Confocal Paraboloidal Coordinates

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

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

$${x^2\over a^2-\nu} + {y^2\over b^2-\nu} = z-\nu,$$ (3)

where $\lambda \in (-\infty ,b^2)$, $\mu \in (b^2,a^2)$, and $\nu \in (a^2,\infty)$.

$$x^2 = {(a^2-\lambda)(a^2-\mu)(a^2-\nu)\over(b^2-a^2)}$$ (4)

$$y^2 = {(b^2-\lambda)(b^2-\mu)(b^2-\nu)\over(a^2-b^2)}$$ (5)

$$z = \lambda+\mu+\nu-a^2-b^2.$$ (6)

The Scale Factors are

$${h_\lambda } = \sqrt{(\mu-\lambda)(\nu-\lambda)\over 4(a^2-\lambda)(b^2-\lambda)}$$ (7)

$${h_\mu} = \sqrt{(\nu-\mu)(\lambda-\mu)\over 4(a^2-\mu)(b^2-\mu)}$$ (8)

$${h_\nu} = \sqrt{(\lambda-\nu)(\mu-\nu)\over 16(a^2-\nu)(b^2-\nu)}.$$ (9)

The Laplacian is

${2(a^2+b^2-2\nu)\over(\mu-\nu)(\nu-\lambda)}{\partial\over\partial\nu} +{4(a^2-\nu)(\nu-b^2)\over(\mu-\nu)(\nu-\lambda)}{\partial^2\over\nu^2}$
$+{2(a^2+b^2-2\mu)\over(\mu-\lambda)(\nu-\mu)}{\partial\over\partial\mu}+{4(a^2-\mu)(\mu-b^2)\over(\mu-\lambda)(\nu-\mu)}{\partial^2\over\partial\mu^2}$
$+{2(2\lambda-a^2-b^2)\over(\mu-\lambda)(\nu-\lambda)}{\partial\over\partial\lam... ...da-b^2)\over(\mu-\lambda)(\nu-\lambda)}{\partial^2\over\partial\lambda^2}.\quad$	(10)

The Helmholtz Differential Equation is Separable.

See also Helmholtz Differential Equation--Confocal Paraboloidal Coordinates

References

Arfken, G. ``Confocal Parabolic Coordinates ($\xi_1$, $\xi_2$, $\xi_3$).'' §2.17 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 119-120, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 664, 1953.