Prolate spheroidal coordinates
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Molecular dynamics simulations of field emission from a prolate spheroidal tip Open
High resolution molecular dynamics simulations with full Coulomb interactions of electrons are used to investigate field emission from a prolate spheroidal tip. The space charge limited current is several times lower than the current calcu…
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Acoustic radiation force exerted on a small spheroidal rigid particle by a beam of arbitrary wavefront: Examples of traveling and standing plane waves Open
The analytical solution of the acoustic radiation force exerted by a beam of arbitrary shape on a small spheroidal rigid particle suspended in an ideal fluid is presented. The particle is assumed to be much smaller than the wavelength, i.e…
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Spheroidal harmonic expansions for the solution of Laplace's equation for a point source near a sphere Open
We propose a powerful approach to solve Laplace's equation for point sources near a spherical object. The central new idea is to use prolate spheroidal solid harmonics, which are separable solutions of Laplace's equation in spheroidal coor…
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Explicit Equations to Transform from Cartesian to Elliptic Coordinates Open
Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical …
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Multiphoton Ionization of One-Electron Relativistic Diatomic Quasimolecules in Strong Laser Fields Open
We perform a theoretical and computational study of relativistic one-electron homonuclear diatomic quasimolecules subject to strong electromagnetic fields linearly polarized along the molecular axis. Several quasimolecules with the nuclear…
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Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid Open
The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differentia…
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A Novel Camera Calibration Method Based on Polar Coordinate Open
A novel calibration method based on polar coordinate is proposed. The world coordinates are expressed in the form of polar coordinates, which are converted to world coordinates in the calibration process. In the beginning, the calibration …
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Transformation between polar and rectangular coordinates of stiffness and dampness parameters in hydrodynamic journal bearings Open
The stiffness and dampness parameters of journal bearings are required in rectangular coordinates for analyzing the stability boundary and threshold speed of oil film bearings. On solving the Reynolds equation, the oil film force is always…
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Biharmonic Coordinates and their Derivatives for Triangular 3D Cages Open
As a natural extension to the harmonic coordinates, the biharmonic coordinates have been found superior for planar shape and image manipulation with an enriched deformation space. However, the 3D biharmonic coordinates and their derivative…
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Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates Open
The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the s…
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Scattering of elastic waves by a spheroidal inclusion Open
An analytical solution is presented for scattering of elastic waves by prolate and oblate spheroidal inclusions. The problem is solved in the frequency domain where separation of variables leads to a solution involving spheroidal wave func…
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Effect of two-center interference on molecular ionization in multiphoton ionization regime Open
Using solution of the full three-dimensional time-dependent Schrödinger equation (TDSE) in prolate spheroidal coordinates, we investigate the orientation dependence of ionization of H2+ in near-infrared laser fields. It is found that, the …
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Spheroidal magnetic stars rotating in vacuum Open
Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force…
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Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates Open
We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expre…
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On generalized prolate spheroidal functions Open
Prolate spheroidal wave functions provide a natural and effective tool for computing with bandlimited functions defined on an interval. As demonstrated by Slepian et al., the so called generalized prolate spheroidal functions (GPSFs) exten…
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Development of a unified high-order nonhydrostatic multi-moment constrained finite volume dynamical core: derivation of flux-form governing equations in the general curvilinear coordinate system Open
In the manuscript we have derived the flux-form atmospheric governing equations in the general curvilinear coordinate system which is used by a high-order nonhydrostatic multi-moment constrained finite volume (MCV) dynamical core, and give…
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Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells Open
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-Schwarzsc…
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New relationships between spherical and spheroidal harmonics and applications Open
This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical…
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New relationships between spherical and spheroidal harmonics and applications Open
This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical…
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The Grad–Shafranov Equation in Cap-Cyclide Coordinates: The Heun Function Solution Open
The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular …
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Coordinate-free compatibility conditions for deformations of material surfaces Open
The study of material surfaces uses notions from classical differential geometry, such as the covariant gradient, the mean and Gaussian curvatures, and the Peterson–Mainardi–Codazzi and Gauss equations. These notions are traditionally intr…
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New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation Open
In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (i…
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Explicit solutions in Cartesian coordinates for an elliptic hole in an infinite elastic plate Open
An explicit analytical solution for an elliptical hole in an infinite elastic plate is derived for uniaxial load from the earliest work on this configuration. This is used along with the biaxial loading case and more recent solutions, avai…
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Epicyclic frequencies of spheroidal stars with non-uniform density Open
We consider the gravitational potential of a rotating star with non-uniform density to derive the orbital and epicyclic frequencies of the particles orbiting the star. We assume that the star is composed of concentric spheroids of constant…
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Asymptotics of prolate spheroidal wave functions Open
Uniform asymptotic approximations are obtained for the prolate spheroidal wave functions, for both the angular functions Ps m n x,γ 2 ( -1 < x < 1 ) and radial functions Ps m n x,γ 2 ( 1 < x < ∞ ).Here γ → ∞ , and the results are uniformly…
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Field theory in normal toroidal coordinates Open
Field theory is widely represented in spherical and cylindrical coordinate systems, since the mathematical apparatus of these coordinate systems has been thoroughly studied. Sources of field with more complex structures require new approac…
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Internal and external harmonics in bi-cyclide coordinates Open
The Laplace equation in three-dimensional Euclidean space is -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion f…
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Investigation of formulas determination of a point’s plane coordinates by the invers linear-angular resection Open
The aim. The study of formulas determination of the point coordinates by the inverse linear-angular intersection method. Previously, we investigated the possibility of using electronic total stations to control the geometric parameters of …
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A Computational Review of The Solar Wind in Oblate Spheroidal Coordinates Perspective Open
The solar as theoretical is known not perfectly round. Many types of research have been trying to approach the states and properties of the solar. Complexities of these issues are very interesting to study and observe. In this paper presen…
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Internal and external harmonics in bi-cyclide coordinates Open
The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé-Wangerin functions called internal and external bi-cyclide harmonics. An expansio…