Global Positioning on Earth

TL;DR

This paper reveals that the global positioning problem on Earth can have multiple solutions even with limited satellites, and introduces methods to determine the correct solution count.

Contribution

It demonstrates the existence of multiple solutions in spherical positioning and provides solution methods to identify the correct one.

Findings

Up to 4 solutions with 3 satellites on a sphere.

Existence of hyperboloid families causing multiple solutions.

Methods to accurately determine the number of solutions.

Abstract

Contrary to popular belief, the global positioning problem on earth may have more than one solutions even if the user position is restricted to a sphere. With 3 satellites, we show that there can be up to 4 solutions on a sphere. With 4 or more satellites, we show that, for any pair of points on a sphere, there is a family of hyperboloids of revolution such that if the satellites are placed on one sheet of one of these hyperboloid, then the global positioning problem has both points as solutions. We give solution methods that yield the correct number of solutions on/near a sphere.