Catenoid

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 [L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, in: Opera omnia I, 24] . Early work on the subject was published also by Meusnier [Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785] . There are only two surfaces of revolution which are also minimal surfaces: the plane and the catenoid [ [http://mathworld.wolfram.com/Catenoid.html Catenoid at MathWorld] ] .

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

[
thumb|right|256px|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 heta ,sinh v ,sin u + sin heta ,cosh v ,cos u

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

z(u,v) = u cos heta + v sin heta ,

for (u,v) in (-pi, pi] imes (-infty, infty), with deformation parameter -pi < heta le pi,

where heta = pi corresponds to a right-handed helicoid, heta = pm pi / 2 corresponds to a catenoid, heta = pm pi corresponds to a left-handed helicoid,

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • catenoid — /kat n oyd /, n. Geom. the surface generated by rotating a catenary about its axis of symmetry. [1875 80; < L caten(a) a chain + OID] * * * …   Universalium

  • catenoid — noun A three dimensional surface formed by rotation of a catenary …   Wiktionary

  • catenoid — 1. Like a chain, such as a chain of fungus spores or a colony of protozoa in which the individuals are joined end to end. SYN: catenulate. 2. Surface of net zero curvature generated by the rotation of a catenary (curve of repose of a suspended… …   Medical dictionary

  • catenoid — adj. linked together …   English contemporary dictionary

  • catenoid — cat·e·noid …   English syllables

  • catenoid — I. ˈ ̷ ̷əˌnȯid, ˈ ̷ ̷ənˌȯid noun ( s) Etymology: Latin catena + English oid : the surface described by the rotation of a catenary about its axis II. adjective Etymology …   Useful english dictionary

  • Helicoid — A helicoid with α=1, 1≤ρ≤1 and π≤θ≤π. The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for… …   Wikipedia

  • Catenary — This article is about the mathematical curve. For other uses, see Catenary (disambiguation). Chainette redirects here. For the wine grape also known as Chainette, see Cinsaut. A hanging chain forms a catenary …   Wikipedia

  • Theorema Egregium — Gauss s Theorema Egregium (Latin: Remarkable Theorem ) is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. Informally, the theorem says that the Gaussian curvature of a surface …   Wikipedia

  • catenary — /kat n er ee/; esp. Brit. /keuh tee neuh ree/, n., pl. catenaries, adj. n. 1. Math. the curve assumed approximately by a heavy uniform cord or chain hanging freely from two points not in the same vertical line. Equation: y = k cosh(x/k). 2. (in… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”