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.