
Re: Calculating the area of a closed 3D path or ring
Posted:
Mar 25, 2013 4:54 PM


On 13/03/2013 2:06 p.m., Math Guy wrote: > 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. > > Each point has an (x,y,z) coordinate. The points are not coplanar. > 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. > > 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. > > I theorize that because the points that define this ring are not > coplanar, that Area A would not be equal to Area B. > > I am looking for a numericalmethods 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 nonoverlapping > triangles will give me "an area". Given 9 perimeter points it is > possible to arrange more than one set of nonoverlapping triangles, > with each set giving it's own total area  but which one is the > "correct" one if they give different results? > > Comments? >
I just learned that determining the minimumarea surface is called Plateau's problem, after the mathematician Joseph Plateau. http://en.wikipedia.org/wiki/Plateau%27s_problem

