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I have a set of points, which I would like to know the locations of. What I have is noisy distance measurements between many of the points, but not necessarily all of them. Additionally, the points are located in 3D space, but we can consider the 2D case here.

My first thought was to set 1 of the points to (x1=0,y1=0) and pick another point as (x1=0,y2=distance between P1 and P2). Then I figured I could incrementally add new points by combining 2 equations like: (x3-x1)^2 + (y3-y1)^2 = D13 and (x3-x2)^2 + (y3-y2)^2 = D23.

My problem is that, I likely have more than 2 distances for each point (for instance for point 10, I may have distances to all 9 previous points) so depending on the order of the points I add and which distances I choose to use I will get different solutions.

Any ideas on how to solve this?


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