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词汇 Prolate spheroidal coordinates
分类 英语词汇 英语翻译词典
释义

Prolate spheroidal coordinates

中文百科

长球面坐标系

In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in Cartesian coordinates; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (x, y, ±z). However, neither of the definitions presented here are degenerate.

长球面坐标系是一种三维正交坐标系。设置二维椭圆坐标系包含于 xz-平面；两个焦点 F_{1} 与 F_{2} 的直角坐标分别为 (0,\ 0,\ - a) 与 (0,\ 0,\ a) 。将椭圆坐标系绕着 z-轴旋转，则可以得到长球面坐标系。（假若，绕着 y-轴旋转，则可以得到扁球面坐标系。）椭圆坐标系的两个焦点，包含于 z-轴。长球面坐标系可以被视为椭球坐标系的极限案例，其两个最短的半轴的长度相同。

英语百科

Prolate spheroidal coordinates 长球面坐标系

The three coordinate surfaces of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ=1, and the blue two-sheet hyperboloid corresponds to ν=45°. The yellow half-plane corresponds to φ=-60°, which is measured relative to the x-axis (highlighted in green). The black sphere represents the intersection point of the three surfaces, which has Cartesian coordinates of roughly (0.831, -1.439, 2.182).
Prolate spheroidal coordinates μ and ν for a=1. The lines of equal values of μ and ν are shown on the x-z plane, i.e. for φ=0. The surfaces of constant μ and ν are obtained by rotation about the z axis, so that the diagram is valid for any plane containing the z axis: i.e. for any φ.
In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in Cartesian coordinates; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (x, y, ±z). However, neither of the definitions presented here are degenerate.

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.
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