The GPS equations and the Problem of Apollonius
Abstract
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.
- Publication:
-
IEEE Transactions on Aerospace and Electronic Systems
- Pub Date:
- July 1996
- DOI:
- Bibcode:
- 1996ITAES..32.1116H
- Keywords:
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- Global Positioning System;
- Nonlinear equations;
- Time measurement;
- Military computing;
- Earth;
- Application software;
- Timing;
- Vehicles;
- Radio frequency;
- Satellite broadcasting