# Thread: Creating poling from segments count and length

1. ## 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

2. Originally Posted by gabon
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 $\displaystyle l_1,\ldots,l_n$ for the lengths, we must have $\displaystyle \sum_{i=1}^n \arcsin\frac{l_i}{2R}=\pi$ (considering the sum of the angles the center views the segments). This can be solved numerically. I doubt there's an explicit expression for $\displaystyle R$. Does anyone know?
Then the angle between the first and second segment is $\displaystyle \arcsin\frac{l_1}{2R}+\arcsin\frac{l_2}{2R}$, and it's the same for the others.

3. 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.

4. Originally Posted by gabon
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 $\displaystyle R$; you can proceed by dichotomy since the function $\displaystyle r \mapsto\sum_{i=1}^n\arcsin\frac{l_i}{2r}$ is strictly decreasing and it is possible to see that $\displaystyle \frac{1}{2}\max_i l_i<R<\frac{n\pi}{2}\max_i l_i$. (Notice I fixed an error in the formulas of my previous post)

5. 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