What is the method used by wikipedia to solve Trilateration problem, is it the Linear Least Square? or is it just a direct solution of a set of linear equation systeme with exact distance? but in this case how this method can find solution even when the three circles does not intersect at one point?

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The article you cite does not mention least-squares or over-determined solutions.

It makes what I consider to be a very bold statement:

In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).

But you can see – even from the article's own diagram – that there are multiple intersection points:

enter image description here

Even in the simpler 2D case, you can see that there are six points of intersection.

The article's solution given appears, to me, to hold only when all three spheres intersect at a point. In the real world, involving measurements and errors, the spheres will intersect, as suggested in the diagram, in a zone of uncertainty. More measurements (e.g., more GPS satellites) and proper estimation techniques (such as least-squares) may converge on a good, or even the "best", solution.

  • Thanks for the clarification, I agree that is not so clear! I found an implementation of the wiki article with Python link but when trying thier code I found that the implementation take in consideration the case where the intersection is not unique? – Noureddine Jun 3 '15 at 16:47
  • How it is done? – Noureddine Jun 3 '15 at 16:53
  • That's a separate (and bigger) question, @Noureddine. You'll need to investigate thoroughly the (so far) two linked questions. As for this question, if my answer is acceptable, please click the check mark. – Martin F Jun 3 '15 at 21:31
  • You mean that is preferable to ask it as new question? – Noureddine Jun 4 '15 at 5:06
  • Yes, but you must first study all those other trilateration (and multilateration) questions. – Martin F Jun 4 '15 at 15:45

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