# Creating poling from segments count and length

• Oct 18th 2008, 11:50 PM
gabon
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
• Oct 19th 2008, 03:23 AM
Laurent
Quote:

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 $l_1,\ldots,l_n$ 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.
• Oct 20th 2008, 10:55 PM
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.
• Oct 21st 2008, 11:36 AM
Laurent
Quote:

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 $R$; 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. (Notice I fixed an error in the formulas of my previous post)
• Oct 21st 2008, 01:26 PM
gabon
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