Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Narasimham Posts: 359 Registered: 9/16/06
Re: Calculating the area of a closed 3-D path or ring
Posted: Mar 14, 2013 7:21 AM

On Mar 13, 3:25 pm, Narasimham <mathm...@gmail.com> wrote:
> On Wednesday, March 13, 2013 6:36:38 AM UTC+5:30, 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.

>
>  ( chip like Pringles brand ?)
>
>
>
>
>
>
>
>
>

> > 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
> > algorithms, 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 liked the problem. The problem of Rado and Plateau are classical, but I found
>
> no easier guide for this problem and gave it up, temporarily at least.
>
> To help you towards its solution, we may still try. But before attempting a numeric solution,scalar invariants are to be first understood.
>
> When curvature and torsion of a closed non-planar rigid loop (arc = s single
>
> parameter) in 3-space are given, you want to find the minimal area.
>
> From differential geometry/surface theory, mean curvature H = (k1 + k2)/2 = 0
>
> ds^2 = E du^2 + 2 F du dv + G dv^2. Two parameters u,v are linked to edge arc
>
> parameter s. u and v should be chosen such that E N + G L = 2 F M , if that
>
> surface should be of minimal area.
>
> Area = Integral sqrt( E G - F^2)du dv
>
> If pressure is introduced across a soap film of rigid boundary,normal curvatures increase, Gauss curvatures also increase.Normal curvature is kn, surface tension = T, then
>
> kn = p/ T.
>
> Normal curvatures are zero along asymptotic directions. In my view they are most natural parameter lines to deal with this problem.
>
> So for each p, there is one minimal sutface that can be defined. p =0 is the minimal area film of _perhaps_ minimal integral curvature. Integ K dA.
>
> Normal curvature  changes with direction si. Now kn = k1 cos(si)^2 + k2 sin(si)^2 ; si = 0 or pi/2 for principal directions. This is from Euler's relation. Positive and negative kn areas areas are partitioned by kn = 0 lines.
>
> Gauss curvature K = k1* k2 < 0 when H = 0 invariably at any saddlle point of soap film.
>
> Hope it can begin.
>
> Narasimham

The simplest case is axisymmetric, si= pi/4 and the lines intersect
orthogonally for a catenoid of revolution.

r = c cosh(z/c) = c cosh( t ) where ( r,t,z ) are in a cylindrical
coordinate system.

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