Elliptic cylindrical 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

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

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

$z^2=\frac{(c^2+\lambda )(c^2+\mu )(c^2+\nu )}{(c^2-b^2)(c^2-a^2)}$

where the following limits apply to the coordinates

$-\lambda <c^2<-\mu <b^2<-\nu <a^2.$

Consequently, surfaces of constant $\lambda$ are ellipsoids

$\frac{x^2}{a^2+\lambda }+\frac{y^2}{b^2+\lambda }+\frac{z^2}{c^2+\lambda }=1,$

whereas surfaces of constant $\mu$ are hyperboloids of one sheet

$\frac{x^2}{a^2+\mu }+\frac{y^2}{b^2+\mu }+\frac{z^2}{c^2+\mu }=1,$

because the last term in the lhs is negative, and surfaces of constant $\nu$ are hyperboloids of two sheets

$\frac{x^2}{a^2+\nu }+\frac{y^2}{b^2+\nu }+\frac{z^2}{c^2+\nu }=1$

because the last two terms in the lhs are negative.

Scale factors and differential operators

For brevity in the equations below, we introduce a function

$S(\sigma )\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}(a^2+\sigma )(b^2+\sigma )(c^2+\sigma )$

where $\sigma$ can represent any of the three variables $(\lambda ,\mu ,\nu )$. Using this function, the scale factors can be written

$h_{\lambda }=\frac{1}{2}\sqrt{\frac{(\lambda -\mu )(\lambda -\nu )}{S(\lambda )}}$

$h_{\mu }=\frac{1}{2}\sqrt{\frac{(\mu -\lambda )(\mu -\nu )}{S(\mu )}}$

$h_{\nu }=\frac{1}{2}\sqrt{\frac{(\nu -\lambda )(\nu -\mu )}{S(\nu )}}$

Hence, the infinitesimal volume element equals

$dV=\frac{(\lambda -\mu )(\lambda -\nu )(\mu -\nu )}{8\sqrt{-S(\lambda )S(\mu )S(\nu )}}d\lambda d\mu d\nu$

and the Laplacian is defined by

${\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{4\sqrt{S(\lambda )}}{(\lambda -\mu )(\lambda -\nu )}\frac{\partial }{\partial \lambda }\left[\sqrt{S(\lambda )}\frac{\partial \mathrm{\Phi }}{\partial \lambda }\right]+$

$\frac{4\sqrt{S(\mu )}}{(\mu -\lambda )(\mu -\nu )}\frac{\partial }{\partial \mu }\left[\sqrt{S(\mu )}\frac{\partial \mathrm{\Phi }}{\partial \mu }\right]+\frac{4\sqrt{S(\nu )}}{(\nu -\lambda )(\nu -\mu )}\frac{\partial }{\partial \nu }\left[\sqrt{S(\nu )}\frac{\partial \mathrm{\Phi }}{\partial \nu }\right]$

Other differential operators such as $\mathrm{\nabla }\cdot \mathbf{F}$ and $\mathrm{\nabla }\times \mathbf{F}$ can be expressed in the coordinates $(\lambda ,\mu ,\nu )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

See also

• Focaloid (shell given by two coordinate surfaces)

References

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Bibliography

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Unusual convention

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

• MathWorld description of confocal ellipsoidal coordinates