinate systems with regard of physical sense of the current task.

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7

Vetter, Philipp, Susan J. Goodbody, and Daniel M. Wolpert. "Evidence for an Eye-Centered Spherical Representation of the Visuomotor Map." Journal of Neurophysiology 81, no. 2 (February 1, 1999): 935–39. http://dx.doi.org/10.1152/jn.1999.81.2.935.

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Evidence for an eye-centered spherical representation of the visuomotor map. During visually guided movement, visual coordinates of target location must be transformed into coordinates appropriate for movement. To investigate the representation of this visuomotor coordinate transformation, we examined changes in pointing behavior induced by a local visuomotor remapping. The visual feedback of finger position was limited to one location within the workspace, at which a discrepancy was introduced between the actual and visually perceived finger position. This remapping induced a change in pointing that extended over the entire workspace and was best captured by a spherical coordinate system centered near the eyes.

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8

Iofa, Mikhail Z. "Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells." Advances in High Energy Physics 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/5632734.

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Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates. We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications to the Israel junction conditions are discussed.

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9

Medvedev, P. A., and M. V. Novgorodskaya. "Development of mathematical model Gauss – Kruger coordinate system for calculating planimetric rectangular coordinates using geodesic coordinates." Geodesy and Cartography 926, no. 8 (September 20, 2017): 10–19. http://dx.doi.org/10.22389/0016-7126-2017-926-8-10-19.

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Algorithms with improved convergence for the calculation of rectangular coordinates in the Gauss – Kruger coordinate system according to the parameters of any ellipsoid were designed. The approach of definition the spherical components in the classic series defined variables x, y, represented by the difference between the degrees of longitude l, followed by the replacement of their sums by formulas of spherical trigonometry. For definition of the amounts of spherical components of the relevant decompositions patterns of transverse-cylindrical sphere plane projection in the condition of the initial data equality on the ellipsoid and sphere radius N were used. Analysis of othertransformation methods of classical expansions in series, used in derivation of both logarithmical and non-logarithmical working formulas is carried outfor comparison with developed algorithms. The technique of algorithms development with usage of hyperbolic tangent function, applied by L. Kruger, Yu. Karelin, A. Schödlbauer is considered and their analysis is carried out. Advantages of Krasovskii – Isotov formulas for six-degree strips are pointed out. The usage of the spherical function sin τ in the expansion made it possible not only to obtain a rapidly convergent series, but also to represent the spherical part of the solution of the problem with the help of trigonometric identities in different types. It is proved that derived for the calculation algorithms with the proposed estimates of their accuracy, are optimal in removing points from the central meridian to l ≤ 6°. For the difference of longitudes l > 6°, the expansions of the unknown quantities into Fourier series should be applied. An example of the calculation of coordinates in the system SK-2011 is given. Theoretical studies have been carried out and shortened formulas with a reliability estimate for the determination of coordinates in the area l ≤ 3° have been proposed.

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10

Zaki, Ahmad, Syafruddin Side, and N. Nurhaeda. "Solusi Persamaan Laplace pada Koordinat Bola." Journal of Mathematics, Computations, and Statistics 2, no. 1 (May 12, 2020): 82. http://dx.doi.org/10.35580/jmathcos.v2i1.12462.

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Penelitian ini mengkaji mengenai persamaan Laplace pada koordinat bola dan menerapkan metode pemisahan variabel dalam menentukan solusi persamaan Laplace Persamaan Laplace merupakan salah satu jenis persamaan diferensial parsial yang banyak digunakan untuk memodelkan permasalahan dalam bidang sains. Bentuk umum persamaan Laplace pada dimensi tiga dimana adalah fungsi skalar dengan menggunakan metode pemisahan variable diperoleh persamaan Laplace dimensi tiga pada koordinat bola. Hasil penelitian ini mendapatkan penyelesaian persamaan Laplace pada koordinat bola dalam bentuk variabel terpisah dengan tidak menggunakan nilai batas. Hubungan koordinat kartesian dan koordinat bola pada persamaan Laplace dapat ditentukan dalam persamaan Laplace dan memperoleh solusi dengan menggunakan koordinat bola.Kata Kunci: Koordinat Bola, Pemisahan Variabel, dan Persamaan Laplace. This study examines Laplace equations on spherical coordinates and applies variable separation methods in determining Laplace equation solutions Laplace equations are one type of partial differential equation that is widely used to model problems in the field of science. The general form of the Laplace equation in the third dimension in which u is a scalar function using the separation method of the variable is obtained by the third dimension Laplace equation on spherical coordinates. The result of this research get solution of Laplace equation on spherical coordinate in the form of separate variable by not using boundary value. The relationship of cartesian coordinates and spherical coordinates to the Laplace equation can be determined in the Laplace equation and obtain solutions using spherical coordinates.Keywords: Spherical Coordinat Variabel Separation, and Laplace Equation.

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11

Chochia, G. A., P. K. Chawdhry, and C. R. Burrows. "A computer algebra-based coordinate reduction technique for multi-body systems." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 214, no. 2 (February 1, 2000): 313–21. http://dx.doi.org/10.1243/0954406001522985.

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This paper presents a heuristic algorithm to solve the linear coordinate reduction problem. If Cartesian coordinates are chosen in the initial formulation the algorithm eliminates two-thirds of dependent coordinates in the planar case and one-half in the spatial case for mechanisms composed of spherical, revolute and universal joints. For an open-loop system composed of spherical joints it eliminates all dependent coordinates. A computer algebra-based implementation in the Maple language is presented. The proposed technique is demonstrated by application to the dynamic analysis of a Peaucellier mechanism.

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12

Poletti, M. A. "Spherical coordinate descriptions of cylindrical and spherical Bessel beams." Journal of the Acoustical Society of America 141, no. 3 (March 2017): 2069–78. http://dx.doi.org/10.1121/1.4978787.

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13

BURNETT, DAVID S. "RADIATION BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION FOR ELLIPSOIDAL, PROLATE SPHEROIDAL, OBLATE SPHEROIDAL AND SPHERICAL DOMAIN BOUNDARIES." Journal of Computational Acoustics 20, no. 04 (November 29, 2012): 1230001. http://dx.doi.org/10.1142/s0218396x12300010.

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One of the most popular radiation boundary conditions for the Helmholtz equation in exterior 3-D regions has been the sequence of operators developed by Bayliss et al.1 for computational domains with spherical exterior boundaries. The present paper extends those spherical operators to triaxial ellipsoidal boundaries by utilizing two mathematical constructs originally developed for ellipsoidal acoustic infinite elements.2 The two constructs are: (i) a radial/angular coordinate system for ellipsoidal geometry, and (ii) a convergent ellipsoidal radial expansion for exterior fields, analogous to the classical spherical multipole expansion. The ellipsoidal radial and angular coordinates are smooth generalizations of the traditional radial and angular coordinates used in spherical, prolate spheroidal and oblate spheroidal systems. As a result, all four coordinate systems and their corresponding radiation boundary conditions are included within this single ellipsoidal system, varying smoothly from one to the other. The geometric flexibility of this system enables the exterior boundary of the computational domain to closely circumscribe objects with a wide range of aspect ratios, thereby reducing the size and cost of 3-D computational models.

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14

GOGBERASHVILI, MERAB. "GRAVITATIONAL FIELD OF SPHERICAL BRANES." Modern Physics Letters A 23, no. 35 (November 20, 2008): 2979–86. http://dx.doi.org/10.1142/s0217732308028405.

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The warped solution of Einstein's equations corresponding to the spherical brane in five-dimensional AdS is considered. This metric represents interiors of black holes on both sides of the brane and can provide gravitational trapping of physical fields on the shell. It is found that the analytic form of the coordinate transformations from the Schwarzschild to co-moving frame that exists only in five dimensions. It is shown that in the static coordinates active gravitational mass of the spherical brane, in agreement with Tolman's formula, is negative, i.e. such objects are gravitationally repulsive.

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15

Dong, Guo Jun, Cheng Shun Han, and Shen Dong. "Solution for Best Fitting Spherical Curvature Radius and Asphericity of Off-Axis Aspherics of Optical Aspheric Surface Component." Key Engineering Materials 364-366 (December 2007): 499–503. http://dx.doi.org/10.4028/www.scientific.net/kem.364-366.499.

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This study aimed to establish the coordinate transformation between the off-axis aspherics coordinate system σ and the axial symmetry aspherics coordinate system σ by transforming coordinates and present the computation models of asphericity in rectangular coordinate system and cylindrical coordinate system respectively. The asphericity expressions in both coordinate systems were applicable to the comparative sphere calculation of Off-axis aspherics with different figures. We selected an Initiation sphere in view of technology, along with equations in a right coordinate system for certain caliber and structure. Then, by numerical computation, we selected the best fitting sphere and simplifed the complex models by choosing a right coordinate system. At last, the solution for asphericity and the best fitting sphere curvature radius of off-axis aspherics were introduced by examples.

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16

Hunter, Ian W. "Spherical-coordinate scanning confocal laser microscope." Optical Engineering 34, no. 7 (July 1, 1995): 2103. http://dx.doi.org/10.1117/12.206583.

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17

Hadjinicolaou, Maria, and Eleftherios Protopapas. "Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System." Mathematical Problems in Engineering 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/9049131.

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When studying axisymmetric particle fluid flows, a scalar function,ψ, is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely,E4ψ=0, whereE2is the Stokes irrotational operator andE4=E2∘E2is the Stokes bistream operator. As it is already known,E2ψ=0in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation acceptsR-separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operatorE4does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, whileE4ψ=0semiseparates variables in the spheroidal coordinate systems and itR-semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equationE′2ψ=0alsoR-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equationE′4ψ=0 R-semiseparates variables. Since the generalized eigenfunctions ofE′2cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions ofE′2in the modified inverted oblate spheroidal coordinate system.

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18

Bschorr, Oskar, and Hans-Joachim Raida. "Spherical One-Way Wave Equation." Acoustics 3, no. 2 (April 28, 2021): 309–15. http://dx.doi.org/10.3390/acoustics3020021.

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The coordinate-free one-way wave equation is transferred in spherical coordinates. Therefore it is necessary to achieve consistency between gradient, divergence and Laplace operators and to establish, beside the conventional radial Nabla operator ∂Φ/∂r, a new variant ∂rΦ/r∂r. The two Nabla operator variants differ in the near field term Φ/r whereas in the far field r≫0 there is asymptotic approximation. Surprisingly, the more complicated gradient ∂rΦ/r∂r results in unexpected simplifications for – and only for – spherical waves with the 1/r amplitude decrease. Thus the calculation always remains elementary without the wattless imaginary near fields, and the spherical Bessel functions are obsolete.

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19

Wang, Shu, and Yongxin Wang. "The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates." Mathematics 8, no. 7 (July 21, 2020): 1195. http://dx.doi.org/10.3390/math8071195.

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This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

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20

Kim, Kun-Woo, Jae-Wook Lee, Jin-Seok Jang, Joo-Young Oh, Ji-Heon Kang, Hyung-Ryul Kim, and Wan-Suk Yoo. "Construction of unwinding equation of motion for thin cable in spherical coordinate system." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 7 (May 11, 2017): 1208–20. http://dx.doi.org/10.1177/0954406217705406.

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The transient-state unwinding equation of motion for a thin cable can be derived by using Hamilton’s principle for an open system, which can consider the mass change produced by the unwinding velocity in a control volume. In general, most engineering problems can be analyzed in Cartesian, cylindrical, and spherical coordinate systems. In the field of unwinding dynamics, until now, only Cartesian and cylindrical coordinate systems have been used. A spherical coordinate system has not been used because of the complexity of derivatives. Therefore, in this study, the unwinding motion of a thin cable was analyzed using a spherical coordinate system in both water and air, and the results were compared with the results in Cartesian and cylindrical coordinate systems. The unwinding motions in the spherical, Cartesian, and cylindrical coordinate systems were nearly same in both water and air. The error related to the total length was within 0.5% in water, and the error related to the maximum balloon radius was also within 0.5 % in air. Therefore, it can be concluded that it is possible to solve the transient-state unwinding equation of motion in a spherical coordinate system.

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21

Li, Qiang, Cun Yun Pan, and Hai Jun Xu. "Contact Characteristics of Spherical Gear and Ring-Rack." Advanced Materials Research 308-310 (August 2011): 2019–24. http://dx.doi.org/10.4028/www.scientific.net/amr.308-310.2019.

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The contact characteristics of spherical gear and ring-rack are researched in this paper. First, the tooth profile surface equation of spherical gear is established on the basis of the transmission theory of spherical gears. After that, the profile surface equation of ring-rack is established. Then, kinematics of the transmission of spherical gear and ring-rack is researched, the appreciate coordinates are established, and the profile surfaces of spherical gear and ring-rack are described in the same coordinate. Then, the contact analysis is conducted between spherical gear and ring-rack, and the contact point and ellipse parameters are acquired, as well as the rules of their evolvements. The conclusions above are benefit to further research on the transmission of spherical gear and ring-rack.

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22

Kartashov, Eduard Mikhailovich, and Sergey Vladimirovich Polyakov. "Generalized model representations of the theory thermal shock for local non-equilibrium processes heat transfer." Keldysh Institute Preprints, no. 100 (2022): 1–28. http://dx.doi.org/10.20948/prepr-2022-100.

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An open problem of the theory of thermal shock in terms of a generalized model of dynamic thermoelasticity under conditions of a locally nonequilibrium heat transfer process was considered. The model simultaneously includes three coordinate systems: Cartesian coordinates - a massive body bounded by a flat surface; spherical coordinates - a massive body with an internal spherical cavity; cylindrical coordinates - a massive body with an internal cylindrical cavity. Three types of intensive heating and cooling are considered: temperature, thermal, medium. The task is set: to obtain an analytical solution, to carry out numerical experiments and to give their physical analysis. As a result, generalized model representations of thermal shock in terms of dynamic thermoelasticity have been developed for locally nonequilibrium heat transfer processes simultaneously in three coordinate systems: Cartesian, spherical, and cylindrical. The presence of curvature of the boundary surface of the thermal shock area substantiates the initial statement of the dynamic problem in displacements using the proposed corresponding "compatibility" equation. The latter made it possible to propose a generalized dynamic model of the thermal reaction of massive bodies with internal cavities simultaneously in Cartesian, spherical and cylindrical coordinate systems under conditions of intense thermal heating and cooling, thermal heating and cooling, heating and cooling by the medium. The model is considered in displacements on the basis of local non-equilibrium heat transfer. An analytical solution for stresses is obtained, a numerical experiment is carried out; the wave nature of the propagation of a thermoelastic wave is described. A comparison with the classical solution is made without taking into account local non-equilibrium. On the basis of the operational solution of the problem, design engineering relations important in practical terms for the upper estimate of the maximum thermal stresses are proposed.

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23

Tolchelnikova, S. A. "Astrometry and Geodesy as a one science. History and assignment of stellar catalogues." Geodesy and Cartography 979, no. 1 (February 20, 2022): 40–53. http://dx.doi.org/10.22389/0016-7126-2022-979-1-40-53.

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According to the resolution of the International Astronomical Union, since 1999 the main reference for spherical coordinates has become the international celestial reference frame ICRF, compiled from observations of quasars and point-like radio galaxies. This decision abolishes the observations necessary for compiling the traditional spherical coordinate system represented by the fundamental catalogs of the FK series. It is useful to recall the principles of astronomers of the 18th–20th centuries who had organized the determination of the absolute coordinates of stars at the state observatories, required for compiling the next fundamental catalog. This will allow natural scientists who study movements by measured coordinates of celestial objects, firstly astronomers and geodesists, to compare the opportunities provided by optical fundamental catalogs and the reference system of radio coordinates for solving the problems of their branch of science. Let us turn to historical experience, which shows that the establishment of main optical coordinate systems was not a volitional decision but was affirmed in the process of searching for the most effective method for solving both fundamental problems and satisfying current needs of each historical era.

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Xing, Shi Tong, Hui Yang, Gang Zheng, and Hai Ma Yang. "The Scattering of Carbon Nanotubes on the Gaussian Beam." Applied Mechanics and Materials 229-231 (November 2012): 171–74. http://dx.doi.org/10.4028/www.scientific.net/amm.229-231.171.

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The first-order approximation description of Gaussian beam in the two parallel Cartesian coordinates was introduced. On the basis of Generalized Mie theory, adopting the relation between the spherical vector wave functions belonging to a rotating Cartesian coordinate system, the electromagnetic fields of Gaussian beam with spherical vector wave functions was deduced at any right coordinates system. Then taking advantage of the cylindrical vector wave functions given by Stratton, the relationship of the spherical vector wave functions expressed in cylindrical vector wave functions was deduced. Finally the electromagnetic fields of infinitely long cylinder was expanded by the cylindrical vector, and the approximate expression of the cylinder to the far zone scattered field was solved.

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25

Gielen, Steffen. "Group Field Theory and Its Cosmology in a Matter Reference Frame." Universe 4, no. 10 (October 2, 2018): 103. http://dx.doi.org/10.3390/universe4100103.

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While the equations of general relativity take the same form in any coordinate system, choosing a suitable set of coordinates is essential in any practical application. This poses a challenge in background-independent quantum gravity, where coordinates are not a priori available and need to be reconstructed from physical degrees of freedom. We review the general idea of coupling free scalar fields to gravity and using these scalars as a “matter reference frame”. The resulting coordinate system is harmonic, i.e., it satisfies the harmonic (de Donder) gauge. We then show how to introduce such matter reference frames in the group field theory approach to quantum gravity, where spacetime is emergent from a “condensate” of fundamental quantum degrees of freedom of geometry, and how to use matter coordinates to extract physics. We review recent results in homogeneous and inhomogeneous cosmology, and give a new application to the case of spherical symmetry. We find tentative evidence that spherically-symmetric group field theory condensates defined in this setting can reproduce the near-horizon geometry of a Schwarzschild black hole.

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26

Bektaş, Sebahattin. "Rigorous spherical bearing with Soldner coordinates and azimuth angles on sphere." Earth Sciences Research Journal 26, no. 3 (November 29, 2022): 205–10. http://dx.doi.org/10.15446/esrj.v26n3.100754.

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Meridian systems, called Soldner coordinates (parallel coordinate) systems, have found wide application in geodesy. In particular, the meridian system constitutes a suitable base for the Gauss-Kruger projection of the ellipsoid and the sphere. Soldner coordinates can be used in Cassini-Soldner projection without any processing. As it is known, the directions of the edges are shown with azimuth angles in the geographic coordinate system and the bearing angles in the Soldner coordinate system. Bearing or azimuth angles are frequently used in geodetic calculations. These angles give the direction of sides in the clockwise direction from a certain initial direction. Both angle values range from 0 to 360 degrees and are usually calculated from the arctan function. But the arctan function returns an angle value between -90 and +90 degrees. Therefore, it is necessary to analyze the quarter for the angle found. For practical computations, the quadrants of the arctangents are determined by the signs of the numerator and denominator in the tangent formulas. Determining the quarter of the angles is done with if…, then…, end..., blocks on the computer. It should be noted that each comparison requires a separate processing time. This study will be given how to calculate both bearing and azimuth angles with direct formulas without any need to examine them. In addition, a solution proposal will be given against the division by zero errors in the bearing and azimuth angles calculations.

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27

Wang, Shi Huan, Zhi Gang Cai, and Li Fang Lin. "Research on Surface Machining System for Light Metal Thin-Walled Parts Based on Spherical Coordinate." Advanced Materials Research 97-101 (March 2010): 1841–44. http://dx.doi.org/10.4028/www.scientific.net/amr.97-101.1841.

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Surface machining system based on spherical coordinate is present for the large thin-walled director sphere part turning, to improve the efficiency and quality of part machining. In this system, a thickness surface B-spline model is set up based on spherical coordinates, the space spiral path generation method is planned, row spacing angle is determined by scallop height and blunt radius of cutting tool, and back cutting depth optimization treatment and space spiral interpolation algorithm are present.

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28

Zhu, Yuanchao, Dazhao Zhang, Yanlin Lai, and Huabiao Yan. "Shape adjustment of "FAST" active reflector." Highlights in Science, Engineering and Technology 1 (June 14, 2022): 391–400. http://dx.doi.org/10.54097/hset.v1i.493.

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Abstract. In this paper, the relevant working principle of "FAST" Chinese Eye is studied, and a mathematical model is established to solve the equation of the ideal paraboloid. The ideal paraboloid model is obtained by rotating the paraboloid around the axis in the two-dimensional plane. On this basis, the specific solutions of each question are discussed, and the parabolic equation, the receiving ratio of the feed cabin to the reflected signal, the numbering information and coordinates of the main cable node and other parameters are obtained. This paper for solving directly above the benchmark of spherical observation of celestial bodies when ideal parabolic equation, according to the geometrical optics to knowledge should be clear all the signals of the incoming signal after the ideal parabolic will converge to the focal point of basic rules, then through converting ideal parabolic model of ideal parabolic equation in a two-dimensional plane, An optimization model was established to minimize the absolute value of the difference between the arc length and the arc length of the parabola in the diameter of 300 meters. The known conditions were substituted into Matlab to solve the equation of the ideal parabola by rotating the parabola around the axis: . In order to determine the ideal paraboloid of the celestial body, a new spatial cartesian coordinate system is first established with the line direction between the celestial body and the spherical center as the axis, so that the observed object is located directly above the new coordinate system. The same model in question 1 is established to obtain the vertex coordinates of the ideal paraboloid at this time. Then the vertex coordinates are converted to the coordinates in the original space cartesian coordinate system by rotation transformation between space cartesian coordinate systems. The solution of its vertex coordinates (-49.5287, -37.0203, -294.1763).

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29

Harry, S. T., and M. A. Adekanmbi. "CONFINEMENT ENERGY OF QUANTUM DOTS AND THE BRUS EQUATION." International Journal of Research -GRANTHAALAYAH 8, no. 11 (December 16, 2020): 318–23. http://dx.doi.org/10.29121/granthaalayah.v8.i11.2020.2451.

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A review of the ground state confinement energy term in the Brus equation for the bandgap energy of a spherically shaped semiconductor quantum dot was made within the framework of effective mass approximation. The Schrodinger wave equation for a spherical nanoparticle in an infinite spherical potential well was solved in spherical polar coordinate system. Physical reasons in contrast to mathematical expediency were considered and solution obtained. The result reveals that the shift in the confinement energy is less than that predicted by the Brus equation as was adopted in most literatures.

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30

HILL, J. M., and Y. M. STOKES. "A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES." ANZIAM Journal 59, no. 3 (January 2018): 335–48. http://dx.doi.org/10.1017/s144618111700058x.

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There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

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31

Wu, Qinghua, Jiacheng Liu, Can Gao, Biao Wang, Gaojian Shen, and Zhiang Li. "Improved RANSAC Point Cloud Spherical Target Detection and Parameter Estimation Method Based on Principal Curvature Constraint." Sensors 22, no. 15 (August 5, 2022): 5850. http://dx.doi.org/10.3390/s22155850.

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Spherical targets are widely used in coordinate unification of large-scale combined measurements. Through its central coordinates, scanned point cloud data from different locations can be converted into a unified coordinate reference system. However, point cloud sphere detection has the disadvantages of errors and slow detection time. For this reason, a novel method of spherical object detection and parameter estimation based on an improved random sample consensus (RANSAC) algorithm is proposed. The method is based on the RANSAC algorithm. Firstly, the principal curvature of point cloud data is calculated. Combined with the k-d nearest neighbor search algorithm, the principal curvature constraint of random sampling points is implemented to improve the quality of sample points selected by RANSAC and increase the detection speed. Secondly, the RANSAC method is combined with the total least squares method. The total least squares method is used to estimate the inner point set of spherical objects obtained by the RANSAC algorithm. The experimental results demonstrate that the method outperforms the conventional RANSAC algorithm in terms of accuracy and detection speed in estimating sphere parameters.

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32

Jin, Gu, Su Xiao, Lai Hanrong, Zhang Bin, and Zhang Yawei. "Bush spherical center detection algorithm based on depth camera 3D point cloud." Journal of Physics: Conference Series 2417, no. 1 (December 1, 2022): 012034. http://dx.doi.org/10.1088/1742-6596/2417/1/012034.

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Automated pruning is an inevitable trend in the improvement of modern gardens. In order to provide necessary information for automatic garden robots and satisfy the requirement of target detection and positioning during pruning, this paper proposed a bush spherical center detection algorithm based on a 3D depth camera point cloud. Firstly, the depth camera collected the bush image, and the results were aligned to the depth image to obtain the 3D point cloud of bush. Then the ROI was extracted by preprocessing, and the 3D point clouds of bush was obtained after filtering and coordinate transformation. Finally, the spherical center coordinates of the bush were extracted by the minimum bounding box method. Four groups of tests on the bush spherical coordinates detection were carried out outdoors. The maximum location error and the minimum location error of the spherical bush center were 10.23mm and 8.65 mm, respectively, and the average location error was 9.51mm. The bush spherical center detection algorithm based on depth camera 3D point clouds proposed in this paper provides a technical reference for the information acquisition of automatic pruning robot.

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33

Tenzer, Robert, and Vladislav Gladkikh. "Application of Möbius coordinate transformation in evaluating Newton's integral." Contributions to Geophysics and Geodesy 41, no. 2 (January 1, 2011): 95–115. http://dx.doi.org/10.2478/v10126-011-0004-1.

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Application of Möbius coordinate transformation in evaluating Newton's integralWe propose a numerical scheme which efficiently combines various existing methods of solving the Newton's volume integral. It utilises the analytical solution of Newton's integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky's truncation coefficients to a spectral representation of Newton's integral. The weak singularity of Newton's integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the Möbius transformation with an enhanced integration grid resolution at vicinity of the observation point.

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34

Singh, S., and A. Saha. "Shannon Information Entropy Sum of a Free Particle in Three Dimensions Using Cubical and Spherical Symmetry." Journal of Scientific Research 15, no. 1 (January 1, 2023): 71–84. http://dx.doi.org/10.3329/jsr.v15i1.60067.

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In this paper, the plane wave solutions of a free particle in three dimensions for Cubical and Spherical Symmetry have been considered. The coordinate space wave functions for the Cubical and Spherical Symmetry are obtained by solving the Schrdinger differential equation. The momentum space wave function is obtained by using the operator form of an observable in the case of Cubical Symmetry. For Spherical Symmetry, the same is obtained by taking the Fourier transform of the respective coordinate space wave function. The wave functions have been used to constitute probability densities in coordinate and momentum space for both the symmetries. Further, the Shannon information entropy has been computed both in coordinate and momentum space respectively for (L is the length of the side of the cubical box) values for Cubical Symmetry and for values in Spherical Symmetry keeping (k is the wave vector and p is the momentum of the free particle) constant. The values obtained for the Shannon information entropies are found to satisfy the Bialynicki-Birula and Myceilski (BBM) inequality at larger values () in case of Cubical Symmetry and for values of and in Spherical Symmetry.

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35

Norris, A. N., and A. L. Shuvalov. "Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2138 (October 12, 2011): 467–84. http://dx.doi.org/10.1098/rspa.2011.0463.

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A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having c ijkl = c ijkl ( r ) in a spherical coordinate system { r , θ , ϕ }. The time-harmonic displacement field u ( r , θ , ϕ ) is expanded in a separation of variables form with dependence on θ , ϕ described by vector spherical harmonics with r -dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as u ( r , θ ), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.

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36

BRIGGS, ANDREW, HORACIO E. CAMBLONG, and CARLOS R. ORDÓÑEZ. "EQUIVALENCE OF THE PATH INTEGRAL FOR FERMIONS IN CARTESIAN AND SPHERICAL COORDINATES." International Journal of Modern Physics A 28, no. 14 (May 30, 2013): 1350047. http://dx.doi.org/10.1142/s0217751x13500474.

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The path integral calculation for the free energy of a spin-1/2 Dirac-fermion gas is performed in spherical polar coordinates for a flat space–time geometry. Its equivalence with the Cartesian-coordinate representation is explicitly established. This evaluation involves a relevant limiting case of the fermionic path integral in a Schwarzschild background, whose near-horizon limit has been shown to be related to black hole thermodynamics.

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37

Costantini, Mauro. "On the coordinate ring of spherical conjugacy classes." Mathematische Zeitschrift 264, no. 2 (January 8, 2009): 327–59. http://dx.doi.org/10.1007/s00209-008-0468-5.

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38

Ermakov, Andrei, and Yury Stepanyants. "Description of Nonlinear Vortical Flows of Incompressible Fluid in Terms of a Quasi-Potential." Physics 3, no. 4 (September 22, 2021): 799–813. http://dx.doi.org/10.3390/physics3040050.

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As it was shown earlier, a wide class of nonlinear 3-dimensional (3D) fluid flows of incompressible viscous fluid can be described by only one scalar function dubbed the quasi-potential. This class of fluid flows is characterized by a three-component velocity field having a two-component vorticity field. Both these fields may, in general, depend on all three spatial variables and time. In this paper, the governing equations for the quasi-potential are derived and simple illustrative examples of 3D flows in the Cartesian coordinates are presented. The generalisation of the developed approach to the fluid flows in the cylindrical and spherical coordinate frames represents a nontrivial problem that has not been solved yet. In this paper, this gap is filled and the concept of a quasi-potential to the cylindrical and spherical coordinate frames is further developed. A few illustrative examples are presented which can be of interest for practical applications.

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39

Li, Yanlin, Nadia Alluhaibi, and Rashad A. Abdel-Baky. "One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes." Symmetry 14, no. 9 (September 15, 2022): 1930. http://dx.doi.org/10.3390/sym14091930.

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E. Study map is one of the most basic and powerful mathematical tools to study lines in line geometry, it has symmetry property. In this paper, based on the E. Study map, clear expressions were developed for the differential properties of one-parameter Lorentzian dual spherical movements that are coordinate systems independent. This eliminates the requirement of demanding coordinates transformations necessary in the determination of the canonical systems. With the proposed technique, new proofs for Euler–Savary, and Disteli’s formulae were derived.

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40

Minnaert, Ben, Giuseppina Monti, and Mauro Mongiardo. "Unified Representation of 3D Multivectors with Pauli Algebra in Rectangular, Cylindrical and Spherical Coordinate Systems." Symmetry 14, no. 8 (August 13, 2022): 1684. http://dx.doi.org/10.3390/sym14081684.

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In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology.

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41

Kartiko, Nugroho, Azli Yahya, Safura Hashim Nor Liyana, M. Daud Razak, Nor Hisham Haji Khamis, Kamal Khalil, Muhammad Arif Abdul Rahim, and Ameruddin Baharom. "Investigation of Workpiece Positioning Methods for Machining Oil-Pocket on Hip-Implant Spherical Surface." Key Engineering Materials 594-595 (December 2013): 535–39. http://dx.doi.org/10.4028/www.scientific.net/kem.594-595.535.

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Oil pocket has been reported that it may improve tribology characteristic and thus prolong the lifespan of the joint. In order to implement it on spherical surface, appropriate positioning system is required. This paper reports the investigation of three axes workpiece positioning system in order to machine oil pocket (micro-pits) on hip implant. A conventional linear x-y-z axes configuration (Cartesian coordinate) and two configuration of spherical coordinate (swing-swing and swing-rotate configuration) are applied in simulation. All machined workpiece are investigated in pits distribution, shape, and machined angle. The inspection concludes that spherical method with swing-rotate configuration is the most suitable method for machining oil pocket on spherical surface.

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42

Zhang, Peng, Cunfeng Zhang, and Wen-wei Su. "Derivative of Spatial Variables in Orthogonal Curvilinear System with Respect to the Ones in Cartesian Coordinate System." Journal of Physics: Conference Series 2187, no. 1 (February 1, 2022): 012039. http://dx.doi.org/10.1088/1742-6596/2187/1/012039.

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Abstract Based on the transformation of the spatial variables between the orthogonal curvilinear coordinate system and the Cartesian coordinate system, the derivation of spatial variables in orthogonal curvilinear system to the ones in Cartesian coordinate system is derived. The derivatives of the spatial variables in cylindrical, spherical, and elliptic cylindrical coordinate system are derived respectively.

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43

Zadorozhnaya, Irina, Igor Zaharov, and Andriy Tevyashev. "The measurement uncertainty of air object spatial coordinates by rho-theta fixing." Ukrainian Metrological Journal, no. 1 (March 31, 2022): 51–56. http://dx.doi.org/10.24027/2306-7039.1.2022.258821.

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The features of measurement uncertainty evaluation of the coordinates of an air object by the rho-theta fixing are discussed. Measurement models are presented that link its coordinates in the local rectangular coordinate system with the spherical coordinates of air object, found using a rangefinder and a goniometer. The models include a correction for determining the location of the base station, a correction for determining the angle of elevation due to inaccuracies in the leveling of the station platform and azimuth, and a correction related to the inaccuracy of the station’s reference to the north. The measurement uncertainty budgets of rectangular coordinates which can be a basis for creation of software for automation of calculation of measurement uncertainties are resulted. Estimates of expanded uncertainties are found by the method of kurtosis. Expressions for the relative standard uncertainties of coordinate measurements are written and an example of their estimation for real data is given.

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44

Xu, Zheng Jie, Li Zeng, and Fang Chen. "Description of Rotor's Position of Magnetic Suspension Spherical Reluctance Motor and Detection Research." Applied Mechanics and Materials 184-185 (June 2012): 114–20. http://dx.doi.org/10.4028/www.scientific.net/amm.184-185.114.

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Detecting rotor position is an important part of work of magnetic suspension spherical reluctance motor． It contains rotor's position and orientation detection． This paper creates three-dimensional coordinate system on spherical reluctance motor， derives the description of the rotor's position and orientation and proposes a method for calculating rotor's position based on radial displacement sensors and angular displacement transducer． It also uses this approach and the spherical geometry relation to calculate orientation matrix of rotor according to the data from the sensors， so we can work out the position of any point through the rotation of coordinate system and position transformation．

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45

Protsenko, V. S., A. I. Solov'yev, and V. V. Tsymbalyuk. "Torsion of elastic bodies bounded by coordinate surfaces of toroidal and spherical coordinate systems." Journal of Applied Mathematics and Mechanics 50, no. 3 (January 1986): 313–21. http://dx.doi.org/10.1016/0021-8928(86)90126-7.

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46

Fong, Li Wei, and I. Heng Chen. "Passive Angle-Only Maneuvering Target Tracking Using Federated Filter in Hybrid Coordinates." Applied Mechanics and Materials 432 (September 2013): 427–31. http://dx.doi.org/10.4028/www.scientific.net/amm.432.427.

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An approach to federated filtering in hybrid coordinates is presented for a maneuvering target tracking through a multi-passive-sensor system. This new design accommodates a group of extended Kalman filters that the sensor-dedicated local processors process angle-only measurements extracted from a number of associated maneuverable aircrafts with onboard direction finders. Each filter utilizes the Reference Cartesian Coordinate (RCC) system for state and state covariance extrapolation and utilizes the modified spherical coordinate system for state and state covariance updating. In addition, a discrete-time state equation with the piecewise constant white target acceleration model is introduced into the RCC system for federated filtering. A weighted least squares estimator is used for the global processor aggregating the local estimates to generate a global estimate in the local inertial Cartesian coordinate system. Simulation results show that proposed approach drastically improves the tracking accuracy.

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47

ZHU, Xuexuan. "Spherical Coordinate-based Reliability Analysis in a Control System." International Journal of Signal Processing, Image Processing and Pattern Recognition 9, no. 5 (May 31, 2013): 77–86. http://dx.doi.org/10.14257/ijsip.2016.9.5.08.

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48

Martel, Karl, and Eric Poisson. "Regular coordinate systems for Schwarzschild and other spherical spacetimes." American Journal of Physics 69, no. 4 (April 2001): 476–80. http://dx.doi.org/10.1119/1.1336836.

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49

Cuccaro, Steven A., Paul G. Hipes, and Aron Kuppermann. "Hyper-spherical coordinate reactive scattering using variational surface functions." Chemical Physics Letters 154, no. 2 (January 1989): 155–64. http://dx.doi.org/10.1016/s0009-2614(89)87279-3.

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50

CHEN, NING, and NANNAN LUO. "CONSTRUCTION OF SPHERICAL PATTERNS FROM PLANAR DYNAMIC SYSTEMS." Fractals 21, no. 01 (March 2013): 1350005. http://dx.doi.org/10.1142/s0218348x13500059.

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We investigated the generation of spherical continuous-tilings of the chaotic attractors or the filled-in Julia sets from the plane mappings. We build three plane mappings, which can be used to construct the continuous patterns on the surfaces of the hexahedron and the unit sphere. We discuss the coordinate transformation for a spatial point between the different coordinate systems and further discuss how to project a spherical point onto a surface of the inscribed hexahedron. We present a method of constructing a spherical pattern with the pattern of a square on the inscribed hexahedron from an arbitrary projective angle, and generate spherical patterns from the three plane mappings. The results show that we can construct a great number of spherical patterns of the chaotic attractors and the filled-in Julia sets from the plane mappings, which are based on the square lattice and meet the requirements of the continuity and the four rotation symmetries on the lattice's boundaries.

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