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
where u in [0,2pi).
The line element is
The first fundamental form has coefficients
and the second fundamental form has coefficients
The principal curvatures are
The mean curvature of the catenoid is
| H=0 |
(13)
|
and the Gaussian curvature is
| [画像: K=-1/(c^2)sech^4(v/c). ] |
(14)
|
The helicoid can be continuously deformed into a catenoid with c=1 by the transformation
where alpha=0 corresponds to a helicoid and alpha=pi/2 to a catenoid.
This deformation is illustrated on the cover of issue 2, volume 2 of The Mathematica Journal.
See also
Catenary, Costa Minimal Surface, Helicoid, Minimal Surface, Surface of RevolutionExplore with Wolfram|Alpha
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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 aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 86, 1986.Geometry Center. "The Catenoid." http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/catenoid/.GRAPE. "Catenoid." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/catenoid.html.GRAPE. "Catenoid-Helicoid Deformation." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/cathel.html.Gray, A. "The Catenoid." §20.4 Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 467-469, 1997.JavaView. "Classic Surfaces from Differential Geometry: Catenoid/Helicoid." http://www.javaview.de/demo/surface/common/PaSurface_CatenoidHelicoid.html.Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 18 1986.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 247-249, 1999.Referenced on Wolfram|Alpha
CatenoidCite this as:
Weisstein, Eric W. "Catenoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Catenoid.html