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

 (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)

and the Gaussian Curvature is
 (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.