Catenoid
From Wikipedia, the free encyclopedia
A catenoid is a three-dimensional shape made by rotating a catenary curve around the <math>x</math> axis.
A catenoid is one of several types of minimal surfaces.
A physical model of a catenoid can be formed by dipping two circles into a soap solution, (popping any film in the center of the circle), and slowly drawing the circles apart.
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. An explicit parameterization of such a deformation is given by the system
<math>x(u,v) = \cos \theta \,\sinh v \,\sin u + \sin \theta \,\cosh v \,\cos u</math>
<math>y(u,v) = -\cos \theta \,\sinh v \,\cos u + \sin \theta \,\cosh v \,\sin u</math>
<math>z(u,v) = u \cos \theta + v \sin \theta \,</math>
for <math>(u,v) \in (-\pi, \pi] \times (-\infty, \infty)</math>, with deformation parameter <math>-\pi < \theta \le \pi</math>.fr:Caténoïde ru:Катеноид
