Catenoid
A Catenary of Revolution. The catenoid and Plane are the only Surfacesof Revolution which are also Minimal Surfaces. The catenoid can be given bythe parametric equations
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
where
. The differentials are | |  | (4) |
 | |  | (5) |
 | |  | (6) |
so the Line Element is
 | |  | |
|  | ||
|  | (7) |
The Principal Curvatures are
 | |  | (8) |
 | |  | (9) |
The Mean Curvature of the catenoid is
 | (10) |
 | (11) |
The Helicoid can be continuously deformed into a catenoid with 
by the transformation
 | |  | (12) |
 | |  | (13) |
 | |  | (14) |
where
corresponds to a Helicoid and 
to a catenoid.See also Catenary, Costa Minimal Surface, Helicoid, Minimal Surface, Surface of Revolution
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. Geometry Center. ``The Catenoid.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/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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