I would like to find a way to define the conformation of an $N$ equilateral polygon.
Consider a $N \in \mathbb{N}$ edge polygon, where each edge is connected by 2 points in $\mathbb{R}^3$. How to define an optimization problem to find the closest "equilateral" conformation to the initial current coordinates. For example, for $N=4$ it would be rectangular, $N=5$ would be a pentagon and $N=6$ would be hexagonal.
To clarify, given $N$ vectors $\{\vec{x}_n\}_1^N$ find the closest conformation that would find the most "equilateral" conformation. Is there a way to define it as an optimization problem? Is there a way to use differential geometry for this matter?
Basically, for initial conditions: $\{\vec{x}_n\}_1^N$ I want to move each of the points the least to form an equilateral polygon.
comment: If it makes stuff easier it's nice even to hear suggestions for the same problem in 2D, namely, $\{\vec{x}_n \in \mathbb{R}^2\}_1^N$