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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 paper, a new inversion procedure for re-covering functions, defined on $\Bbb R^{2}$, from the spherical mean transform, which integrates functions on a prescribed family of circles, where $\Lambda$ consists of circles whose centers belong to a given ellipse E on the plane, was introduced.
Abstract: The aim of this paper is to introduce a new inversion procedure for re- covering functions, defined on $\Bbb R^{2}$, from the spherical mean transform, which integrates functions on a prescribed family $\Lambda$ of circles, where $\Lambda$ consists of circles whose centers belong to a given ellipse E on the plane. The method presented here follows the same procedure which was used by S. J. Norton in [22] for recovering functions in case where $\Lambda$ consists of circles with centers on a circle. However, at some point we will have to modify the method in [22] by using expansion in elliptical coordinates, rather than spherical coordinates, in order to solve the more generalized elliptical case. We will rely on a recent result obtained by H.S. Cohl and H.Volkmer in [8] for the eigenfunction expansion of the Bessel function in elliptical coordinates.
01 Jan 2014
TL;DR: In this paper, the authors examined the often overlooked problem of representing an optical field discretely using cylindrical coordinates, and they showed that the choice of coordinate system depends on the optical problem being examined.
Abstract: The Fresnel transform is a diffraction integral used in optics to calculate the propagation of a wave field in the paraxial domain. It is possible to express the Fresnel transform in Cartesian or cylindrical coordinates. Often the choice of coordinate system depends on the optical problem being examined. In this proceeding we examine the often overlooked problem of representing an optical field discretely using cylindrical coordinates.
05 Jul 2022
TL;DR: This is the first work to test training of PINNs by modifying PDEs according to the boundary shape and it is demonstrated that PINNs with Cartesian coordinate shows better approximation accuracy.
Abstract: In this work, we investigate some coordinate systems to solve partial differential equations (PDEs) using a neural network. We approximate the solution using physics-informed neural networks (PINNs) both before and after the coordinate transformation for two cases: a coordinate system with periodicity and without periodicity. We demonstrate that PINNs with Cartesian coordinate shows better approximation accuracy. This implies in PINNs training the Cartesian coordinate system is superior to the other coordinate systems derived by coordinate transformation. To the best of our knowledge, this is the first work to test training of PINNs by modifying PDEs according to the boundary shape.
TL;DR: In this article, the flow pattern of a fluid of grade four between two elliptic tubes is determined using elliptic coordinates, and a comparison between the position of the maximum of the axial velocity in the present case and in the case of two concentric circular tubes shows a basic difference.
Abstract: Using elliptic coordinates, the flow pattern of a fluid of grade four between two elliptic tubes is determined. A comparison between the position of the maximum of the axial velocity in the present case and in the case of two concentric circular tubes shows a basic difference. In the elliptic case the maximum is shifted towards the external wall, while in the case of concentric circular tubes the shift is in the direction of the internal wall. The secondary flow shows dissymmetry with reference to the intermediate line\(\tilde \psi _4 = 0\), which itself lies nearer to the external wall.
25 May 2021
TL;DR: In this paper, the authors proposed a target tracking controller based on the binocular coordination principle, which simplifies the design of the control law by leveraging rich physical meanings of elliptic coordinates.
Abstract: This paper concentrates on the collaborative target tracking control of a pair of tracking vehicles with formation constraints. The proposed controller requires distance measurements between tracking vehicles and the target. Its novelty lies in two aspects: 1) the elliptic coordinates are used to represent an arbitrary tracking formation, which can be deduced from inter-agent distances, and 2) the regulation of the tracking vehicle system obeys a binocular coordination principle, which simplifies the design of the control law by leveraging rich physical meanings of elliptic coordinates. The tracking system with the proposed controller is proven to be exponentially convergent when the target is stationary. When the target drifts with a small velocity, the desired tracking formation is achieved within a small margin proportional to the magnitude of the target's drift velocity. Simulation examples are provided to demonstrate the tracking performance of the proposed controller.