Global positioning: The uniqueness question and a new solution method

Quadrics of revolutionWe provide a new algebraic solution procedure for the global positioning problem in n dimensions using m satellites.We also give a geometric characterization of the situations in which the problem does not have a unique solution.This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures.We fill a gap in the literature by giving a proof for the long-held belief that when m ě n `2, the solution is unique for almost all user positions.Even better, when m ě 2n `2, almost all satellite configurations will guarantee a unique solution for all user positions.Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness.Some of our results are obtained using tools from algebraic geometry.