Construct polygon from random segments

Question

Given an arbitrary amount of ordered segments, with arbitrary lengths is there a way to determine if they can be formed into a simple polygon?

And if so, is it possible to work out the angles needed to do so?

Thanks

Answer 1

As @Mark Bennet points out, there are lots of possible configurations (for $n>3$); here's one:

Chain the segments end-to-end with their vertices on the circumference of a circle whose diameter is at least as long as the longest segment. Simply adjust the size of the circle until the segment chain makes a single closed loop.

The worst case is the limit of the configuration in which the largest segment "separates" the other segments from the center, with the center moving off to Infinity. In the limit, the segment sub-chains on either side of the largest segment are "folded flat" onto that largest segment as the circle becomes a line; if the ends of the chain don't meet in that circumstance, then it wasn't meant to be.

With this cyclic construction, you can determine the angles between segments by joining all the endpoints to the center and analyzing the various isosceles triangles.

Answer 2

If I read the question right, the order of the (say $n>2$) segments is given, but the orientation is not. If the longest segment is shorter than the sum of the others then you can make a non-degenerate simple polygon (in a plane).

Since the order is cyclical, make the longest segment the first. Then the next $n-2$ can be made into a straight line (piece 2) with the final piece as a third side - and you can construct a triangle. This configuration, incidentally, minimises the angle between the first two pieces, and maximises the angle between the last piece and the first one. You can put a bound on the angles, therefore, but unless you have only three pieces you cannot determine the angles.

It is clearly possible to purturb such a polygon into a non-planar configuration and there are non-convex possibilities as well ($n>3$ in both cases).