Let me know if this is mathoverflow material, and I’ll wend my way over there. I’m hoping someone will recognise this and point me in the right direction…
I’m trying to map out related nodes. I’ve figured out how to calculate the minimum distances between all the points, and now I need to know how to turn those into actual co-ordinates in 2D space.
So, given a point Pn (where n > 1), a set of points [P1..Pn-1], and a set of distances [d1..dn-1] where each d represents the minimum distance between Pn and d’s respective point, how do I calculate the best valid co-ordinate set [x,y] for Pn?
When I say ‘best’ valid co-ordinate set, I mean the set that brings Pn closest to all the other points without violating any of the constraints.
My first thought was to stick P1 at [0,0], P2 at [0,d] (d1 for P2) and then for P3 I would put it at [0,y] where y is the minimum distance that P3 has to be at to satisfy its d1 and d2 and then move it in a circle around P2 at a radius of d2 for as long as it still satisfies d1.
That would have to be repeated for all points, which sounds like it would take ages.
Does this problem ring any bells with anyone? I’m not sure what formula or algorithms I’m looking for.
Update
I hadn’t thought about what ‘closest’ means. I knew what I meant when I wrote it, but I hadn’t thought about calculating it!
Minimum sum of squares sounds like it will do the job.
It sounds a lot as if you’re trying to do some form of multidimensional scaling.