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Crossed Trough

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CrossedTrough

The crossed trough is the surface

z=x^2y^2.
(1)

The coefficients of its first fundamental form are

E=1+4x^2y^4
(2)
F=4x^3y^3
(3)
G=1+4x^4y^2
(4)

and of the second fundamental form are

e=(2y^2)/(sqrt(1+4x^2y^2(x^2+y^2)))
(5)
f=(4xy)/(sqrt(1+4x^2y^2(x^2+y^2)))
(6)
g=(2x^2)/(sqrt(1+4x^2y^2(x^2+y^2))).
(7)

The Gaussian and mean curvatures are

K=-(12x^2y^2)/([1+4x^2y^2(x^2+y^2)]^2)
(8)
H=(x^2+y^2-8x^4y^4)/([1+4x^2y^2(x^2+y^2)]^(3/2)).
(9)

The Gaussian curvature can be written implicitly as

K(x,y,z)=-(12a^6x^2y^2)/((a^6+4x^4y^2+4x^2y^4)^2).
(10)

See also

Monkey Saddle

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References

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 286, 1993.

Cite this as:

Weisstein, Eric W. "Crossed Trough." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CrossedTrough.html

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