Minimal area surfaces

Forum|Forum|9 years ago
September 8, 2016
10 replies
4096 views

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?

Other

F

Fred_Kohlhepp

23-Emerald I

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.

A

awhite-3

Author

1-Visitor

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

R

RichardJ

19-Tanzanite

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.