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Hyperbolic Cylinder

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HyperbolicCylinder

The hyperbolic cylinder is a quadratic surface given by the equation

(x^2)/(a^2)-(y^2)/(b^2)=-1.
(1)

It is a ruled surface.

It can be given parametrically by

x=asinhu
(2)
y=bcoshu
(3)
z=v.
(4)

The coefficients of the first fundamental form are

E=a^2cosh^2u+b^2sinh^2u
(5)
F=0
(6)
G=1,
(7)

and of the second fundamental form are

e=-(ab)/(sqrt(a^2cosh^2u+b^2sinh^2u))
(8)
f=0
(9)
g=0.
(10)

The Gaussian and mean curvatures are

K=0
(11)
H=-(ab)/(2(a^2cosh^2u+b^2sinh^2u)^(3/2)).
(12)

See also

Cylinder, Elliptic Paraboloid, Hyperbola, Paraboloid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 210-211, 1987.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 12, 1999.

Cite this as:

Weisstein, Eric W. "Hyperbolic Cylinder." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCylinder.html

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