The crossed trough is the surface
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(1)
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The coefficients of its first fundamental form are
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(2)
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(3)
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(4)
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and of the second fundamental form are
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(5)
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(6)
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(7)
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The Gaussian and mean curvatures are
 |  | ![-(12x^2y^2)/([1+4x^2y^2(x^2+y^2)]^2)](/images/equations/CrossedTrough/Inline21.svg) |
(8)
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 |  | ![(x^2+y^2-8x^4y^4)/([1+4x^2y^2(x^2+y^2)]^(3/2)).](/images/equations/CrossedTrough/Inline24.svg) |
(9)
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The Gaussian curvature can be written implicitly as
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(10)
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