The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position

Lionheart, William R.B. and Moses, Peter J.C and Kimberling, Clark (2019) The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position. The Journal of Navigation. ISSN 1469-7785

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Abstract

Navigation methods, traditional and modern, use lines of position in the plane. Standard Gaussian assumptions about errors leads to a constant sum of squared distances from the lines defining a probability contour. It is well known these contours are a family of ellipses centred on the most probable position and they can be computed using algebraic methods. In this paper we show how the most probable position, the axes and foci of ellipses can be found using geometric methods. This results in a ruler and compasses construction of these points and this gives insight into the way the shape and orientation of the probability contours depend on the angles between the lines of position. We start with the classical case of three lines of position with equal variances, we show how this can be extended to the case where the variances in the errors in the lines of position differ, and we go on to consider the case of four lines of position using a methodology that generalises to an arbitrary number of lines.

Item Type: Article
Uncontrolled Keywords: Navigation; cocked hat; lines of position; least squares; symmedian point; probability ellipse; triangle centre; incentre of similitude; ruler and compass constructions; Lemoine point
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 51 Geometry (See also algebraic geometry)
MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 85 Astronomy and astrophysics
Divisions: Manchester Institute for the Mathematical Sciences
Depositing User: Prof WRB Lionheart
Date Deposited: 16 Aug 2019 23:27
Last Modified: 21 Aug 2019 20:33
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2725

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