I'm trying to come up with a sort of trilateration algorithm that, given n >= 3 circles, finds the point of intersection. The radii come from samplings of electromagnetic magnitudes, therefore there won't be any real point where all circles intersect, but only an approximation.
What is the best way to get this approximate intersection, that can also be easily translated into a relatively simple algorithm?
On
I suppose that you know the coordinates $(x_i,y_i,z_i)$ for each of the $n$ points as well as the distance $d_i$ to the point $(X,Y,Z)$ you are looking for.
The rigorous method, even if not too complex, requires nonlinear regression which in turn requires good estimates for starting.
There is another one which is simple I descrided here. It si very simple and does not require anything beside linear regression or matrix operation. It works well. If you want more rigor, you have a very good starting point.
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I found this paper that deals with the problem, especially in the section about multilateration.
http://web.ist.utl.pt/ist150077/doc/localization.pdf
It wasn't hard to implement, provided you find a good library for linear algebra to lean on.
The Wikipedia article on Trilateration gives a nice introduction.
One might run that procedure for radii plus minus some error margin to sample the intersection area for the case with uncertainty.
Also have a look at the reference here.