Catenoid

A Catenary of Revolution. The catenoid and Plane are the only Surfaces of Revolution which are also Minimal Surfaces. The catenoid can be given by the parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle c \cosh\left({v\over c}\right)\cos u$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle c \cosh\left({v\over c}\right)\sin u$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle v,$	(3)

where $u\in[0,2\pi)$ . The differentials are

$\displaystyle dx$	$\textstyle =$	$\displaystyle \sinh\left({v\over c}\right)\cos u\,dv-\cosh\left({v\over c}\right)\sin u\,du$	(4)
$\displaystyle dy$	$\textstyle =$	$\displaystyle \sinh\left({v\over c}\right)\sin u\,dv+\cosh\left({v\over c}\right)\cos u\,du$	(5)
$\displaystyle dz$	$\textstyle =$	$\displaystyle du,$	(6)

so the Line Element is

$\displaystyle ds^2$	$\textstyle =$	$\displaystyle dx^2+dy^2+dz^2$
	$\textstyle =$	$\displaystyle \left[{\sinh^2\left({v\over c}\right)+1}\right]\,dv^2+\cosh^2\left({v\over c}\right)\,du^2$
	$\textstyle =$	$\displaystyle \cosh^2\left({v\over c}\right)\,dv^2+\cosh^2\left({v\over c}\right)\,du^2.$	(7)

$\displaystyle \kappa_1$	$\textstyle =$	$\displaystyle -{1\over c}\mathop{\rm sech}\nolimits ^2\left({v\over c}\right)$	(8)
$\displaystyle \kappa_2$	$\textstyle =$	$\displaystyle {1\over c}\mathop{\rm sech}\nolimits ^2\left({v\over c}\right).$	(9)

The Mean Curvature of the catenoid is

$\begin{displaymath} H=0 \end{displaymath}$

(10)

$\begin{displaymath} K=-{1\over c^2}\mathop{\rm sech}\nolimits ^4\left({v\over c}\right). \end{displaymath}$

(11)

$\begin{figure}\begin{center}\BoxedEPSF{HelicoidCatenoid.epsf}\end{center}\end{figure}$

The Helicoid can be continuously deformed into a catenoid with by the transformation

$\displaystyle x(u,v)$	$\textstyle =$	$\displaystyle \cos\alpha\sinh v\sin u+\sin\alpha\cosh v\cos u$	(12)
$\displaystyle y(u,v)$	$\textstyle =$	$\displaystyle -\cos\alpha\sinh v\cos u+\sin\alpha\cosh v\sin u$	(13)
$\displaystyle z(u,v)$	$\textstyle =$	$\displaystyle u\cos\alpha+v\sin\alpha,$	(14)

where $\alpha=0$ corresponds to a Helicoid and $\alpha=\pi/2$ to a catenoid.

References

do Carmo, M. P. ``The Catenoid.'' §3.5A in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.

Fischer, G. (Ed.). Plate 90 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 86, 1986.

Gray, A. ``The Catenoid.'' §18.4 Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 367-369, 1993.

Meusnier, J. B. ``Mémoire sur la courbure des surfaces.'' Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.