The GPS equations and the Problem of Apollonius

By relating the Global Positioning System (GPS) problem of location to the ancient Problem of Apollonius, this work presents a closed solution to the pseudorange positioning problem for two and three dimensions. The positioning problem, given by a set of nonlinear equations, has been reduced to the solution of a quadratic equation. The resulting expressions yield either two or one physically meaningful solutions for both the two- and three-dimensional problems. Expressions for the boundary curves and surfaces that separate the one-solution domain from the two-solution domains are also given. Asymptotic lines and planes for the boundary curves and surfaces have also been derived.