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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)

The Principal Curvatures are
$\displaystyle \kappa_1$ $\textstyle =$ $\displaystyle -{1\over c}\mathop{\rm sech}\nolimits ^2\left({v\over c}\right)$ (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
\end{displaymath} (10)

and the Gaussian Curvature is
K=-{1\over c^2}\mathop{\rm sech}\nolimits ^4\left({v\over c}\right).
\end{displaymath} (11)


The Helicoid can be continuously deformed into a catenoid with $c=1$ 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.

See also Catenary, Costa Minimal Surface, Helicoid, Minimal Surface, Surface of Revolution


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.

Geometry Center. ``The Catenoid.''

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.

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