The coordinates are the asymptotic angle of confocal hyperbolic cylinders symmetrical about the x-axis. The coordinates are confocal elliptic cylinders centered on the origin.
(1)
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(2)
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(3)
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where , , and . They are related to Cartesian coordinates by
(4)
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(5)
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The scale factors are
(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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The matrices of Christoffel symbols of the second kind in the sense of Misner et al. (1973) are given by
(13)
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(14)
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(15)
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The Jacobian is
(16)
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The Laplacian is
(17)
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Let
(18)
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(19)
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(20)
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Then the new scale factors are
(21)
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(22)
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(23)
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The Helmholtz differential equation is separable in elliptic cylindrical coordinates.