Applications of numerical analysis in navigation

Williams, Roy (1994). Applications of numerical analysis in navigation. PhD thesis The Open University.



Part one of the thesis contains analysis of the methods of computation in navigation. We start with loxodromic navigation and, although this subject is well documented, we make a positive attempt to analyse the subject matter using the methods of differential geometry. We then turn to the problem of shortest path curves and set out an alternative method of solving the problem of navigating along the arc of a great circle on the surface of a sphere which can be generalised to other surfaces. In particular, a contribution made by the thesis is an analysis of the problem of navigating along shortest path geodesic arcs on the surface of a spheroid which introduces an algebraic representation of the geodesic curve by solving Clairaut' s equation using a cylindrical transformation. We are therefore able to compute the the coordinates of the positions of points along the path of the geodesic and the length of arc along the path of the. geodesic curve can then be computed step by step between these points by a numerical method - the Direct Cubic Spline method which was first introduced by this author in the Bachelor of Philosophy thesis in 1982 and is developed further in part 2 of this thesis. We apply this method also to the special problem of computing the distance along the shortest path between nearly antipodean points on the surface of a spheroid. We analyse the problem of computing an observer's position on the surface of the Earth using astronomical observations and show how a position locus is distorted when it is transferred over the surface. We offer a method of computing the observer's position by a series of observations of a single astronomical body taken over a comparitively short period of time and which does not necessarily include an observation at the tide of meridian passage of the body.

In part two of the thesis we discuss the Direct Spline approximation to integrals and give some error bounds. The Direct Cubic Spline is a step by step method of fitting a cubic spline to the integral of a function directly which computes the value of the integral of the function step by step between the data points which need not be evenly spaced. We extend the idea to splines of higher order and give the formula from which they may be obtained but we show that, except for a particular special form of the direct quartic spline. the higher order direct splines do not yield algorithm for computing integrals which are as efficient as the Direct Cubic Spline.

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