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

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.