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This paper describes George Biddell Airy's almost completely unknown method of approximating an orthodromic arc (great circle arc) using a circular arc in the normal aspect Mercator projection of a sphere. In addition, it is demonstrated that the centre of the circle can be defined in at least two different ways, which yields slightly different results. Airy's approach is built upon in this paper. The method of computing coordinates of Airy's circle arc centre is described. The formulae derived in the paper can be used to calculate the length of Airy's approximation of the orthodromic arc connecting two points on the sphere and on the Mercator chart. Moreover, the actual length of the orthodromic arc on the sphere and on the Mercator chart can be computed using the formulae derived in this paper. The purpose of the paper is not to suggest an application of Airy's method in navigation, but to analyse Airy's proposal and to show that a great circle arc on a Mercator chart is close to a circular arc for distances which are not too great. This property can be useful in education, having in mind that the stereographic projection is the only one that maps any circle on a sphere onto a circle in the projection plane.

The Great Elliptic Sailing (GES), which can reduce sailing distance, is important to navigators. Whether a Great Ellipse (GE) is worth using depends on whether the distance saved is significant. Otherwise, the Rhumb Line (RL) is easier to steer. We propose a simple criterion to evaluate the difference in distance between the GE and the RL. The criterion is that the GE is worth using when the vertex lies between the departure and destination. In order to take the advantage of shorter distance, the GE is usually approximated as a series of waypoints. Unlike currently practised methods, we propose the Longitude Bisection Method (LBM) which determines waypoints with varying intervals. This approach can establish the appropriate number of waypoints to approximate the GE effectively. The proposed criterion and the LBM are demonstrated in practical examples.

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.

In this note the Middle Latitude Rule is derived. Namely, to reach Great Circle vertex in two steps such that the total distance is a minimum, an initial rhumb line course equal to the orthodromic course at Middle latitude is to be used. The shortest distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point.

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.

Many problems in navigation can be best viewed and solved as problems in analytical geometry. We only need to understand the geometry of two ‘navigable’ surfaces; the sphere and the ellipsoid of revolution. The ellipsoidal model is generated by revolving an ellipse about its minor axis and this model is used as a global model for the surface of the Earth. The eccentricity of the meridian ellipse is small (≈0·082) so we sometimes refer to this surface as a ‘spheroid’ since the surface is still ‘sphere-like’. The physical Earth is, in fact, referred to as a ‘geoid’ whose surface is that which approximates global mean sea level. The mathematical representation of the geoid is not trivial and the ellipsoid of revolution is an extremely good approximation to it.

About ten years ago this author wrote the software for a suite of navigation programmes which was resident in a small hand-held computer. In the course of this work it became apparent that the standard text books of navigation were perpetuating a flawed method of calculating rhumb lines on the Earth considered as an oblate spheroid. On further investigation it became apparent that these incorrect methods were being used in programming a number of calculator/computers and satellite navigation receivers. Although the discrepancies were not large, it was disquieting to compare the results of the same rhumb line calculations from a number of such devices and find variations of some miles when the output was given, and therefore purported to be accurate, to a tenth of a mile in distance and/or a tenth of a minute of arc in position. The problem has been highlighted in the past and the references at the end of this show that a number of methods have been proposed for the amelioration of this problem. This paper summarizes formulae that the author recommends should be used for accurate solutions. Most of these may be found in standard geodetic text books, such as, but also provided are new formulae and schemes of solution which are suitable for use with computers or tables. The latter also take into account situations when a near-indeterminate solution may arise. Some examples are provided in an appendix which demonstrate the methods. The data for these problems do not refer to actual terrestrial situations but have been selected for illustrative purposes only. Practising ships' navigators will find the methods described in detail in this paper to be directly applicable to their work and also they should find ready acceptance because they are similar to current practice. In none of the references cited at the end of this paper has the practical task of calculating, using either a computer or tabular techniques, been addressed.

In an earlier paper, the writer attempted a quantitative study of a flow-planning system for aircraft intending to fly across Europe. This paper aims to offer each aircraft a clearance giving a reasonable expectation that, throughout cruising flight at least, there will be no conflict with other traffic. Such a clearance is sometimes termed a ‘tube of flight’. Reference i assumed a pattern of traffic flying along rhumb lines, each flight having a starting position chosen at random in a region roughly the size of Europe. Flight was in a random direction and the flight path length was chosen at random from a plausible distribution.

This article addresses the mathematical foundations of rhumblines or loxodrome curves. These curves are critical to navigation and small-scale charting by virtue of the fact that they provide an efficient routeing from one point on a surface to another by means of a constant ‘course angle‘. This article will develop the necessary mathematical relations for the construction of such a curve, then apply the relations to both spherical and oblate-spheroidal surfaces. The purpose of this article is to produce a superior oblate-spheroidal loxodrome curve, which better models curves or routes of constant course on the actual, approximately oblatespheroidal, Earth.

A curve that cuts all meridians of a rotating surface at the same angle is called a loxodrome. If the shape of the Earth is approximated by a sphere, then the loxodrome is a logarithmic spiral that cuts all meridians at the same angle and asymptotically approaches the Earth's poles but never meets them. Since maritime surface navigation defines the course as the angle between the current meridian and the longitudinal direction of the ship, it may be concluded that the loxodrome is the curve of the constant course, which means that whenever navigating on an unchanging course we are navigating according to a loxodrome.

Continued discussions on techniques of rhumb-line sailing might well be considered irrelevant in today's climate of navigational technology. However, there is apparently one group of navigators who have a need for rhumb-line sailing by computation as opposed to direct measurement from a chart: this group comprises the navigators of merchant ships.It may well be argued that traditional methods are adequate and that, although imprecise, the resulting errors are swamped by the errors of the data used. This argument is satisfactory for established navigators but, with the great improvements that have taken place in nautical education, does not hold for the teaching of newcomers.

The navigator is frequently faced with the task of finding the course and distance along a rhumb-line between two places on the Earth's surface.The following paragraphs tell of a two-part investigation into the methods of calculating rhumb-line courses and distances.Part I deals with a comparison of the methods used and the results are presented in graphic form.Part II considers the various corrections that may be applied to mean latitude sailing to get better results.

IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e. a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal.

I am reminded that this was the title of a paper by Professor W. M. Smart, published elsewhere in 1946, dealing with the rhumb line on the spheroid, by references in recent issues of this Journal to a paper of mine entitled ‘Loxodro-mic distances on the Terrestrial Spheroid’ which appeared in this Journal 32 years ago: the subject crops up every few years.

Turner has chosen in a recent paper to criticize Moss for inaccurate methods and observes that the inherent inconsistencies are ‘educationally objectionable’. However, he who criticizes must be prepared to be criticized in turn, and when a computer program of the standard evidenced in the printout of PROGRAM 2 is put forward as a general solution to the rhumb-line problem, and one which will enable the navigator to compute his tracks rapidly and accurately, we must needs examine it closely. Such an examination shows that, at the very least, the programmed solution offered is ‘computerizationally objectionable’.

A student of navigation who refers to the standard textbooks will be confronted with a multiplicity of sailings which are said to include plane sailing, parallel sailing, middle or mid-latitude sailing, mean latitude sailing, traverse sailing, short distance sailing using rhumb-line formulae, mercator sailing, great-circle sailing, approximate great-circle sailing, composite sailing and composite greatcircle sailing. To the student it may seem a reasonable assumption that so many differently named sailings are based upon as many different ways of taking a ship from one position to another, which is of course a nonsense.

If Turner is right that ‘the underlying theory of the traditional approach (to rhumb-line sailing) is obscure’ and that there is a ‘lack of ready availability of a table of distances of parallels of latitude from the equator, it is certainly not the fault of this Journal which precisely 20 years earlier published a paper which gave:1. The correct mathematical theory of rhumb-line sailing on an oblate spheroid.2. The name ‘meridional distance’ to what Turner now calls the L(φ) function.3. A table to reduce latitude to meridional distance.4. A rule of thumb procedure to calculate rhumb-line distances correctly with no more labour than that which has always been used to do it wrongly.5. A method (with table) for use on the spheroid when the track angle is nearly 90° and the method Turner discusses is impracticable.6. A survey of methods and tables then current.