```Date: Mar 12, 2013 9:03 PM
Author: Math Guy
Subject: Calculating the area of a closed 3-D path or ring

Looking for some thoughts about how to understand this problem.A closed loop (an irregular ring) is defined by a set of n points inspace.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 thearea of this ring.a) if a membrane was stretched across the ring, what would the area ofthe membrane be?  Think of the membrane as a film of soap - whichbecause of suface tension would conform itself to the smallest possiblesurface 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 wouldbe Area B.I theorize that because the points that define this ring are notco-planar, that Area A would not be equal to Area B.I am looking for a numerical-methods formula or algorythm to calculatethe "area" of such a ring, and because I believe there are two differentareas that can be imagined, there must be two different formulas oralgorythms, and thus I'm looking for both of them.If I am wrong, and there is only one "area" that can result from such aring, then I am looking for that formula.I can imagine that summing the area of individual non-over-lappingtriangles will give me "an area".  Given 9 perimeter points it ispossible 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?
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