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Confocal ellipsoidal coordinates : ウィキペディア英語版
Ellipsoidal coordinates
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (\lambda, \mu, \nu) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
==Basic formulae==

The Cartesian coordinates (x, y, z) can be produced from the ellipsoidal coordinates
( \lambda, \mu, \nu ) by the equations
:
x^ = \frac + \mu \right) \left( a^ + \nu \right)} \right) \left( a^ - c^ \right)}

:
y^ = \frac + \mu \right) \left( b^ + \nu \right)} \right) \left( b^ - c^ \right)}

:
z^ = \frac + \mu \right) \left( c^ + \nu \right)} \right) \left( c^ - a^ \right)}

where the following limits apply to the coordinates
:
- \lambda < c^ < - \mu < b^ < -\nu < a^.

Consequently, surfaces of constant \lambda are ellipsoids
:
\frac + \lambda} + \frac + \lambda} + \frac + \lambda} = 1,

whereas surfaces of constant \mu are hyperboloids of one sheet
:
\frac + \mu} + \frac + \mu} + \frac + \mu} = 1,

because the last term in the lhs is negative, and surfaces of constant \nu are hyperboloids of two sheets
:
\frac + \nu} + \frac + \nu} + \frac + \nu} = 1

because the last two terms in the lhs are negative.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Ellipsoidal coordinates」の詳細全文を読む



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