A "'catenoid "'is a type of surface in topology, arising by rotating a catenary curve about an axis.
12.
Because they are members of the same associate family of surfaces, one can bend a catenoid into a portion of a helicoid without stretching.
13.
Many explicit examples of minimal surface are known explicitly, such as the catenoid, the helicoid, the Scherk surface and the Enneper surface.
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It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same.
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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.
16.
Euler proved in 1744 that the catenary is the curve which, when rotated about the-axis, gives the surface of minimum surface area ( the catenoid ) for the given bounding circles.
17.
This differs from what is normally called a catenoid in that the catenary is rotated about its central axis, forming a surface with the topology of a bowl rather than that of a cylinder.
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In the associate family, these square catenoids open up ( similar to the way the catenoid opens up to a helicoid ) to form gyrating ribbons, then finally become the Schwarz D surface.
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In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation, and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing.
20.
When the cross section with this metric is embedded in euclidean three-space the image is the catenoid \ mathcal C shown above, with \ rho measuring the distance from the central circle at the throat, of radius n, along a curve on which \ varphi is fixed ( one such being shown ).
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