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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.
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TL;DR: In this article, several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables, which can give the exact solutions of some unsolved problems in quantum mechanics and diffraction theory.
Abstract: Several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables. Such coordinate systems can give the exact solutions of some unsolved problems in quantum mechanics and diffraction theory.
01 Aug 1981
TL;DR: In this article, the exact configured vehicle's approximation equations used in their precision microwave landing system navigation experiments were used for use in a terminal area air traffic model with deterministic aircraft traffic.
Abstract: Formulas and their approximation were developed to map geodetic position to an Earth tangent plane with an airport centered rectangular coordinate system. The transformations were developed for use in a terminal area air traffic model with deterministic aircraft traffic. The exact configured vehicle's approximation equations used in their precision microwave landing system navigation experiments.
19 Sep 2016
TL;DR: In this article, anisotropic elliptic inclusions in a static and uniform excitation field were investigated and two classes of radial anisotropy, porcupic and bulbic, were found to exhibit field concentration and cloaking.
Abstract: This contribution concerns anisotropic elliptic inclusions in a static and uniform excitation field. In particular, the discussion considers anisotropy which is axial in a way that is defined by the elliptic coordinate system. This type of axial anisotropy has been referred to as “radial anisotropy”, although other varieties of radial anisotropy do exist for other geometries. In case of an ellipse, there are at least two methods to create radial anisotropy, and the two methods lead to qualitatively different outcomes. The terms “porcupic” and “bulbic” have been used to refer to these two classes of radial anisotropy. The porcupic inclusions are found to exhibit field concentration and the bulbic ones are found to exhibit cloaking. The contribution discusses both types of inclusions first in isolation and then as layered structures.
TL;DR: In this article, a new class of invariant optical fields which are exact solutions of the Helmholtz equation in parabolic coordinates is introduced, and the spatial distribution of these PIOFs is described in terms of the even and odd Parabolic cylinder functions Pe and Po.
Abstract: From group theory, it is known that the three-dimensional Helmholtz equation is separable in four orthogonal cylindrical coordinate systems: rectangular (i.e. Cartesian), circular, elliptic, and parabolic systems [1]. Invariant Optical Fields (often referred to as nondiffracting beams) have been demonstrated theoretical and experimentally for plane waves in Cartesian coordinates, for Bessel beams in circular cylindrical coordinates [2,3], and for Mathieu beams in elliptic coordinates [4,5]. We introduce in this work a new class of invariant optical fields which are exact solutions of the Helmholtz equation in parabolic coordinates. The spatial distribution of these PIOFs is described in terms of the even and odd Parabolic cylinder functions Pe and Po, namely
01 Sep 1980
TL;DR: In this article, it was shown that the formulae for the wave frequencies in both coordinate systems are the same to a good degree of approximation, and that the results obtained with the cylindrical coordinate system approximate quite well those obtained with a spherical coordinate system.
Abstract: In earlier studies of long low-frequency atmospheric waves in the polar atmosphere a cyclindrical coordinate system had been used with the planez=0 tangential to the earth's surface at the pole It had been found by numerical examples that the results obtained with a cylindrical coordinate system approximate quite well those obtained with a spherical coordinate system Here, it is shown that the formulae for the wave frequencies in both coordinate systems are the same to a good degree of approximation