Catenoid

A catenoid is a surface in 3-dimensional Euclidean space arising by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744.[1] Early work on the subject was published also by Jean Baptiste Meusnier.[2] There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[3]

The catenoid may be defined by the following parametric equations:

\( x=c \cosh \frac{v}{c} \cos u \)

\( y=c \cosh \frac{v}{c} \sin u \)

\( z=v \)

where u and v are real parameters and c is a non-zero real constant.

In cylindrical coordinates:

\( \rho =c \cosh \frac{z}{c} \)

Where c is a real constant.

A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.

The catenoid may be also defined approximately by the Stretched grid method as a facet 3D model
Helicoid transformation
Animation showing the deformation of a helicoid into a catenoid.

Because they are members of the same associate family of surfaces, one can bend a catenoid into a portion of a helicoid without stretching. In other words, one can make a (mostly) continuous and isometric deformation of a catenoid to a portion of the helicoid such that every member of the deformation family is minimal (having a mean curvature of zero). A parametrization of such a deformation is given by the system

\( x(u,v) = \cos \theta \,\sinh v \,\sin u + \sin \theta \,\cosh v \,\cos u \)

\( y(u,v) = -\cos \theta \,\sinh v \,\cos u + \sin \theta \,\cosh v \,\sin u \)

\( z(u,v) = u \cos \theta + v \sin \theta \, \)

for \( (u,v) \in (-\pi, \pi] \times (-\infty, \infty) \), with deformation parameter \( -\pi < \theta \le \pi, \)

where \( \theta = \pi \)corresponds to a right-handed helicoid, \( \theta = \pm \pi / 2 \)corresponds to a catenoid, and \( \theta = 0 \) corresponds to a left-handed helicoid.

References

L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, in: Opera omnia I, 24
Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785

Catenoid at MathWorld

External links

Hazewinkel, Michiel, ed. (2001), "Catenoid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Caténoïde" at Encyclopédie des Formes Mathématiques Remarquables (in French)
Animated 3D WebGL model of a catenoid

Mathematics Encyclopedia