fom
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Re: Calculating the area of a closed 3D path or ring
Posted:
Mar 14, 2013 1:20 AM


On 3/12/2013 8:06 PM, Math Guy wrote: > > 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?
Right away, this seemed related to problems from the calculus of variations. Better men than I have already told you how difficult a good answer will be, and, I hope that someone who has actually faced something similar gives you an answer or at least helps you to define your needs more carefully.
Given that, it might be sufficient to find the barycenter or centroid or whatever one cares to call it (someone at wikipedia redirected barycenter to center of mass), and calculate the area of the triangles formed from the corners of the cyclic polygonal line forming the ring to the barycenter.
With the formula for that area, do the necessary analysis with directional derivatives to see how your area function changes with respect to variation from the barycenter.
Or, perhaps, since the number of triangles is finite, construct a weighted function based on the calculated areas for each triangle in relation to the barycenter (in other words, a baseline against which to define "correctness" if that is appropriate) and then see how the weighted function varies with respect to variation from the barycenter.
If by some lucky chance there were a particular line through the barycenter  or even a manageably finite number of lines  that are identified by the analysis of variation, you could examine a parametrized function that calculates the area of a pyramidal cone whose vertex lies on those lines. Then you might find an extreme on one of those lines that is not at the barycenter.
With respect to those same lines, one could consider a different notion of "correctness". The extrema that might be of interest in this case would be based on a least squares minimization. What would be minimized would be the angular differences of the normal lines of the triangles relative to the line that is being used to parametrize the function. The idea of this would be to make the surface as "orthogonal" as possible to whatever line seemed interesting enough to pursue further. With this additional notion, "correctness" might lead to a point different from the barycenter for different reasons.
It may be that an "interesting" direction actually lies in one of the triangles. Then you would want to consider moving off of the barycenter in that direction and repeating the analysis with a new point along that line.
It may be that an "interesting" direction is not "orthogonal" enough for the analysis above to even make sense. With that same idea, however, you might try to find a hyperplane for your ring based on a least squares minimization of the angles the segments of your polygonal line make with hyperplanes. Take the basis of calculating the areas of a pyramidal cone as the line passing through the barycenter normal to such a hyperplane if one can be found.
As I tried to imply above, I am not the one who should be replying to you. But, you asked for numerical methods. So, even though this is a variational problem, answers involving the calculus of variations will not help you directly. There probably are instances of people who have converted problems like this into numerical approximation methods (but, you would need to better explain your problem for them to recognize that their knowledge is directly applicable). If so, I certainly hope one can help you.
But, if that does not happen, I hope some of the above suggestions may be of help.

