Helicoid and Catenoid:

 
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Nov 09, 2014 1:40 am    Post subject: Helicoid and Catenoid: Reply with quote

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"
]
}
}


https://www.facebook.com/pages/MathMod/529510253833102
More at : http://en.wikipedia.org/wiki/Catenoid
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Abderrahman
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