Английская Википедия:Catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution).[1] It is a minimal surface, meaning that it occupies the least area when bounded by a closed space.[2] It was formally described in 1744 by the mathematician Leonhard Euler.
Soap film attached to twin circular rings will take the shape of a catenoid.[2] Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa.
Geometry
The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix.[2] It was found and proved to be minimal by Leonhard Euler in 1744.[3][4]
Early work on the subject was published also by Jean Baptiste Meusnier.[5][4]Шаблон:Rp There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[6]
The catenoid may be defined by the following parametric equations: <math display=block>\begin{align} x &= c \cosh \frac{v}{c} \cos u \\ y &= c \cosh \frac{v}{c} \sin u \\ z &= v \end{align}</math> where <math>u \in [-\pi, \pi)</math> and <math>v \in \mathbb{R}</math> and <math>c</math> is a non-zero real constant.
In cylindrical coordinates: <math display=block>\rho =c \cosh \frac{z}{c},</math> where <math>c</math> is a real constant.
A physical model of a catenoid can be formed by dipping two circular rings 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
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 <math display=block>\begin{align} 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 \end{align}</math> for <math>(u,v) \in (-\pi, \pi] \times (-\infty, \infty)</math>, with deformation parameter <math>-\pi < \theta \le \pi</math>, where:
- <math>\theta = \pi</math> corresponds to a right-handed helicoid,
- <math>\theta = \pm \pi / 2</math> corresponds to a catenoid, and
- <math>\theta = 0</math> corresponds to a left-handed helicoid.
References
Further reading
External links
- Шаблон:Springer
- Catenoid – WebGL model
- Euler's text describing the catenoid at Carnegie Mellon University
- Calculating the surface area of a Catenoid
- Minimal Surface of Revolution
