From Apollonius to Newton to GPS

Joseph Hoshen

Abstract:	This paper shows that a closed solution for the GPS equation can be determined from the geometrical construction for the ancient Problem of Apollonius dating to the third century BC. Newton in his Principia (1687) proved a lemma solving the Problem of Apollonius in two dimensions. Remarkably, this lemma is a statement of the two-dimensional GPS-like- equations. Deriving insight from the solution of the Problem of Apollonius, the positioning problem, given by a set of non-linear 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 limes and planes for the boundary curves and surfaces have also been derived.
Published in:	Proceedings of the 51st Annual Meeting of The Institute of Navigation (1995) June 5 - 7, 1995 Antlers Doubletree Hotel Colorado Springs, CO
Pages:	129 - 134
Cite this article:	Hoshen, Joseph, "From Apollonius to Newton to GPS," Proceedings of the 51st Annual Meeting of The Institute of Navigation (1995), Colorado Springs, CO, June 1995, pp. 129-134.
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