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Toroidal coordinates : ウィキペディア英語版
Toroidal coordinates

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.
==Definition==

The most common definition of toroidal coordinates (\sigma, \tau, \phi) is
:
x = a \ \frac \cos \phi

:
y = a \ \frac \sin \phi

:
z = a \ \frac

where the \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to opposite sides of the focal ring
:
\tau = \ln \frac}.

The coordinate ranges are -\pi<\sigma\le\pi and \tau\ge 0 and 0\le\phi < 2\pi.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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