Confocal Ellipsoidal Coordinates

The confocal ellipsoidal coordinates (called simply ellipsoidal coordinates by Morse and Feshbach 1953) are given by the equations

$${x^2\over a^2+\xi}+{y^2\over b^2+\xi}+{z^2\over c^2+\xi}=1$$ (1)

$${x^2\over a^2+\eta}+{y^2\over b^2+\eta}+{z^2\over c^2+\eta}=1$$ (2)

$${x^2\over a^2+\zeta}+{y^2\over b^2+\zeta}+{z^2\over c^2+\zeta}=1,$$ (3)

where $-c^2<\xi<\infty$, $-b^2<\eta<-c^2$, and $-a^2<\zeta<-b^2$. Surfaces of constant $\xi$ are confocal Ellipsoids, surfaces of constant $\eta$ are one-sheeted Hyperboloids, and surfaces of constant $\zeta$ are two-sheeted Hyperboloids. For every $(x,y,z)$, there is a unique set of ellipsoidal coordinates. However, $(\xi,\eta,\zeta)$ specifies eight points symmetrically located in octants. Solving for $x$, $y$, and $z$ gives

$$x^2={(a^2+\xi)(a^2+\eta)(a^2+\zeta)\over (b^2-a^2)(c^2-a^2)}$$ (4)

$$y^2={(b^2+\xi)(b^2+\eta)(b^2+\zeta)\over (a^2-b^2)(c^2-b^2)}$$ (5)

$$z^2={(c^2+\xi)(c^2+\eta)(c^2+\zeta)\over (a^2-c^2)(b^2-c^2)}.$$ (6)

The Laplacian is

$\nabla^2\Psi=(\eta-\zeta)f(\xi){\partial\over\partial\xi}\left[{f(\xi){\partial\Psi\over\partial\xi}}\right]$
$+(\zeta-\xi)f(\eta){\partial\over\partial\eta}\left[{f(\eta){\partial\Psi\over... ...tial\over\partial\zeta}\left[{f(\zeta){\partial\Psi\over\partial\zeta}}\right],$
	(7)

where

$$f(x)\equiv \sqrt{(x+a^2)(x+b^2)(x+c^2)}.$$ (8)

Another definition is

$${x^2\over a^2-\lambda} + {y^2\over b^2-\lambda} + {z^2\over c^2-\lambda} = 1$$ (9)

$${x^2\over a^2-\mu} + {y^2\over b^2-\mu} + {z^2\over c^2-\mu} = 1$$ (10)

$${x^2\over a^2-\nu} + {y^2\over b^2-\nu} + {z^2\over c^2-\nu} = 1,$$ (11)

where

$$\lambda < c^2 < \mu < b^2 < \nu < a^2$$ (12)

(Arfken 1970, pp. 117-118). Byerly (1959, p. 251) uses a slightly different definition in which the Greek variables are replaced by their squares, and $a=0$. Equation (9) represents an Ellipsoid, (10) represents a one-sheeted Hyperboloid, and (11) represents a two-sheeted Hyperboloid. In terms of Cartesian Coordinates,

$$x^2 = {(a^2-\lambda)(a^2-\mu)(a^2-\nu)\over (a^2-b^2)(a^2-c^2)}$$ (13)

$$y^2 = {(b^2-\lambda)(b^2-\mu)(b^2-\nu)\over (b^2-a^2)(b^2-c^2)}$$ (14)

$$z^2 = {(c^2-\lambda)(c^2-\mu)(c^2-\nu)\over (c^2-a^2)(c^2-b^2)}.$$ (15)

The Scale Factors are

$${h_\lambda } = \sqrt{(\mu-\lambda)(\nu-\lambda)\over 4(a^2-\lambda)(b^2-\lambda) (c^2-\lambda)}$$ (16)

$${h_\mu} = \sqrt{(\nu-\mu)(\lambda-\mu)\over 4(a^2-\mu)(b^2-\mu)(c^2-\mu)}$$ (17)

$${h_\nu} = \sqrt{(\lambda-\nu)(\mu-\nu)\over 4(a^2-\nu)(b^2-\nu)(c^2-\nu)}.$$ (18)

The Laplacian is

$$\nabla^2 = 2{a^2b^2+a^2c^2+b^2c^2-2\nu(a^2+b^2+c^2)+3\nu^2\over(\mu-\nu)(\nu-\lambda)}{\partial\over\partial\nu}$$

$$\phantom{=} +{4(a^2-\nu)(b^2-\nu)(c^2-\nu)\over(\mu-\nu)(\nu-\lambda)}{\partial^2\over\partial\nu^2}\hfill$$

$$\phantom{=} +2{a^2b^2+a^2c^2+b^2c^2-2\mu(a^2+b^2+c^2)+3\mu^2\over(\nu-\mu)(\mu-\lambda)}{\partial\over\partial\mu}$$

$$\phantom{=} +{4(a^2-\mu)(b^2-\mu)(c^2-\mu)\over(\mu-\lambda)(\nu-\mu)}{\partial^2\over\partial\mu^2}$$

$$\phantom{=} +2{-(a^2b^2+a^2c^2+b^2c^2)+2\lambda(a^2+b^2+c^2)-3\lambda^2\over(\mu-\lambda)(\nu-\lambda)}{\partial\over\partial\lambda}$$

$$\phantom{=} +{4(a^2-\lambda)(b^2-\lambda)(c^2-\lambda)\over(\mu-\lambda)(\nu-\lambda)}{\partial^2\over\partial\lambda^2}.$$ (19)

Using the Notation of Byerly (1959, pp. 252-253), this can be reduced to

$$\nabla^2=(\mu^2-\nu^2){\partial^2\over\partial\alpha^2}+(\la... ...\beta^2} +(\lambda^2-\mu^2){\partial^2\over\partial\gamma^2},$$ (20)

where

$$\alpha = c\int_c^\lambda {d\lambda\over\sqrt{(\lambda^2-b^2)(\lambda^2-c^2)}}$$

$$= F\left({{b\over c}, {\pi\over 2}}\right)-F\left({{b\over c},\sin^{-1}\left({c\over\lambda}\right)}\right)$$ (21)

$$\beta = c\int_b^\mu {d\mu\over\sqrt{(c^2-\mu^2)(\mu^2-b^2)}}$$

$$= F\left[{\sqrt{1-{b^2\over c^2}}, \sin^{-1}\left({\sqrt{1-{b^2\over\mu^2}\over 1-{b^2\over c^2}}\,}\right)}\right]$$ (22)

$$\gamma = c\int_0^\nu{d\nu\over\sqrt{(b^2-\nu^2)(c^2-\nu^2)}}$$

$$= F\left({{b\over c}, \sin^{-1}\left({\nu\over b}\right)}\right).$$ (23)

Here, $F$ is an Elliptic Integral of the First Kind. In terms of $\alpha$, $\beta$, and $\gamma$,

$$\lambda = c\mathop{\rm dc}\nolimits \left({\alpha, {b\over c}}\right)$$ (24)

$$\mu = b\mathop{\rm nd}\nolimits \left({\beta,\sqrt{1-{b^2\over c^2}}}\right)$$ (25)

$$\nu = b\mathop{\rm sn}\nolimits \left({\gamma,{b\over c}}\right),$$ (26)

where $\mathop{\rm dc}\nolimits$, $\mathop{\rm nd}\nolimits$ and $\mathop{\rm sn}\nolimits$ are Jacobi Elliptic Functions. The Helmholtz Differential Equation is separable in confocal ellipsoidal coordinates.

See also Helmholtz Differential Equation--Confocal Ellipsoidal Coordinates

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Elliptical Coordinates.'' §21.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.

Arfken, G. ``Confocal Ellipsoidal Coordinates $(\xi_1, \xi_2, \xi_3)$.'' §2.15 in Mathematical Methods for Physicists, 2nd ed. New York: Academic Press, pp. 117-118, 1970.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.