Showing papers on "Elliptic coordinate system published in 2015"

Simultaneous separation for the Neumann and Chaplygin systems

Abstract: The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.

Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids

Abstract: In this paper we use symmetric coordinate systems for the hexagonal and the triangular grids (that are dual of each other). We present new coordinate systems by extending the symmetric coordinate systems that are appropriate to address elements (cells) of cell complexes. Coordinate triplets are used to address the hexagon/triangle pixels, their sides (the edges between the border of neighbour pixels) and the points at the corners of the hexagon/triangle pixels. Properties of the coordinate systems are detailed, lines (zig-zag lines) and lanes (hexagonal stepping lanes) are defined on the triangular (resp. hexagonal) grid by fixing a coordinate value. The bounding relation of the cells can easily be captured by the coordinate values. To illustrate the utility of these coordinate systems some topological algorithms, namely collapses and cuts are presented.

Boundary Methods for Dirichlet Problems of Laplace's Equation in Elliptic Domains with Elliptic Holes

Abstract: Recently, the null field method (NFM) is proposed by J.T. Chen with his groups. In NFM, the fundamental solutions (FS) with the field nodes Q outside of the solution domains are used in the Green formulas. In this paper, the NFM is developed for the elliptic domains with elliptic holes. First, the FS is expanded by the infinite series in elliptic coordinates. When the Fourier approximations of the boundary conditions on the elliptic boundaries are chosen, the explicit algebraic equations are derived, and the semi-analytic solutions can be found. Next, the interior field method (IFM) is developed, which is equivalent to the NFM when the field nodes approach the domain boundary. Moreover, the collocation Trefftz method (CTM) is also employed by using the particular solutions in elliptic coordinates. The CTM is the simplest algorithm, has no risk of degenerate scales, and can be applied to non-elliptic domains. Numerical experiments are carried out for elliptic domains with one elliptic hole by the IFM, the NFM and the CTM. In summary, for Laplace׳s equation in elliptic domains, a comparative study of algorithms, errors, stability and numerical results is explored in this paper for three boundary methods: the NFM, the IFM and the CTM.

Third-order superintegrable systems with potentials satisfying only nonlinear equations

Abstract: The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinates are shown to have potentials with only non-movable singularities.

The squeeze-film air damping of circular and elliptical micro-torsion mirrors

Abstract: This paper proposes an analytical solution to calculate the squeeze-film air damping of circular and elliptical micro-torsion mirrors To derive the expressions of squeeze-film air-damping torque, the nonlinear Reynolds equation, which governs the air behavior of torsion mirror, is solved by the method of eigenfunction expansions in polar coordinate and elliptical coordinate, respectively The series solutions are integrated and summed up to deduce the damping torque of circular and elliptical torsion mirrors The formulas of circular mirror and elliptical mirror are deduced independently, and their results match when the eccentricity of the elliptical mirror approaches zero Besides, the results of the formulas are consistent with numerical simulation Both of them verifies the damping torque formulas in this paper

On a discretization of confocal quadrics. I. An integrable systems approach

Abstract: 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-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting orthogonal Koenigs nets). Our construction is based on an integrable discretization of the Euler–Poisson–Darboux equation and leads to discrete nets with the separability property, with all two-dimensional subnets being Koenigs nets, and with an additional novel discrete analogue of the orthogonality property. The coordinate functions of our discrete nets are given explicitly in terms of gamma functions.

2D Quaternionic Time-Harmonic Maxwell System in Elliptic Coordinates

Abstract: In this paper we consider the 2D time–harmonic Maxwell equations in elliptic coordinates through certain quaternionic perturbed Dirac operator. The main goal is aimed to analyze an electromagnetic Dirichlet problem for a curvilinear polygon with rectifiable boundary in \({\mathbb{R}^2}\). In addition, we provide an integral representation formula for electromagnetic fields that resembles the classical Stratton-Chu formula. The importance of the problem for applications makes it worthy of consideration.

Vertical mode expansion method for analyzing elliptic cylindrical objects in a layered background.

Abstract: The vertical mode expansion method (VMEM) [J. Opt. Soc. Am. A31, 293 (2014)] is a frequency-domain numerical method for solving Maxwell's equations in structures that are layered separately in a cylindrical region and its exterior. Based on expanding the electromagnetic field in one-dimensional vertical modes, the VMEM reduces the original three-dimensional problem to a two-dimensional (2D) problem on the vertical boundary of the cylindrical region. However, the VMEM has so far only been implemented for structures with circular cylindrical regions. In this paper, we develop a VMEM for structures with an elliptic cylindrical region, based on the separation of variables in the elliptic coordinates. A key step in the VMEM is to calculate the so-called Dirichlet-to-Neumann (DtN) maps for 2D Helmholtz equations inside or outside the ellipse. For numerical stability reasons, we avoid the analytic solutions of the Helmholtz equations in terms of the angular and radial Mathieu functions, and construct the DtN maps by a fully numerical method. To illustrate the new VMEM, we analyze the transmission of light through an elliptic aperture in a metallic film, and the scattering of light by elliptic gold cylinders on a substrate.

Hyperspherical calculations of ultralow-energy collisions in Coulomb three-body systems

Abstract: Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the $R$-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.

On a discretization of confocal quadrics

Abstract: Confocal quadrics lie at the basis of the system of elliptic coordinates. We suggest a discretization which respects two crucial properties of elliptic coordinates: separability and Koenigs parametrization of all two-dimensional coordinate surfaces (that is, parametrization along conjugate lines having equal Laplace invariants). Actually, one even has an isothermic parametrization, which means an additional orthogonality property. Our discretization is based on the discrete Euler-Darboux equations and leads to discrete nets with the separability property and with all two-dimensional subnets being Koenigs. The coordinate functions of our discrete nets are given explicitly in terms of gamma function.

Formulation of Toupin–Mindlin strain gradient theory in prolate and oblate spheroidal coordinates

Abstract: The Toupin–Mindlin strain gradient theory is reformulated in orthogonal curvilinear coordinates, and is then applied to prolate and oblate spheroidal coordinates for the first time. The basic equations, boundary conditions, the gradient of the displacement, strain and strain gradient tensors of this theory are derived in terms of physical components in these two coordinate systems, which have a potential significance for the investigation of micro-inclusion and micro-void problems. As an example, using these formulae, we formulate and discuss the boundary-value problem of a spheroidal cavity embedded in a strain gradient elastic medium subjected to uniaxial tension. In addition, the previous results given by Zhao and Pedroso (Int. J. Solids. Struct. (2008) 45, 3507–3520) in cylindrical and spherical coordinates are amended.

Methods and systems for local principal axis rotation angle transform

Abstract: A method for processing synthetic aperture radar (SAR) data. The method includes the step of receiving SAR data that has been collected to provide a representation of a target scene, and dividing the data into a plurality of sub-blocks each having a plurality of pixels, each of the plurality of pixels having a coordinate and an amplitude. A transformation performed on each of the sub-blocks includes the steps of: (i) computing a mean coordinate; (ii) subtracting the mean coordinate from the pixel's actual coordinate to arrive at a modified coordinate; (iii) multiplying the modified coordinate by the amplitude to arrive at an amplitude-modified coordinate; (iv) creating a covariance matrix using the amplitude-modified coordinates; (v) performing a singular value decomposition on the covariance matrix to arrive at a vector; and (vi) associating an angle with the calculated vector.

Third-order superintegrable systems with potentials satisfying nonlinear equations

Abstract: The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions Systems separating in parabolic or elliptic coordinated are shown to have potentials with only non-movable singularities

A transmission line model for propagation in elliptical core optical fibers

Abstract: The calculation of mode propagation constants of elliptical core fibers has been the purpose of extended research leading to many notable methods, with the classic step index solution based on Mathieu functions. This paper seeks to derive a new innovative method for the determination of mode propagation constants in single mode fibers with elliptic core by modeling the elliptical fiber as a series of connected coupled transmission line elements. We develop a matrix formulation of the transmission line and the resonance of the circuits is used to calculate the mode propagation constants. The technique, used with success in the case of cylindrical fibers, is now being extended for the case of fibers with elliptical cross section. The advantage of this approach is that it is very well suited to be able to calculate the mode dispersion of arbitrary refractive index profile elliptical waveguides. The analysis begins with the deployment Maxwell’s equations adjusted for elliptical coordinates. Further algebraic analysis leads to a set of equations where we are faced with the appearance of harmonics. Taking into consideration predefined fixed number of harmonics simplifies the problem and enables the use of the resonant circuits approach. According to each case, programs have been created in Matlab, providing with a series of results (mode propagation constants) that are further compared with corresponding results from the ready known Mathieu functions method.

On a discretization of confocal quadrics. I. An integrable systems approach

Abstract: 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-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting orthogonal Koenigs nets). Our construction is based on an integrable discretization of the Euler-Poisson-Darboux equation and leads to discrete nets with the separability property, with all two-dimensional subnets being Koenigs nets, and with an additional novel discrete analog of the orthogonality property. The coordinate functions of our discrete nets are given explicitly in terms of gamma functions.

Avoiding the Coordinate Singularity Problem in the Numerical Solution of the Dirac Equation in Cylindrical Coordinates

Abstract: A new numerical method is developed for the solution of the Dirac equation for 3D axisymmetric geometries using cylindrical coordinates. It is based on a split-step scheme in coordinate space, which can be parallelized very efficiently. A new technique to circumvent the coordinate singularity at r = 0 using Poisson’s integral solution of the wave equation for the radial operator is used. The general strategy is to interpolate the solution using cubic Hermite polynomials and to integrate exactly the Poisson solution. The result of this procedure gives a nonstandard finite difference scheme on a time staggered grid. The numerical method is then utilized to evaluate the ground state of an electron bound in a Coulomb potential.

Circular Effect Generation for Color Images

Abstract: This paper presents a novel circular effect generation approach. In general, pixel location in an image can be represented with i and j coordinate. To manipulate (i,j) coordinate system easier, we transform (i,j) coordinate system into (ρ,θ) coordinate system. Two parameters are used for generating circular effect: R and T. After applying selected R and T in (ρ,θ) coordinate system, (ρ2,θ2) are obtained. Finally, (ρ2,θ2) signals are inverse-transformed into (i,j) coordinate system and (i2,j2) is obtained. Experimental results introduce performance comparison.

New Approach to Determine 6DOF Position and Orientation of a Non-Orthogonal Coordinate System on the Object Using its Image

Abstract: In this paper, a new method for determining position and orientation of a coordinate system using its image is presented. This coordinate system is a three dimensional non-orthogonal system in respect to the two dimensional and orthogonal camera coordinate system. In real world, it’s exactly easy to select three directions on an object so that they don’t be orthogonal and on a plane. The image of this non-orthogonal coordinate system on the camera image plane is a two dimensional coordinate system. This image is obtained by a nonlinear mapping between three dimensional worlds coordinate and two dimensional image coordinate. In this paper, we review geometric relationships between a direction of a vector on the object and its image that was presented in paper [18] months ago. Then, using these relationships for three arbitrary non-orthogonal directions which are not on a plane, a system of 15 nonlinear equations is established, and by solving it, nine unknowns are extracted. Because of the importance of the sign of these unknowns to determine true lengths and angels, it’s essential to run this system of nonlinear equations in eight cases and then best answer with right signs can be extracted. The results of this theory have been examined using simulation and programs. In paper [18] we have to select three orthogonal vectors on an object. Since world is 3D, in some cases it is exactly difficult to choose all three directions with proper length and maybe we have to choose third vector (which is in depth) with a short length and it increase errors in