TOPICS
Search

Catenoid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
catenoid

A catenoid is the surface of revolution generated by revolving a catenary about its axis. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations

x=ccosh(v/c)cosu
(1)
y=ccosh(v/c)sinu
(2)
z=v,
(3)

where u in [0,2pi).

The line element is

ds^2=cosh^2(v/c)dv^2+c^2cosh^2(v/c)du^2.
(4)

The first fundamental form has coefficients

E=c^2cosh^2(v/c)
(5)
F=0
(6)
G=cosh^2(v/c),
(7)

and the second fundamental form has coefficients

e=-c
(8)
f=0
(9)
g=1/c.
(10)

The principal curvatures are

kappa_1=1/csech^2(v/c)
(11)
kappa_2=-1/csech^2(v/c).
(12)

The mean curvature of the catenoid is

H=0
(13)

and the Gaussian curvature is

K=-1/(c^2)sech^4(v/c).
(14)
HelicoidCatenoid

The helicoid can be continuously deformed into a catenoid with c=1 by the transformation

x(u,v)=cosalphasinhvsinu+sinalphacoshvcosu
(15)
y(u,v)=-cosalphasinhvcosu+sinalphacoshvsinu
(16)
z(u,v)=ucosalpha+vsinalpha,
(17)

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.

Catenoid-like forms occur in architectural applications of minimal surfaces (Bock Hyeng et al. 2025).

See also

Catenary, Costa Minimal Surface, Critical Catenoid, Helicoid, Minimal Surface, Minimal Surface of Revolution, Surface of Revolution

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Bock Hyeng, C. A.; Krivoshapko, S. N.; Kouamou Nguessi, A.; Yamb Bell, E.; and Bahel, B. "Application of Curvilinear Analytical Surfaces in Forms of Architectural Objects and Machine Building Products." Int. J. Archit. Arts Appl. 11, 19-35, 2025. https://doi.org/10.11648/j.ijaaa.20251101.13.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

Catenoid

Cite this as:

Weisstein, Eric W. "Catenoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Catenoid.html

Subject classifications