I have $n$ points with pairwise distances known, $d_{i,j}: 0<i,j<n$. Their coordinates, $\vec{x}_i \in \mathbb{R}^k: 0 <i<n$, are unknown. I can set up $n^2 - n$ equations to solve for coordinates:
$$ (\vec{x}_i - \vec{x}_j) \cdot (\vec{x}_i - \vec{x}_j)^T=d_{i,j}^2, i \neq j $$
Assuming coordinates are $k$-dimensional, this system of equations will have a single solution if $(n^2-n)/2 \geq k \cdot n$ i.e. enough equations to cover the number of variables ($n$ points times $k$ dimensions). Otherwise there will be infinitely many solutions.
In either case, are their algorithms for getting coordinates (or one possible solution) only from pairwise distances? I want to apply this in a python codebase - do existing libraries implement this?
