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.

The Transfer Model between the Xi'an 80 and WGS-84 Coordinate Systems

Abstract: Based on coinciding 174 triangular points from coordinate system 1980 with GPS-A.B networks, this paper takes 3-,4-,7-, parameter transfer model to accomplish calculation from Xi'an 80 coordinate system to WGS-84 coordinate system. After all, accuracy analysis is discussed in this Paper.

Curvature Coordinates in Cosmology

Abstract: We develop cosmological theory from first principles starting with curvature coordinates ($R$,$T$) in terms of which the metric has the form $d{s}^{2}(R,T)=\frac{d{R}^{2}}{A(R,T)}+{R}^{2}d{\ensuremath{\Omega}}^{2}\ensuremath{-}B(R,T)d{T}^{2}$ The Einstein field equations, including cosmological constant, are given for arbitrary ${T}_{\ensuremath{ u}}^{\ensuremath{\mu}}$, and the timelike geodesic equations are solved for radial motion We then show how to replace $T$ with a new time coordinate $\ensuremath{\tau}$ that is equal to the time measured by radially moving geodesic clocks Cosmology is brought into the picture by setting ${T}_{\ensuremath{ u}}^{\ensuremath{\mu}}$ equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting $\ensuremath{\tau}$ be the time measured by clocks coincident with the fluid particles We solve the field equations in terms of ($R$,$\ensuremath{\tau}$) coordinates to get the metric coefficients in terms of the pressure and density of the fluid The metric on the subspace $\ensuremath{\tau}=\mathrm{const}$ is equal to $d{R}^{2}+{R}^{2}d{\ensuremath{\Omega}}^{2}$, and so is flat, with $R$ having the physical significance that it is a measure of proper distance in this subspace As specific examples, we consider the de Sitter and Einstein---de Sitter universes On an ($R$,$\ensuremath{\tau}$) spacetime diagram, all trajectories in an Einstein---de Sitter universe are emitted from $R=0$ at the "big bang" at $\ensuremath{\tau}=0$ Further, a light signal coming toward $R=0$ at some time $\ensuremath{\tau}g0$ will, in its past history, have started from $R=0$ at $\ensuremath{\tau}=0$, and have turned around on the line $2R=3\ensuremath{\tau}$ A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle The turnaround line $2R=3\ensuremath{\tau}$ marks the transition where an $R=\mathrm{const}$ line changes from spacelike to timelike in character We show how to apply the techniques developed here to the inhomogeneous problem of a Schwarzschild mass imbedded in a given universe in the paper immediately following this one

Boundary conditions of high-order accuracy at the poles of curvilinear coordinate systems

Abstract: We develop a method of imposing improved difference boundary conditions at singularity points of polar, cylindrical, and spherical coordinate systems with the aim to apply them to high-order accuracy schemes for stationary and nonstationary problems. We consider two main types of boundary value problems, viz. symmetric and arbitrary problems that have no symmetry. We obtain difference equations, which are consistent in order of accuracy and are of simple form, along the axis (at the centre) of symmetry. These are special approximations of the original equations in Cartesian coordinates. We study a problem of realizing the boundary conditions in implicit schemes of approximate factorization and in iterative processes. The author [4,5] constructed schemes of fourth-order accuracy for second-order equations in any orthogonal coordinate systems. In order to apply these results to coordinate systems with singularities (i.e. with lines or ambiguous points of the transformations of coordinates, in the following for brevity they are referred to as poles) it is necessary to impose difference boundary conditions with sufficient accuracy on nodes corresponding to these singularities. Thus, for example, when formulating the, boundary value problem for a cylinder it is necessary to impose difference boundary conditions on its axis, which would correspond to the used scheme in order of approximation. For the symmetric case the region of singularity is the line r = 0 in the rectangular range of the radius and height (r, z\\ and for an arbitrary problem it is the plane r = 0 in the three-dimensional range (r, ζ, φ\\ where φ is a vectorial angle. In polar coordinates the set r = 0 is a point or a line depending on whether a problem is symmetric or nonsymmetric, and in spherical coordinates it is a point or a plane. A common feature is the ambiguity of coordinates transformation at the poles (i.e. on the set r = 0) and the absence of boundary conditions in initial boundary value problems for a differential equation, whereas it is necessary to impose boundary conditions for a difference problem. For symmetric problems in the case of second-order accuracy schemes the problems of imposing boundary conditions, convergence and implementation of algorithms are properly developed (see, e.g. the monographs [2,6]). In the case of high-order accuracy schemes it is rather difficult to find out satisfactory relations at the poles even for symmetric boundary value problems. In this respect nonsymmetric boundary value problems are much more complicated even for conventional schemes, to say nothing of high-order accuracy schemes. This paper deals with the solution of these problems. We develop methods of imposing boundary conditions of high-order accuracy at the poles of some orthogonal coordinate systems most generally employed. We also describe methods of realizing these * Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia The work was supported by the program Integration Basic Research', Siberian Branch of the Russian Academy of Sciences (43) and the Russian Foundation for the Basic Research (97-01-00819).

A note on elliptic coordinates on the Lie algebra e(3)

Abstract: We introduce elliptic coordinates on the dual space to the Lie algebra e(3) and discuss the separability of the Clebsch system in these variables. The proposed Darboux coordinates on e*(3) coincide with the usual elliptic coordinates on the cotangent bundle of the two-dimensional sphere at the zero value of the corresponding Casimir function.

Уравнения кинематики беспилотного летательного аппарата в эллиптической системе координат при наведении по разностно-дальномерной навигационной информации

Abstract: This paper discusses the problem of determining a kinematics (in terms of transfer function, as far as possible) of parameters of the motion of an aircraft expressed in the curvilinear coordinate system and control accelerations expressed in a rectangular coordinate system. Examples of curvilinear coordinate systems using in practice can be polar, biangular, two-center bipolar, elliptic, parabolic cylindrical, spherical, ellipsoidal, coordinate systems. A technique for obtaining a kinematic link for the control problem of an unmanned aerial vehicle in the elliptic coordinate system was described. It allowed to obtain simpler view of the kinematic link which could provide navigation an aircraft along the hyperbola deriving from the time difference of arrival navigation system. It can. As a result, it is possible to reduce the number of the navigation radio beacons.