Bispherical Coordinates

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A system of Curvilinear Coordinates defined by

$\displaystyle x$	$\textstyle =$	$\displaystyle {a\sin\xi\cos\phi\over\cosh\eta-\cos\xi}$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {a\sin\xi\sin\phi\over\cosh\eta-\cos\xi}$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle {a\sinh\eta\over\cosh\eta-\cos\xi}.$	(3)

The Scale Factors are

$\displaystyle h_\xi$	$\textstyle =$	$\displaystyle {a\over \cos\eta-\cos\xi}$	(4)
$\displaystyle h_\eta$	$\textstyle =$	$\displaystyle {a\over \cosh\eta-\cos\xi}$	(5)
$\displaystyle h_\phi$	$\textstyle =$	$\displaystyle {a\sin\xi\over \cosh\eta-\cos\xi}.$	(6)

The Laplacian is

$\nabla^2= \left({-\cos u\cot^2 u+3\cosh v\cot^2 u-3\cosh^2 v\cot u\csc u+\cosh^3 v\csc^2 u\over \cosh v-\cos u}\right){\partial\over\partial\phi^2}$

$\quad +(\cos u-\cosh v)\sinh v{\partial\over\partial v}+(\cosh^2 v-\cos u)^2{\partial^2\over\partial v^2}$

$\quad +(\cosh v-\cos u)(\cosh v\cot u-\sin u-\cos u\cot u){\partial\over\partial u}$

$\quad +(\cosh^2 v-\cos u)^2{\partial^2\over\partial u^2}.$ (7)

In bispherical coordinates, Laplace's Equation is separable, but the Helmholtz Differential Equation is not.

References

Arfken, G. ``Bispherical Coordinates ( $\xi$ , $\eta$ , $\phi$ ).'' §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.

$\nabla^2= \left({-\cos u\cot^2 u+3\cosh v\cot^2 u-3\cosh^2 v\cot u\csc u+\cosh^3 v\csc^2 u\over \cosh v-\cos u}\right){\partial\over\partial\phi^2}$
$\quad +(\cos u-\cosh v)\sinh v{\partial\over\partial v}+(\cosh^2 v-\cos u)^2{\partial^2\over\partial v^2}$
$\quad +(\cosh v-\cos u)(\cosh v\cot u-\sin u-\cos u\cot u){\partial\over\partial u}$
$\quad +(\cosh^2 v-\cos u)^2{\partial^2\over\partial u^2}.$	(7)