Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

Abstract: A new representation of the Coulomb three-body wavefunction via the well-known solutions of the separable Coulomb two-centre problem j(ξ, η) = Xj(ξ)Yj(η) is obtained, where Xj(ξ) and Yj(η) are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the asymptotic conditions of the scattering problem below the three-particle breakup. That is, the wavefunction of two colliding clusters in any open channel is the asymptotics of the single, corresponding to that channel, term of the suggested expansion. The effect is achieved due to a new relation between three internal coordinates of a three-body system and the parameters of j(ξ, η). It ensures the orthogonality of j(ξ, η) on a sphere of constant hyperradius, ρ = const, in place of the surface R = |x2 − x1| = const appearing in the traditional Born–Oppenheimer approach. The independent variables ξ and η are the orthogonal coordinates on this sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of a stereographic projection. For the total angular momentum J ≥ 0 the products of j and the Wigner D-functions form a hyperspherical Coulomb spheroidal (HSCS) basis on a five-dimensional hypersphere, ρ being a parameter. The system of the differential equations and the boundary conditions for the radial functions fJi(ρ), the coefficients of the HSCS decomposition of the three-body wavefunction, are presented.

Coordinate Transformation Based on Direction Cosine Parameters between the Object Coordinate System and the Word Coordinate System

Abstract: A object coordinate system is built by three points in object and the method of solving the direction cosines of the axises of the coordinate system is presented. The coordinate transformation matrix based on the direction cosine parameters from the object coordinate system to the world coordinate system is derived by vector algebra. The coordinate transformation matrix from the world coordinate system to the object coordinate system is derived by finding the converse matrix. As deductions of the coordinate transformation, two special coordinate transformation and their matrices are given. The two transformation matrices between the object coordinate system and the world coordinate system presented in this article are universal coordinate transforma- tion matrices for two arbitrary coordinate systems.

Perfectly Matched Layers in the Cylindrical and Spherical Coordinates for Elastic Waves in Solids

Abstract: The material constants of perfectly matched layers (PMLs) in the cylindrical and spherical coordinates in the frequency domain are presented. Using the coordinate transformation laws of tensors on manifolds, the quotient rule, and complex coordinate stretching, we obtain the material parameters of PMLs in the real coordinate. Our results show that PML parameters for elastic waves may be determined by the same procedure in the Cartesian coordinates. However, this rule has been determined for PML material constants derived from the analytic continuation in the cylindrical and spherical coordinates by Zheng and Huang in 2002. Our derivation based on differential forms shows that this rule holds for PML parameters in any orthogonal coordinate system.

Parallel coordinate order for high-dimensional data

Abstract: Visualization of high‐dimensional data is counter‐intuitive using conventional graphs. Parallel coordinates are proposed as an alternative to explore multivariate data more effectively. Ho...

Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering - extended version

Abstract: Understanding the spectral properties of boundary integral operators in acoustic scattering has important practical implications, such as for the analysis of the stability of boundary element discretisations or the convergence of iterative solvers as the wavenumber k grows. Yet little is known about spectral decompo- sitions of the standard boundary integral operators in acoustic scattering. Theoretical results are mainly available on the unit disk, where these operators diagonalise in a simple Fourier basis. In this paper we investigate spectral decompositions for more general smooth domains. Based on the decomposition of the acoustic Green’s function in elliptic coordinates we give spectral decompositions on ellipses. For general smooth domains we show that approximate spectral decompositions can be given in terms of circle Fourier modes transplanted onto the boundary of the domain. An important underlying question is whether or not the operators are normal. Based on previous numerical investigations it appears that the standard boundary integral operators are normal only when the domain is a ball and here we prove that this is indeed the case for the acoustic single layer potential. We show that the acoustic single, double and conjugate double layer potential are normal in a scaled inner product on the ellipse. On more general smooth domains the operators can be split into a normal component plus a smooth perturbation. Numerical computations of pseudospectra are presented to demonstrate the nonnonnormal behaviour on general domains.