Funnel
Funnel
The funnel surface is a regular surface and surface of revolution defined by the Cartesian equation
| z=1/2aln(x^2+y^2) |
(1)
|
and the parametric equations
x(u,v) = ucosv
(2)
y(u,v) = usinv
(3)
z(u,v) = alnu
(4)
for u>0 and v in [0,2pi). The coefficients of the first fundamental form are
the coefficients of the second fundamental form are
the area element is
| dA=sqrt(a^2+u^2)du ^ dv, |
(11)
|
and the Gaussian and mean curvatures are
K = [画像:-(a^2)/((a^2+u^2)^2)]
(12)
H = [画像:(a^3)/(2u(a^2+u^2)^(3/2)).]
(13)
The Gaussian curvature can be given implicitly as
Both the surface area and volume of the solid are infinite.
See also
Dini's Surface, Gabriel's Horn, Pseudosphere, Sinclair's Soap Film ProblemExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
FunnelCite this as:
Weisstein, Eric W. "Funnel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Funnel.html