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Elliptic Cylinder

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An elliptic cylinder is a cylinder with an elliptical cross section.

The elliptic cylinder is a quadratic ruled surface.

The parametric equations for the laterals sides of an elliptic cylinder of height h, semimajor axis a, and semiminor axis b are

x=acosu
(1)
y=bsinu
(2)
z=v,
(3)

where u in [0,2pi) and v in [0,h].

The volume of the elliptic cylinder is

V=piabh.
(4)

The coefficients of the first fundamental form are

E=b^2cos^2u+a^2sin^2u
(5)
F=0
(6)
G=1
(7)

and of the second fundamental form are

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

The area element is

dA=sqrt(a^2sin^2u+b^2cos^2u)dudv.
(11)

The Gaussian and mean curvatures are

K=0
(12)
H=-(ab)/(2(a^2sin^2u+b^2cos^2u)^(3/2)).
(13)

See also

Cone, Cylinder, Elliptic Cone, Elliptic Paraboloid, Quadratic Surface, Ruled Surface

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 12, 1999.

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Elliptic Cylinder

Cite this as:

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

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