Elliptic coordinate system
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Fourth order superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials Open
A study is presented of two-dimensional superintegrable systems separating in\nCartesian coordinates and allowing an integral of motion that is a fourth order\npolynomial in the momenta. All quantum mechanical potentials that do not\nsatis…
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Parametric upconversion of Ince–Gaussian modes Open
The Ince–Gaussian (IG) mode, a recently discovered type of structured Gaussian beam, corresponds to eigenfunctions of the paraxial wave equation in elliptical coordinates. This propagation-invariant mode is of significance in various domai…
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Temperature distribution in an elliptical body with an internal heat source with partial adiabatic isolation Open
The article calculates the temperature field in an elliptical body with internal heat dissipation. In this case, the boundary conditions are boundary conditions of the third kind. The solution is located at the transition to the elliptic c…
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A Continuous Coordinate System for the Plane by Triangular Symmetry Open
The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the tr…
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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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Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems Open
We propose the integrable N-dimensional Calogero-Coulomb-Stark and two-center\nCalogero-Coulomb systems and construct their constants of motion via the Dunkl\noperators. Their Schr\\"odinger equations decouple in parabolic and elliptic\nco…
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Vector Differential Calculus Open
This paper treats the fundamentals of the vector differential calculus part of universal geometric calculus. Geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for te…
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Coordinate transformation and its uncertainty under consideration of a non-orthogonal coordinate base Open
Nanopositioning and nanomeasuring machines are 3D coordinate measuring systems with nanometer precision at measurement volumes in the cubic centimeter range whose coordinate base is formed by an interferometer system with a common mirror c…
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Investigation on the effects of number of common points in 2D transformation problem Open
Coordinate transformation from one datum to another is the basic problem in geodesy. Generally, the problem may be expressed by converting coordinates in a cartesian coordinate system with defined origin provided by the intersection of two…
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Experimental observation of three-dimensional non-paraxial accelerating beams Open
We experimentally realize three-dimensional non-paraxial accelerating beams associated with different coordinate systems. They are obtained by Fourier transforming a phase-modulated wave front in an aberration-compensated system. The phase…
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Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials Open
We consider a two dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e potent…
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Development of Parameter Transformation of Indonesian Geospatial Reference System 2013 Open
DGN95 is a static geospatial reference system, in which the change in the value of coordinates towards time as a result of tectonic plate movement and deformation of the earth’s crust, is not considered. Changes in the value of coordinates…
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Temperature distribution in an elliptical body without internal heat sources under boundary conditions of the third kind with partial adiabatic isolation Open
The article deals with the distribution of the temperature field in an elliptical body without internal heat sources. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found when moving to an …
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New approach to an underwater discharge simulation using elliptic coordinates Open
In this study, a system of two equations is developed that allows the time evolution of the underwater spark channel to be calculated numerically from a given power input. The proposed mathematical model utilizes the elliptical coordinates…
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Different Approaches to Coordinate Transformation Parameters Determination of Nonhomogeneous Coordinate Systems Open
During reconstruction and restoration of city geodetic networks, there is quite a common problem that is related to the nonhomogeneity of existing geodetic networks. In any city, local authorities operate with their coordinate systems. Suc…
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Non-uniform torsion of functionally graded anisotropic bar of an elliptical cross section Open
An analytical solution is given to the problem of non-uniform torsion of an elliptical cylinder made of functionally graded anisotropic linear elastic material. The material moduli of the considered anisotropic non-homogeneous elastic bar …
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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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Derivation of Jacobian formula with Dirac delta function Open
We demonstrate how to make the coordinate transformation or change of variables from Cartesian coordinates to curvilinear coordinates by making use of a convolution of a function with Dirac delta functions whose arguments are determined by…
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Methodology of determining of the transfer function of engagement kinematics of accelerations of an aircraft and its elliptic coordinates used for thr guidance based on time difference of arrival Open
This paper discusses the problem of determining a kinematics (in terms of transfer function, as far as possible) of parameters of the motion of an aircraft expressed in the curvilinear coordinate system and control accelerations expressed …
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K teorii ploscadej giperboliceskoj ploskosti polozitel'noj krivizny Open
A hyperbolic plane ? of positive curvature is the projective model of the de Sitter plane. In article the ways of measurement of the figures areas of the plane ? are offered. The cyclic orthogonal coordinate systems are described. One fami…
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On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability Open
It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential e…
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Necessary and Sufficient Conditions for the Separability and the R-Separability of the Irrotational Stokes Equation and Applications Open
In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric syst…
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Elliptic Calogero-Moser system, crossed and folded instantons, and bilinear identities Open
Affine analogues of the Q-functions are constructed using folded instantons partition functions. They are shown to be the solutions of the quantum spectral curve of the N-body elliptic Calogero-Moser (eCM) system, the quantum Krichever cur…
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A Fourth Order FDFD Approach for the Analysis of Sectorial Elliptic Waveguides Open
We present a fourth order frequency domain finite difference approach (FDFD) in curvilinear coordinates for the computation of the modes of sectorial and ridged elliptic waveguides. The use of an elliptic mesh allows to avoid usual the sta…
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Exact conic section arc elements in 2D and 2.5D FEM using a coordinate transformation Open
This study introduces the conic section arc elements in 2D and 2.5D finite element method (FEM). Elements are obtained by deforming an edge in a standard triangular element through a coordinate transformation. This allows to completely eli…
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Off-axis elliptic Gaussian beams with an intrinsic orbital angular momentum Open
We discuss paraxial light beams composed of decentered Gaussian beams, with their phase selected in a special way so that their superposition is invariant as it propagates in free space, retaining its cross-section shape. By solving a syst…
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On a discretization of confocal quadrics. I. An integrable systems approach Open
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-…
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Assessment of the Impact of Probe Head Hysteresis on the Accuracy of Five-Axis Coordinate Measuring Systems Open
One of the factors affecting the accuracy of Coordinate Measuring Systems is hysteresis of a probe head. It is important especially in case of touch trigger probes and should be taken into account during the determination of probing strate…
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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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2D Fourier finite element formulation for magnetostatics in curvilinear coordinates with a symmetry direction Open
We present a numerical method for the solution of three-dimensional linear magnetostatic problems by embedding their geometry into a symmetric domain and Fourier expansion in the symmetry coordinate, leading to a set of decoupled two-dimen…