The funnel surface is a regular surface and surface of revolution defined by the Cartesian equation
 |
(1) |
and the parametric equations
 |  |  |
(2) |
 |  |  |
(3) |
 |  |  |
(4) |
for 
and 
. The coefficients of the first
fundamental form are
 |  |  |
(5) |
 |  |  |
(6) |
 |  |  |
(7) |
the coefficients of the second fundamental form are
 |  |  |
(8) |
 |  |  |
(9) |
 |  |  |
(10) |
the area element is
 |
(11) |
and the Gaussian and mean curvatures are
 |  |  |
(12) |
 |  |  |
(13) |
The Gaussian curvature can be given implicitly as
 |
(14) |
Both the surface area and volume of the solid are infinite.
See also
Dini's Surface, Gabriel's Horn, Pseudosphere, Sinclair's Soap Film Problem
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References
Gray, A. "The Funnel Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 423-426, 1997.
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Cite this as:
Weisstein, Eric W. "Funnel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Funnel.html