TOPICS
Search

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

E=1+(a^2)/(u^2)
(5)
F=0
(6)
G=u^2,
(7)

the coefficients of the second fundamental form are

e=-a/(usqrt(a^2+u^2))
(8)
f=0
(9)
g=(au)/(sqrt(a^2+u^2)),
(10)

the area element is

 dA=sqrt(a^2+u^2)du ^ dv,
(11)

and the Gaussian and mean curvatures are

K=-(a^2)/((a^2+u^2)^2)
(12)
H=(a^3)/(2u(a^2+u^2)^(3/2)).
(13)

The Gaussian curvature can be given implicitly as

 K(x,y,z)=-(a^2)/((a^2+e^(2z/a))^2).
(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

Explore with Wolfram|Alpha

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.

Cite this as:

Weisstein, Eric W. "Funnel." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Funnel.html

Subject classifications