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单词 Toroidal coordinates
释义

Toroidal coordinates

中文百科

圆环坐标系

圆环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 ( - a,\ 0,\ 0) 与 (a,\ 0,\ 0) 。将双极坐标系绕着 z-轴旋转，则可以得到圆环坐标系。双极坐标系的两个焦点，变为一个半径为 a 的圆圈，包含于圆环坐标系的 xy-平面。称这圆圈为焦圆，又称为参考圆。

英语百科

Toroidal coordinates 圆环坐标系

Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis separating its two foci. The foci are located at a distance 1 from the vertical z-axis. The red sphere is the σ = 30° isosurface, the blue torus is the τ = 0.5 isosurface, and the yellow half-plane is the φ = 60° isosurface. The green half-plane marks the x-z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.996, −1.725, 1.911).
Rotating this two-dimensional bipolar coordinate system about the vertical axis produces the three-dimensional toroidal coordinate system above. A circle on the vertical axis becomes the red sphere, whereas a circle on the horizontal axis becomes the blue torus.
Geometric interpretation of the coordinates σ and τ of a point P. Observed in the plane of constant azimuthal angle φ, toroidal coordinates are equivalent to bipolar coordinates. The angle σ is formed by the two foci in this plane and P, whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F_1 and F_2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. The focal ring is also known as the reference circle.
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更新时间：2025/6/20 8:33:50