Minimal Surfaces: Catenoid

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Hold shift key and drag (or use mouse wheel) to adjust the separation between the two rings.

A soap film is formed between two parallel rings of a fixed radius separated by a fixed distance. To minimize the surface-tension energy of the soap film, its total area seeks a minimum value. The derivation of the shape of the film involves a problem in the calculus of variations. When the separation between the two rings gets too large, the film collapses to disks within the two rings.

Click here for an explanation/derivation of the shape of this surface.
Better yet, check this out: Weisstein, Eric W. "Minimal Surface of Revolution." From MathWorld--A Wolfram Web Resource.

LOQO was used to compute the minimal surface shown at right. To get a better feel for the surface, set it in motion by dragging and releasing the object.

Here are some other interesting minimal surfaces:

Updated 2012 Dec 2