fom
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Re: Calculating the area of a closed 3D path or ring
Posted:
Mar 14, 2013 5:48 AM


On 3/12/2013 8:06 PM, Math Guy 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.
As in the case of the other reply, I am the last person who should be answering your question.
What you are describing with this statement is typically called a "saddle point" (someone should add an entry at wikipedia for "potato chip point")
http://en.wikipedia.org/wiki/Saddle_point
There are specific analytic functions and specific methods for curvilinear approximation that would be helpful to you if you had constraints to choose such functions and if you were fortunate enough to find someone who knew methods that might apply.
Since neither of those conditions are satisfied by me, let us think about the problem in some other way.
If you look at the picture on the wikipedia page, you will see that the saddle is enclosed in a convex rectilinear region. So, you would want to do something along those lines in order to frame the problem.
Next, notice that a line segment connecting the maximum points of the concave up curve and a line segment connecting the minimum points of the concave down curve give you a basis for constructing a tetrahedral convex domain. You may need to add points at the extrema once the rectilinear region is determined in order to do this.
Suppose now that you find the centroid of the tetrahedral domain.
If you now form triangles from the line segments of the cyclic polygonal line forming your ring to the centroid of the tetrahedral domain, you will have some structure based on the saddle point geometry you are claiming to approximate with your statement (my other reply did not even try to address this since my first thought was simply any "area" for any configuration of n points in space based on triangular surfaces  that is why I snipped the beginning of your post).
Once again, you need to look at variations with respect to the centroid. Your polygonal boundary may not be as symmetrical as a "horse saddle". If it is, you might be restricted to vary only along the normal to the "top" and "bottom" surfaces of the rectilinear boundary which passes through the centroid.
If you want to vary off of such a line for any reason, consider forming an ovoid region inscribed in the tetrahedron, having the same ratio of axes as the rectilinear region, and oriented to have its axes coherently aligned with the rectilinear region. Then your variations will have some constraint within the tetrahedral region that reflects the external constraint used to orient your tetrahedral geometry. This will prevent you from varying too far from your intended "shape".
Actually, once you have determined the ovoid that could be inscribed, use a smaller ovoid (the golden ratio is always the aesthetic choice since there is no notion of "correctness" available to us) so that your variations are sufficiently distant from the tetrahedral boundary.
http://en.wikipedia.org/wiki/Golden_ratio
Good luck.

