A cone with elliptical cross section. The parametric equations for an elliptic cone of height , semimajor axis , and semiminor axis are
where and .
The elliptic cone is a quadratic ruled surface, and has volume
(4)
The coefficients of the first fundamental form
second fundamental form coefficients
The lateral surface area can then be calculated as
where is a complete elliptic integral of the second kind and assuming .
The Gaussian curvature is
(14)
and the mean curvature is
(15)
See alsoCone,
Elliptic Cylinder,
Elliptic Paraboloid,
Hyperbolic Paraboloid,
Quadratic Surface,
Ruled Surface Explore with Wolfram|Alpha ReferencesBeyer, 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|AlphaElliptic Cone Cite this as:Weisstein, Eric W. "Elliptic Cone." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticCone.html
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