I'll return to my idea in post #23
Let n_1, n_2, ...n_k be a set of k "angles" such that their sum is 180(k-2)
Let m_1, m_2, ...m_k, be defined by
m_i = 180 - n_i for 1<=i<=k ( some m_i may be negative)
the sum of all m_i is 360
rotations when m_i>0 will be clockwise and when m_i<0 will be anti-clockwise
Start with a line segment A_0 to A_1
at A_1 rotate through m_1 and draw a line segment A_1 to A_2
at A_2 rotate through m_2 and draw a line segment A_2 to A_3
repeat until
at A_k rotate through m_k and draw a line segment A_k to A_{k+1}
As the sum of all m_i's is 360 A_k to A_{k+1} will be parallel to A_0 to A_1
Following Status X, pick two non-parallel sides whose length you adjust. This corresponds to shifting the final point by a linear combination of two linearly indpendent vectors and so arrange for A_{k+1} and A_0 to be coincident.
Any problems with the above please let me know
However you may end up with a complex polygon as in
http://en.wikipedia.org/wiki/Polygon
I am fairly certain you can take a complex polygon and as above by repeatedly picking two non-parallel sides whose lengths to adjust you can transform the complex polygon into a simple one, either concave or convex