I have a set of 2D points in which each pair has a known Euclidean distance between them. How can I go about determining an arrangement of them?
I understand there is not a unique solution in general, but for the sake of my question, assume one point is fixed at the origin.
Mathematically, we have $P = \{p \vert p \in R^2\}$ and $D = \{d \in P \times P \vert \| d\| \text{is known} \}$. How can I find a valid arrangement? (Forgive my rustiness with proper set notation)
Note: I feel like least squares may be the best solution. This is related to bundle adjustment in photogrammetry.
This solution assumes that a solution exists; to check whether a solutions exists requires more iteration.
Notice that you can fix two points, not just one. If you have the first point at the origin, you can also a set a second point at $(d_{12},0)$ where $d_{12}$ is the distance from the first and second points. If there are only 3 points, there are 2 locations that 3rd point can be (if there are any). For more than 3 points, you can iterate this process, picking a pair of points to locate the third. Since for each step there are 2 possible locations, you may run into a situation that is impossible for a previous choice, so you have to go back and make the opposite choice.