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Topic: Calculating the area of a closed 3-D path or ring
Replies: 23   Last Post: Mar 25, 2013 4:54 PM

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 Gib Bogle Posts: 42 Registered: 3/28/11
Re: Calculating the area of a closed 3-D path or ring
Posted: Mar 25, 2013 4:54 PM

On 13/03/2013 2:06 p.m., 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 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 the
> area of this ring.
>
> a) if a membrane was stretched across the ring, what would the area of
> the membrane be? Think of the membrane as a film of soap - which
> because of suface tension would conform itself to the smallest possible
> surface 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 would
> be Area B.
>
> I theorize that because the points that define this ring are not
> co-planar, that Area A would not be equal to Area B.
>
> I am looking for a numerical-methods formula or algorythm to calculate
> the "area" of such a ring, and because I believe there are two different
> areas that can be imagined, there must be two different formulas or
> algorythms, and thus I'm looking for both of them.
>
> If I am wrong, and there is only one "area" that can result from such a
> ring, then I am looking for that formula.
>
> 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?
>
>

I just learned that determining the minimum-area surface is called
Plateau's problem, after the mathematician Joseph Plateau.
http://en.wikipedia.org/wiki/Plateau%27s_problem

Date Subject Author
3/12/13 Math Guy
3/13/13 Ray Koopman
3/13/13 Narasimham
3/14/13 Narasimham
3/13/13 Shmuel (Seymour J.) Metz
3/13/13 Frederick Williams
3/13/13 Brian Q. Hutchings
3/14/13 fom
3/14/13 fom
3/14/13 Math Guy
3/15/13 Ray Koopman
3/15/13 Math Guy
3/15/13 fom
3/16/13 Ray Koopman
3/16/13 fom
3/16/13 Math Guy
3/16/13 fom
3/16/13 Ray Koopman
3/15/13 Peter Spellucci
3/16/13 Math Guy
3/17/13 Ray Koopman
3/17/13 Math Guy
3/18/13 Ray Koopman
3/25/13 Gib Bogle