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Math Help - Creating poling from segments count and length

  1. #1
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    Question Creating poling from segments count and length

    Hi guys, I know the length of some segments and their order and I have no clue how to determine the position the segments should have to create a polygon with the largest area (more similar to a circle). Basically is like having a long line divided by segments and I would to join the two ends.

    Can anyone help me?


    Thanks, chr
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  2. #2
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    Quote Originally Posted by gabon View Post
    Hi guys, I know the length of some segments and their order and I have no clue how to determine the position the segments should have to create a polygon with the largest area (more similar to a circle). Basically is like having a long line divided by segments and I would to join the two ends.
    In fact, the configuration maximizing the area is the one that is convex and inscribed in a circle: look here for a proof.

    It remains to find the radius of the circle. Writing l_1,\ldots,l_n for the lengths, we must have \sum_{i=1}^n \arcsin\frac{l_i}{2R}=\pi (considering the sum of the angles the center views the segments). This can be solved numerically. I doubt there's an explicit expression for R. Does anyone know?
    Then the angle between the first and second segment is \arcsin\frac{l_1}{2R}+\arcsin\frac{l_2}{2R}, and it's the same for the others.
    Last edited by Laurent; October 21st 2008 at 11:36 AM. Reason: mistake in the formulas
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  3. #3
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    Hi Laurent, you got the problem right, I have those N sticks with different length and I would like to create the maximized area joining them. The good thing is that I know their order. And yes, basically I need to find each angle based on the length of the side. Maybe if I find the relation between the average angle and average length, I could find how a difference in length affect the angle.
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  4. #4
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    Quote Originally Posted by gabon View Post
    Hi Laurent, you got the problem right, I have those N sticks with different length and I would like to create the maximized area joining them. The good thing is that I know their order. And yes, basically I need to find each angle based on the length of the side. Maybe if I find the relation between the average angle and average length, I could find how a difference in length affect the angle.
    Hi,
    did you understand what I wrote? Because I thought it answered your question. It depends what you're looking for: do you want to write a computer program that draws the polygon? If so, my post gives you a solution. What remains is to find R; you can proceed by dichotomy since the function r \mapsto\sum_{i=1}^n\arcsin\frac{l_i}{2r} is strictly decreasing and it is possible to see that \frac{1}{2}\max_i l_i<R<\frac{n\pi}{2}\max_i l_i. (Notice I fixed an error in the formulas of my previous post)
    Last edited by Laurent; October 21st 2008 at 12:09 PM.
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  5. #5
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    Hi Laurent, yes I have to write a computer program. As you can/will see, I've not that confidence with Geom, especially the sintax

    Btw, in you previous answer you say basically that (l1/2R) + arcsin(l2/2R) + arcsin(ln/2R) = PI, right?

    I don't see any fix on that formula in your second post.


    Thanks a lot for your help, chr
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