> Looking for some thoughts about how to understand this problem. > > A closed loop (an irregular ring) is defined by a set of n points in > space.
How is the curve determined by those points? Is it polygonal?
> Each point has an (x,y,z) coordinate. The points are not co-planar. > Typically, this ring would approximate the perimeter of a horse saddle, > or a potato chip. The number of points (n) is typically from 6 to 12 > (usually 9) but will never be more than 16. > > The way I see it, there are two ways to understand the concept of the > area of this ring. > > a) if a membrane was stretched across the ring, what would the area of > the membrane be? Think of the membrane as a film of soap - which > because of suface tension would conform itself to the smallest possible > surface area. This would be Area A.
You would have to solve the minimal surface equation. Note that the solution is not unique in general, i.e. there are curves which are boundary to different different compact minimal surfaces, so even your "Area A" is not uniquely determined.
> b) if the ring represented an aperture through which some material (gas, > fluid) must pass, or the flux of some field (electric, etc). This would > be Area B.
This is not specified clearly enough.
> I theorize that because the points that define this ring are not > co-planar, that Area A would not be equal to Area B. > > I am looking for a numerical-methods formula or algorythm to calculate > the "area" of such a ring, and because I believe there are two different > areas that can be imagined, there must be two different formulas or > algorythms, and thus I'm looking for both of them. > > If I am wrong, and there is only one "area" that can result from such a > ring, then I am looking for that formula. > > I can imagine that summing the area of individual non-over-lapping > triangles will give me "an area". Given 9 perimeter points it is > possible to arrange more than one set of non-over-lapping triangles, > with each set giving it's own total area - but which one is the > "correct" one if they give different results? > > Comments?
Solving the minimal surface equation is non-trivial, but I guess you could find codes doing that, at least in simple cases.