A catenoidA catenoid obtained from the rotation of a catenary
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.
^Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Springer Science & Business Media. p. 141. ISBN 9783642116988.
^ abGullberg, Jan (1997). Mathematics: From the Birth of Numbers. W. W. Norton & Company. p. 538. ISBN 9780393040029.
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning...
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. It was described by Euler in 1774...
cartographic projection necessarily distorts at least some distances. The catenoid and the helicoid are two very different-looking surfaces. Nevertheless...
of the catenary, and the minimal surface of revolution will thus be a catenoid. Solutions based on discontinuous functions may also be defined. In particular...
with this metric is embedded in euclidean three-space the image is the catenoid C {\displaystyle {\mathcal {C}}} shown above, with ρ {\displaystyle \rho...
less likely to cause it to buckle because in an inverted paraboloid or catenoid the pressures are nearer to being exclusively compressive. The individual...
In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to...
no sharp edge at the gluing. If done with care the result will be the catenoid C {\displaystyle {\mathcal {C}}} pictured at right, or something similar...
in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings...
(surfaces of revolution which are also minimal surfaces): the plane and the catenoid. A surface of revolution given by rotating a curve described by y = f (...
their mean curvature is zero at every point. The best-known examples are catenoids and helicoids, although many more have been discovered. Minimal surfaces...
P surface. In the associate family, these square catenoids "open up" (similar to the way the catenoid "opens up" to a helicoid) to form gyrating ribbons...
defined a sequence of capillary shapes known as (1) nodoid with 'neck', (2) catenoid, (3) unduloid with 'neck', (4) cylinder, (5) unduloid with 'haunch' (6)...
which has zero mean curvature at all points. Classic examples include the catenoid, helicoid and Enneper surface. Recent discoveries include Costa's minimal...
along an ellipse. Some examples of associate surface families are: the catenoid and helicoid family, the Schwarz P, Schwarz D and gyroid family, and the...
the rock formation to wide European notice. Leonhard Euler discovers the catenoid and proves it to be a minimal surface. By July – Northampton General Hospital...
based on the Weierstrass representation. The H surface is similar to a catenoid with a triangular boundary, allowing it to tile space. Susquehanna University...
the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. Let sn ( u , k ) {\displaystyle \operatorname...
incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such...
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