Bispherical coordinates

Paraboloidal coordinates are a three-dimensional orthogonal coordinate system $(\lambda ,\mu ,\nu )$ that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

File:Parabolic coordinates 3D.png

Coordinate surfaces of the three-dimensional paraboloidal coordinates.

Basic formulae

The Cartesian coordinates $(x,y,z)$ can be produced from the ellipsoidal coordinates $(\lambda ,\mu ,\nu )$ by the equations

$x^2=\frac{(A-\lambda )(A-\mu )(A-\nu )}{B-A}$

$y^2=\frac{(B-\lambda )(B-\mu )(B-\nu )}{A-B}$

$z=\frac{1}{2}(A+B-\lambda -\mu -\nu )$

where the following limits apply to the coordinates

$\lambda <B<\mu <A<\nu$

Consequently, surfaces of constant $\lambda$ are elliptic paraboloids

$\frac{x^2}{\lambda -A}+\frac{y^2}{\lambda -B}=2z+\lambda$

and surfaces of constant $\nu$ are likewise

$\frac{x^2}{\nu -A}+\frac{y^2}{\nu -B}=2z+\nu$

whereas surfaces of constant $\mu$ are hyperbolic paraboloids

$\frac{x^2}{\mu -A}+\frac{y^2}{\mu -B}=2z+\mu$

Scale factors

The scale factors for the paraboloidal coordinates $(\lambda ,\mu ,\nu )$ are

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

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

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

Hence, the infinitesimal volume element equals

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

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.

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.
  
  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
• 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 Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• 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

External links

• MathWorld description of confocal paraboloidal coordinates