Unduloid and elliptic catenary

Hello beautiful Rhino community!

I am looking for a way to create Unduloids from elliptic catenaries. They look like reasonably simple surfaces to generate with Rhino/Grasshopper, but sadly I lack the maths skills to implement what is described on these Wikipedia pages.

Thanks for your input!

no math needed, just read the bolt words and ignore the rest:

In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and no…

yellow line: representing an elipse, but for explanation purposes, this will do. mentionned as elliptic catenary above.
black line:
fixed line above. axis around which the yellow curve is revolved.(revolution above)
green circle:
not used in practice, but represents the way the yellow line is moved around the axis to obtain the surface. in an abstract way this can represent the revolution itself.

hope thats a bit more clear for you. you can find the command _-revolve in rhino and follow the instructions in the command line and you’ll easily understand.

Thanks @benedict !

For this specific purpose I need the ellipse to be a roulette curve as described in the second link.
So the question is more about roulette curves the revolution surfaces, I think.

I am looking for an elliptic catenary which is created through rolling a line around a catenary curve as described here.

You can get a roulette of any curve like this
roulette.gh (14.4 KB)


Wow, thanks @DanielPiker !
So simple and effective.

I am slightly embarrased I could not figure this myself. Orientated planes is the bread and butter of my operation.