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INTRODUCTION
The loxodrome is a transcendental curve spiralling between the north pole and the south pole. The orthodrome (Great Circle) is the intersection of the sphere with a plane containing the centre of the sphere. An intrinsic property of these two curves can be found by Napier's Rules of Circular Parts that have been widely used in navigation. The following problem will demonstrate their advantage. Finding the infinitesimal distance dD O that the vessel must cover in Great Circle sailing for the orthodromic course to change by a small value dD O illustrates the problem. Referring to Figure 1, from a spherical right angled triangle TVP with any two parts given any third can be found as follows:
(1)
[formula omitted: see PDF] Having differentiated the expression (1) with respect to the geographic latitude ([phi]), as C O =C O ([phi]) and D O =D O ([phi]) obtains:
(2)
[formula omitted: see PDF] From the same triangle:
(3)
[formula omitted: see PDF] and
(4)
[formula omitted: see PDF] Thus, inserting Equations (3) and (4) in Equation (2) it becomes:
(5)
[formula omitted: see PDF] When [phi] equals [phi] V , dCO =-dD O tan[phi] V . For [phi]=0, dC O =0 showing no course change at the intersection with the equator (node point). Finally, transposing Equation (5) gives an answer to the above problem i.e. for the unit course alteration the unit distance is defined:
(6)
[formula omitted: see PDF]
Since the orthodrome is a curved line whose true direction changes continually (except for a meridian or the equator), a number of points...