Catenoid: Definition from Answers.com

A catenoid

A catenoid is a three-dimensional shape made by rotating a catenary curve around the $x$ axis. 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 Meusnier.^[2] There are only two surfaces of revolution which are also minimal surfaces: the plane and the catenoid.^[3]

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

Animation showing the deformation of a helicoid into a catenoid. Generated with Mac OS X Grapher.

One can bend a catenoid into the shape of a helicoid without stretching. In other words, one can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal. 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 $θ = π$ corresponds to a right-handed helicoid, $\theta = \pm \pi / 2$ corresponds to a catenoid, and $θ = 0$ 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

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