Can this be triangulated?

Question

Given N people with their phones that can sense the signal strength of every other phone knowing what phone it is. Phones don't know their absolute location (underground). There is a formula that gives a distance for a signal strength. Also these phones can talk to each other without limitation. Is it possible to triangulate the spatial disposition of each phone (an angle and distance to each of them). If so, how?

Answer 1

Angles are hopeless because any rotation of the plane preserves the strength of every received signal. Similarly for an inversion (mirror image). Apart from that, however, relative positions and angles are determined because every triangle is determined up to isometry by the lengths of it sides and therefore every polygon is determined up to isometry by the lengths of its sides plus diagonals, by dividing it into triangles in various ways.

Answer 2

This is an optimization problem, in which all people need to be positioned such that they are properly distanced from everyone else. Because you don't know their orientation, any "positioning" will have to be relative either to everyone else or to an arbitrary anchor.

I suggest you take a look at force directed drawing algorithms, which should solve your problem, which will give you the best solution when some data is noisy, or not all distances between phones are known, etc.

But from a theoretical standpoint, simple trigonometry should do the trick. You'd have to repeat it for every phone knowing its distance to three "anchor phones" in which phone to phone distance is unnecessary. Note- distance to three anchor phones and not two because there is only one intersection of three circles (where 'r1!=r2') but two intersections for two circles.