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I know it sounds absurd, but say we can move the point $X$ (the one we want to get the location of) and get as many measurements as we can to the other reference points. Is it theoretically possible to create a 3D model of the (approximate) relative positions of the reference points?

The input is a series of distance vectors and the output should be their positions in a freely chosen coordinate system.

Food for thought: Say in probe position $q_0$ (unknown) we get the distance vector $(2,3)$ to the set of reference points $(p_0,p_1)$ and in the probe position $q_1$ (also unknown) we get $(3,2)$. We can get close to no information, but we can at least deduce that we moved towards point $p_1$ and away from $p_0$, but maybe if we had more reference points we could get a better picture.

If it cannot be solvable, would it if we had distances and/or orientation changes between pairs of probe positions?

Edit: I modelled the problem in Octave by trying to solve an equation system of distances given that the unknowns are the reference and the probing locations. For this test I only modelled in 2d, guessing that a lower number of dimensions would lead to an easier problem.

I modelled 3 reference points with fixed locations, so that's 6 unknown variables for the solver (their X and Y coordinates). For each added probing location, I get 3 more distance equations for the equation system and 2 more unknowns. If I have 6 probing locations, my equation system has 3x6=18 equations and 2*(3+6)=18 unknowns. Assuming the equations are linearly independent (strong assumption!), in theory I could solve for all unknowns, I guess.

Result: No success. Either the equations are not linearly independent or my script was buggy, but I'm leaning towards the former.

Gonzalo
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  • Tl;dr: I agree with your conclusion on your results.

    One probing location gives you 3 distance equations, but these distance equations always involve the 2 new unknowns, and 2 of the unknowns of interest. Disregarding the fact that they are not linear, it's unlikely you'd reach a solution with that. Also about the "simpler" version where you have information on the pairs of probe position, do you mean every possible pair of probe positions? Or just some of them, say, pairs of consecutive probe positions?

     – N.Bach Oct 26 '17 at 00:20

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If your points $p_0$ and $p_1$ are not defined, then you can't find out unknown position of $ q_0$ using only $d(q_0, p_0)$ and $d(q_0, p_1)$ because there will be infinite number of choices - here's how to imagine it:

You can pick out any $q_0$ with probability $1/|\mathbb R^3|$, then you can imagine 2 spheres around that $q_0$ with radiuses $p_0$ and $p_1$. And any combination of those radiuses will give you particular lengths chosen, say (2, 3) or (10, 20) or whatever.

You may consider this as 'approximation'.

O K
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  • I think you proved that one cannot determine the position of the reference points from a single distance vector, but what I think is that multiple distance vectors from multiple probe locations may help. Also you can have more than 2 reference points (typically 3 reference points would give you a certain position in 3d space, but their locations must be known beforehand). – Gonzalo Oct 24 '17 at 19:09
  • @Gonzalo If it's in 3D space, shouldn't that be 4 points (whose location we know beforehand)? If you intersect 3 known spheres, in general you have the choice between two points, so you'd need a 4th sphere to know exactly the position. To me, 3 points would be triangulation problem in 2D. – N.Bach Oct 24 '17 at 19:13
  • @N.Bach Yes, you're right. It's 4 spheres in 3D! – Gonzalo Oct 24 '17 at 19:22
  • @Gonzalo Any probe locations for $p_0$ and $p_1$ (satisfying given distance) will lay on the spheres I described. You have multiple choices for those ref points. If you freeze some ref points(pick on random, or positions known beforehand), then for 3d space you need 3 such points to obtain a basis... But then your question has no reason. Original question was: "Is it theoretically possible to create a 3D model of the (approximate) relative positions of the reference points?". So, I guess, you wanted to predict these 'reference points' position given current known point and distance to them – O K Oct 25 '17 at 13:55
  • @OleksandrKhryplyvenko Well, my actual target problem is to determine the positions of the probe locations given that I can build a map of the reference points. I edited my answer with an experiment that did not succeed. – Gonzalo Oct 25 '17 at 15:06