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I am able to measure my distance to a set of (about 6 or 7) fixed but unknown points from many positions.

The difference in position between measurements is also unknown.

I believe that I should be able to work out the relative position of the fixed points, and therefore where I measured from and the path I took.

I have looked at the wiki page for trilateration, but it only gives examples working from known points.

Any help?

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Let's say the distances are 1,2,3,4,5,6. Now construct the following sets of points in $\mathbb{R}^2$: you are on position $(0,0)$, the next point is on position $(1,0)$, the next one on position $(2,0)$, etc... on the other hand, the following is also compatible with your distances: $(0,0),(0,1),(2,0),(0,3),(4,0),(0,5),(6,0)$. See? –  Raskolnikov Dec 12 '10 at 12:25
I understand that, but it doesn't get me any closer to a solution. I will need to use differences in the distance sets of two or more readings. –  FlightOfStairs Dec 12 '10 at 13:49
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1 Answer

For each distance measured, you can write an equation for the distance between that point on your path and that fixed point, both of which are unknown. In two dimensions (on a flat plane), each unknown point has two unknown coordinates. If you have $n$ fixed points and you measure from $k$ points on your path, you will have $nk$ distances and equations and $2n+2k$ unknowns. As long as $nk\ge 2n+2k$, this should yield a system that can be solved (if the distances are exact), but it probably helps to have $nk$ be more than $2n+2k$ because the system when $nk=2n+2k$ might have multiple solutions (because the equations are nonlinear). (If you are working in three dimensions, then $2n+2k$ changes to $3n+3k$.)

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