
Re: Calculating the area of a closed 3D path or ring
Posted:
Mar 13, 2013 3:44 AM


On Mar 12, 6:06 pm, Math Guy <M...@guy.com> 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?
A simple first approximation (and probably a lower bound to the area you want) would be to shift the origin to the centroid and then take the area inside irregular ring defined by the projections of the points into the space of their first two principal axes.

