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The cocked hat: formal statements and proofs of the theorems

Barany, I; Steiger, W; Toledo, S; (2021) The cocked hat: formal statements and proofs of the theorems. The Journal of Navigation , 74 (3) pp. 713-722. 10.1017/S0373463321000011. Green open access

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Abstract

Navigators have been taught for centuries to estimate the location of their craft on a map from three lines of position, for redundancy. The three lines typically form a triangle, called a cocked hat. How is the location of the craft related to the triangle? For more than 80 years navigators have also been taught that, if each line of position is equally likely to pass to the right and to the left of the true location, then the likelihood that the craft is in the triangle is exactly 1/4. This is stated in numerous reputable sources, but was never stated or proved in a mathematically formal and rigorous fashion. In this paper we prove that the likelihood is indeed 1/4 if we assume that the lines of position always intersect pairwise. We also show that the result does not hold under weaker (and more reasonable) assumptions, and we prove a generalisation to lines.

Type: Article
Title: The cocked hat: formal statements and proofs of the theorems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1017/S0373463321000011
Publisher version: https://doi.org/10.1017/S0373463321000011
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Science & Technology, Technology, Physical Sciences, Engineering, Marine, Oceanography, Engineering, pilotage, stochastic error, history
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10129645
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