Traditionally, on a great circle, the latitude or longitude of a waypoint is found by inspection. In this paper, using an elementary knowledge of vector algebra including linear combination of vectors and vector basis, we provide an easy method for finding the equation of a great circle path as a parameterized curve. By use of this vector function of distance travelled, the latitude and longitude of waypoints can be found based on the distance from departure point along a great circle. The approach is intended to appeal to the navigator who is interested in the mathematics of navigation and who, nowadays, solves his navigation problems with a personal computer.

In this paper, equations are established to solve problems of Rhumb Line Sailing (RLS) on an oblate spheroid. Solutions are provided for both the inverse problem and the direct problem, thereby providing a complete solution to RLS. Development of these solutions was achieved in part by means of computer based symbolic algebra. The inverse solution described attains a high degree of accuracy for distance and azimuth. The direct solution has been obtained from a solution for latitude in terms of distance derived with the introduction of an inverse series expansion of meridian arc-length via the rectifying latitude. Also, a series to determine latitude at any longitude has been derived via the conformal latitude. This was achieved through application of Hermite's Interpolation Scheme or the Lagrange Inversion Theorem. Numerical examples show that the algorithms are very accurate and that the differences between original data and recovered data after applying the inverse or direct solution of RLS to recover the data calculated by the direct or inverse solution are very small. It reveals that the algorithms provided here are suitable for programming implementation and can be applied in the areas of maritime routing and cartographical computation in Graphical Information System (GIS) and Electronic Chart Display and Information System (ECDIS) environments. K E Y WO R D S 1. Rhumb Line (RL).2. Rectifying Latitude. 3. Conformal Latitude.First published online: 30 March 2012. I N T R O D U C T I O N .In marine and air navigation, ships and aircraft sailing or flying on fixed compass headings may travel along Rhumb Lines (RL), hence knowledge of RL calculation is important. Mercator's projection (a normal aspect cylindrical conformal projection) has the unique property that RLs on the Earth's surface are projected as straight lines on the map. Methods of calculating the course and the distance between two points from knowledge of their latitudes and longitudes, or calculating the latitude and the longitude of the arrival point from the course and the distance from a known departure point, are called 'sailings'. A RL appears as a straight line on a Mercator chart and as a spiral curve (Loxodrome) on a surface of an oblate spheroid. Both of these cut all the meridians at the same angle (Thomas, 1952; Williams, 1998). The distance for a RL on the navigation sphere is within 0·5% of the distance on the RL on the WGS84 spheroid (Earle, 2006) Figure 1.

This paper presents the similarities between equations used for great circle sailing and 2D linear equations. Great circle sailing adopts spherical triangle equations and vector algebra to solve problems of distance, azimuth and waypoints on the great circle; these equations are sophisticated and deemed hard for those unfamiliar with them, whereas on the other hand, 2D linear equations can be solved easily with basic algebra and trigonometry definitions. By pointing out the similarities, readers can quickly comprehend great circle equations and grasp just how similar they are to the corresponding 2D linear equations. K E Y

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions. K E Y

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