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Elliptic Cone

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EllipticCone

A cone with elliptical cross section. The parametric equations for an elliptic cone of height h, semimajor axis a, and semiminor axis b are

x=a(h-u)/hcosv
(1)
y=b(h-u)/hsinv
(2)
z=u,
(3)

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

The elliptic cone is a quadratic ruled surface, and has volume

V=1/3piabh.
(4)

The coefficients of the first fundamental form

E=(h^2+a^2cos^2v+b^2sin^2v)/(h^2)
(5)
F=((a^2-b^2)(h-u)cosvsinv)/(h^2)
(6)
G=((h-u)^2(a^2sin^2v+b^2cos^2v))/(h^2),
(7)

second fundamental form coefficients

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

The lateral surface area can then be calculated as

S=int_0^(2pi)int_0^hsqrt(EG-F^2)dudv
(11)
=2asqrt(b^2+h^2)E(sqrt((1-(b^2)/(a^2))/(1+(b^2)/(h^2))))
(12)
=2bsqrt(a^2+h^2)E(sqrt((1-(a^2)/(b^2))/(1+(a^2)/(h^2)))),
(13)

where E(k) is a complete elliptic integral of the second kind and assuming 0<b<a.

The Gaussian curvature is

K=0,
(14)

and the mean curvature is

M= (sqrt(2)abh^2(h^2+a^2cos^2v+b^2sin^2v))/((h-u)[2a^2b^2+(a^2+b^2)h^2+(b^2-a^2)h^2cos(2v)]^(3/2)).
(15)

See also

Cone, Elliptic Cylinder, Elliptic Paraboloid, Hyperbolic 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. 226, 1987.Fischer, G. (Ed.). Plate 68 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 63, 1986.

Referenced on Wolfram|Alpha

Elliptic Cone

Cite this as:

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

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