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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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fom

Posts: 1,968
Registered: 12/4/12
Re: Calculating the area of a closed 3-D path or ring
Posted: Mar 14, 2013 5:48 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 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.


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.












Date Subject Author
3/12/13
Read Calculating the area of a closed 3-D path or ring
Math Guy
3/13/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/13/13
Read Re: Calculating the area of a closed 3-D path or ring
Narasimham
3/14/13
Read Re: Calculating the area of a closed 3-D path or ring
Narasimham
3/13/13
Read Re: Calculating the area of a closed 3-D path or ring
Shmuel (Seymour J.) Metz
3/13/13
Read Re: Calculating the area of a closed 3-D path or ring
Frederick Williams
3/13/13
Read Re: Calculating the area of a closed 3-D path or ring
Brian Q. Hutchings
3/14/13
Read Re: Calculating the area of a closed 3-D path or ring
fom
3/14/13
Read Re: Calculating the area of a closed 3-D path or ring
fom
3/14/13
Read Re: Calculating the area of a closed 3-D path or ring
Math Guy
3/15/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/15/13
Read Re: Calculating the area of a closed 3-D path or ring
Math Guy
3/15/13
Read Re: Calculating the area of a closed 3-D path or ring
fom
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
fom
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
Math Guy
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
fom
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/15/13
Read Re: Calculating the area of a closed 3-D path or ring
Peter Spellucci
3/16/13
Read Re: Calculating the area of a closed 3-D path or ring
Math Guy
3/17/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/17/13
Read Re: Calculating the area of a closed 3-D path or ring
Math Guy
3/18/13
Read Re: Calculating the area of a closed 3-D path or ring
Ray Koopman
3/25/13
Read Re: Calculating the area of a closed 3-D path or ring
Gib Bogle

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