Topic
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.
Papers published on a yearly basis
Papers
More filters
Posted Content•
21 Sep 2020
TL;DR: The tracking system with the proposed controller is proven to be exponentially convergent when the target is stationary and the desired tracking formation is achieved within a small margin proportional to the magnitude of the target's drift velocity.
Abstract: This paper concentrates on the collaborative target tracking control of a pair of tracking vehicles with formation constraints. The proposed controller requires only 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 without singularity, 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.
03 Aug 2021
TL;DR: In this article, the authors use differential geometry to derive the transformations between Cartesian and elliptic coordinates, and show that in elliptic and Cartesian coordinates, a constant vorticity flow reduces to a flow with closed elliptic streamlines.
Abstract: Vortices (flows with closed elliptic streamlines) are exact nonlinear solutions to the compressible Euler equation. In this contribution, we use differential geometry to derive the transformations between Cartesian and elliptic coordinates, and show that in elliptic coordinates a constant vorticity flow reduces to $\dot{\mu}=0$ and $\dot{
u}={\rm const}$ along the streamline $\mu_0$ that matches the vortex eccentricity.
TL;DR: In this article , the authors presented formulae for obtaining coordinates for cylindrical projections of a triaxial ellipsoid for global mapping the surfaces of celestial bodies, the figures of which the International Astronomical Union recommends to approximate by the mathematical surface of the mentioned geometric body.
Abstract:
Cylindrical projections of a triaxial ellipsoid are used for global mapping the surfaces of celestial bodies, the figures of which the International Astronomical Union recommends to approximate by the mathematical surface of the mentioned geometric body. Its reference surface and projections are not available in GIS packages. In this regard, we are facing the problem of representing projection formulae in finite elementary functions or reducing the integrals included in them to elliptic ones. This will simplify direct and inverse transformations of such map projections for inclusion in GIS software. The authors present formulae for obtaining coordinates for cylindrical projections of a triaxial ellipsoid. Determining the horizontal coordinate for all those projections is reduced to the calculation of an elliptic integral of the second kind, as well as determining the vertical coordinate of a cylindrical projection which preserves lengths along the meridians. To determine that coordinate in an equal-area cylindrical projection, original formulae were obtained that enables representing the corresponding integral in elementary functions. For the vertical coordinate in the cylindrical projection of the meridian section similar formulae deduced in previous studies are presented. The definition of a vertical coordinate in a projection preserving the angle between the meridian and the parallel is reduced to the calculation of elliptic integrals of the first, second, and third kind. Thus, the formulae derived in the article can be used when including cylindrical projections of a triaxial ellipsoid in geoinformation technologies.
TL;DR: In this paper , an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions is presented.
Abstract: 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 elliptic coordinate system. The author has obtained an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.
01 Jan 2023
TL;DR: In this article , a unidirectional fiber reinforced polymer (FRP) lamina is modeled as a local coordinate system of an anisotropic material which is required to be defined when mechanical properties of a lamina are formulated.
Abstract: A unidirectional fiber reinforced polymer (FRP) lamina as illustrated in Fig. 1 consists of fibers along the main direction of the lamina (1-axis), and matrix all around the fibers. The 2-axis is perpendicular to 1-axis in the lamina plane and 3-axis is normal to this plane. The 1–2–3 system is the local coordinate system of an anisotropic material which is required to be defined when mechanical properties of a lamina are formulated. All the formulations presented here are in the lamina local coordinate system unless it is stated that the global coordinate system (x–y–z system) is considered.