# The GPS equations and the Problem of Apollonius

@article{Hoshen1996TheGE, title={The GPS equations and the Problem of Apollonius}, author={Joseph Hoshen}, journal={IEEE Transactions on Aerospace and Electronic Systems}, year={1996}, volume={32}, pages={1116-1124} }

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…

## 49 Citations

On the Apollonius solutions to the GPS equations

- Mathematics1999 IEEE Africon. 5th Africon Conference in Africa (Cat. No.99CH36342)
- 1999

The physical significance of the complete 16 solutions of the problem of Apollonius to the GPS problem when reflected signals are considered is demonstrated.

An Efficient GPS Position Determination Algorithm

- Engineering
- 2003

: A direct, closed-form and efficient new position determination algorithm that exploits the closed-form solution of the GPS trilateration equations and works in the presence of pseudorange…

Efficient GPS Position Determination Algorithms

- Computer Science
- 2012

The stand-alone user GPS algorithm is a direct, closed- form, and efficient new position determination algorithm that exploits the closed-form solution of the GPS trilateration equations and works in the presence of pseudorange measurement noise for an arbitrary number of satellites in view.

TDOA-based Localization in Two Dimensions: the Bifurcation Curve

- Computer ScienceFundam. Informaticae
- 2014

In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane,…

A comprehensive analysis of the geometry of TDOA maps in localization problems

- MathematicsArXiv
- 2014

The TDOA map is defined from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space, and the identifiability of the model is studied, giving a complete analytical characterization of the image of this map and its invertibility.

Performance Comparison of Positioning Algorithms for Complex GPS Systems

- Computer Science2012 32nd International Conference on Distributed Computing Systems Workshops
- 2012

In this paper, we first present a classification for existing GPS positioning algorithms based on their features of mathematical operations in the calculating process and the accuracy of the…

Where on earth am I? Don’t worry, GPS satellites will guide you

- Physics
- 1998

Since ancient times navigators have been taking the help of celestial objects to find angles between the horizontal and the lines of sight to the celestial objects, in order to determine their…

Global Navigation Satellite Systems: Signal, Theory and Applications

- Environmental Science
- 2012

Different GNSS applications are demonstrated and evaluated in hybrid positioning, multi-sensor integration, height system, Network Real Time Kinematic (NRTK), wheeled robots, and status and engineering surveying.

On the localizability of cooperative positioning: Two user model

- Computer Science2014 Sixth International Conference on Wireless Communications and Signal Processing (WCSP)
- 2014

The localizability of users is analyzed and an exact and algebraic solution for two user cooperative positioning model is provided and it is theoretically proved that with the help of clock offset and altitude difference information, two users who can't be positioned by themselves can be positioned through cooperation.

Geometry of locating sounds from differences in travel time: isodiachrons.

- PhysicsThe Journal of the Acoustical Society of America
- 2004

A new geometrical shape, called an isodiachron, is described, which is the locus of points corresponding to a constant difference in travel time along straight paths between the animal and two receivers and can be used to better understand locations of calling animals and other sounds in the sea, Earth, and air.

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