Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T05:42:00.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilities in a Gaussian Cocked Hat

Published online by Cambridge University Press:  27 March 2019

Robin G. Stuart
Robin G. Stuart*
Affiliation:
(Valhalla, New York, USA)
*
(E-mail: robinstuart@earthlink.net)
Get access Rights & Permissions [Opens in a new window]

Abstract

A round of three celestial sights yields three lines of position along which the observer's true position could lie. Due to measurement errors, the lines of position do not intersect at a point but rather form a triangle called the “cocked hat”. The probability that this encloses the observer's true position is well known to be 25% which is the average over all possible cocked hats that could arise when the sights are made. It does not apply to any specific set of sights and in that case the probabilities depend on the statistical distribution of the measurement errors. With fixed azimuths for the observed celestial bodies and assuming a normal distribution for the errors in their measured altitudes, a closed form analytic expression is derived for the probability of the observer's position falling inside the cocked hat and this is related back to the global average. Probabilities for exterior regions bounded by the lines of position are also obtained. General results are given that apply for any number of lines of position.

Keywords

NavigationSextantShip DomainUncertainty
Type
Research Article
Information
The Journal of Navigation , Volume 72 , Issue 6 , November 2019 , pp. 1496 - 1512
Copyright
Copyright © The Royal Institute of Navigation 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. https://hdl.handle.net/2027/mdp.39015004486885Google Scholar
Adams, D. (1979). The Hitchhiker's Guide to the Galaxy. Pan Books, London.Google Scholar
Admiralty Manual of Navigation. (1938). Admiralty Manual of Navigation. Her Majesty's Stationery Office, London. https://hdl.handle.net/2027/uc1.31822005681556Google Scholar
Cook, I. (1993). Random Cocked Hats. The Journal of Navigation, 46, 132–135.Google Scholar
Daniels, H.E. (1951). The theory of position finding. Journal of the Royal Statistical Society, 13 B, 186–207.Google Scholar
DiDonato, A.R., Jarnagin, M.P. Jr. and Hageman, R.K. (1980). Computation of the Integral of the Bivariate Normal Distribution over Convex Polygons. SIAM Journal on Scientific and Statistical Computing, 1, 179–186.Google Scholar
DiDonato, A.R. and Hageman, R.K. (1982). A Method for Computing the Integral of the Bivariate Normal Distribution Over an Arbitrary Polygon. SIAM Journal on Scientific and Statistical Computing, 3, 434–446.Google Scholar
Gething, P.J.D. (1992). The Cocked Hat. The Journal of Navigation, 45, 143.Google Scholar
Gradshteyn, I.S. and Rhyzhik, I.M. (1965). Table of Integrals Series and Products. Academic Press, New York.Google Scholar
Kaplan, G.H. (2019). Fix Probabilities from LOP Geometry. Submitted to The Journal of Navigation (Under Review).Google Scholar
Lee, L. (1991). The Cocked Hat. The Journal of Navigation, 44, 433.Google Scholar
Meyer, C. (2013) Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution. Journal of Statistical Software, 52, 1–14. https://www.jstatsoft.org/index.php/jss/article/view/v052i10/v52i10.pdfGoogle Scholar
Pettofrezzo, A. (1978). Matrices and Transformations, Dover.Google Scholar
Stansfield, R.G. (1947). Statistical theory of D.F. fixing. Journal of the Institution of Electrical Engineers-Part IIIA: Radiocommunication 94(15), 762–770.Google Scholar
Swift, D. W. (1992). The Cocked Hat. The Journal of Navigation, 45, 143–146.Google Scholar
Whitworth, W.A. (1866) Trilinear Coordinates, Deighton, Bell and Co., Cambridge. https://archive.org/details/ trilinearcoordin029731mbpGoogle Scholar
Williams, J.E.D. (1991). The Cocked Hat. The Journal of Navigation, 44, 269–271.Google Scholar