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nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Nov 09, 2014 1:40 am    Post subject: Helicoid and Catenoid:

Hi,
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.
The attache images are for a Catenoid, an intermediate figure and Helicoid. The script shows the transformation from the Catenoid to the Helicoid.

CatenoidHelicoid by taha_ab, on Flickr

 Quote: { "Param3D": { "Name": [ "HelicoidCatenoid" ], "Component": [ "HelicoidCatenoid" ], "Fx": [ "cos(sin(t)*pi)*sinh(v/2)*sin(u) + sin(sin(t)*pi)*cosh(v/2)*cos(u)" ], "Fy": [ "-cos(sin(t)*pi)*sinh(v/2)*cos(u) + sin(sin(t)*pi)*cosh(v/2)*sin(u)" ], "Fz": [ "(u*cos(sin(t)*pi) + v*sin(sin(t)*pi))/2" ], "Umax": [ "pi" ], "Umin": [ "-pi" ], "Vmax": [ "pi" ], "Vmin": [ "-pi" ] } }

More at : http://en.wikipedia.org/wiki/Catenoid
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Cheers,
Abderrahman
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