Date: Mar 14, 2013 1:20 AM Author: fom Subject: Re: Calculating the area of a closed 3-D path or ring On 3/12/2013 8:06 PM, Math Guy wrote:

>

> 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?

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.