Minimal area surfaces

[awhite-3]

Is Mathcad able to find the surface of minimum area defined by a hexagonal perimeter whose vertices are not necessarily all in the same plane?

[Fred_Kohlhepp]

You would need to define the surface; if the vertices are not co-planar then the shape of the surface must be a three dimensional function.  And the verticies will have three coordinates.

Mathcad is capable of the math.

[awhite-3]

Thanks for your help, Fred..

I was not necessarily looking for a closed expression; an STL file (or equivalent) would be fine for feeding a 3D printer.   The sides of the perimeter are actually well defined in the interesting case where the hexagon is a closed perimeter formed by the six edges of a regular octahedron which connect a top face to a bottom face.   Dipping this frame in a soap solution yields a beautiful minimum area surface which is only approximated by stretching (stretch) nylon over the same frame.  I've made a garden sculpture using 14 such surfaces (cast in bronze) welded into a column 7 feet tall.   But I would like to get an accurate representation in file form for 3D printing.

Alan White

[RichardJ]

The surface of minimum area is what (I think) you would get if you stretched an elastic sheet between the vertices and edges. If those vertices can be at arbitrary  positions (other than the fact that they form a hexagon when viewed from one direction) then I would be surprised if you could derive a mathematical expression that describes such a surface. Mathcad is certainly not going to derive it for you. I think the best approach would be finite element analysis. You could set that up in Mathcad (after all, it's just math) but it would be easier to use a piece of software designed for that.

[Fred_Kohlhepp]

"Soap bubble" math?

[awhite-3]

Yes, soap bubble math.... :-}

[Fred_Kohlhepp]

Never figured out how to do that . . .

Three dimensional catenary, strain in a membrane.

C'mon Valery!!

[awhite-3]

Richard,

You are correct; stretching nylon fabric does a pretty good job and a soap bubble does in even better job.   But neither of these surfaces lend themselves readily to 3D scanning techniques.   See reply to Fred K.

[awhite-3]

Can you suggest some software?

[RichardJ]

Here's a list of free FEA software. I have used the first one, LISA, a little, but for a thermal problem.

http://www.freebyte.com/cad/fea.htm

[VladimirN]

Thanks for this link, Richard.