Funnel

The funnel surface is a Regular Surface defined by the Cartesian equation

$$z={\textstyle{1\over 2}}\ln(x^2+y^2)$$ (1)

and the parametric equations

$$x(r,\theta) = r\cos\theta$$ (2)

$$y(r,\theta) = r\sin\theta$$ (3)

$$z(r,\theta) = \ln r.$$ (4)

See also Gabriel's Horn, Pseudosphere, Sinclair's Soap Film Problem

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 325-327, 1993.