Navigation  
Geodesic based trajectories in navigation
The paper presents the current and uniform approaches to sailing calculations highlighting recent developments. We published the first part of the paper in May 11. Here we present the concluding part. 





ECDIS APPROACH
In the course of navigation programmes for ECDIS purposes 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 a few 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. This paper presents and recommends the guidelines that should be used for the 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 paper also takes into account situations when a nearindeterminate solution may arise. The data for these problems do not refer to actual terrestrial situations but have been selected for illustrative purposes.
The references also present the review of different approaches to contact formulae for the computation of the position, the distance, and the azimuth along a great ellipse. The proposed alternative formulae are to be primarily used for accurate sailing calculations on the ellipsoid in a GIS environment as in ECDIS and other ECS. Among the ECDIS requirements is the need for a continuous system with a level of accuracy consistent with the requirements of safe navigation. At present, this requirement is best fulfilled by the Global Positioning System (GPS). The GPS system is referenced to World Geodetic System 1984 Datum. Using the ellipsoid model instead of the spherical model attains more accurate calculation of sailing on the Earth. Therefore, we aim to construct a computational procedure for solving the length of the arc of a geodesic path, the waypoints and azimuths along it. We aspire to provide the straightforward formulae involving the great elliptic sailing based on two scenarios. The first is that the departure point and the destination point are known. The second is that the departure point and the initial azimuth are given (direct and inverse geodetic problems on reference ellipsoids).
As a minimum, an ECDIS system must be able to perform the following calculations and conversions [Weintrit, 2009]:
− geographical coordinates to display coordinates, and display coordinates to geographical coordinates;
− transformation from local datum to WGS84;
− true distance and azimuth between two geographical positions;
− geographic position from a known position given distance and azimuth (course);
− projection calculations such as great circle and rhumb line courses and distances;
− “RLGC” difference between the rhumb line and great circle in sailing along the great circle (or great ellipse?).
The ECDIS allows the navigator to create waypoints and routes including setting limits of approach and other cautionary limits. Both rhumb line and great circle routes can be defined. Routes can be freely exchanged between the ECDIS and GPS or ARPA. Route checking facility allows the intended route to be automatically checked for safety against limits of depth and distance as defined by the navigator.
The mariner can calculate and display both a rhumb line and a great circle line and verify that no visible distortion exists between these lines and the chart data. Authors predict the early end of the era of the rhumb line. This line in the natural way will go out of use. Nobody after all will be putting the navigational triangle to the screen of the ECDIS. Our planned route need not be a straight line on the screen. So, why hold this line still in the use? Each ship’s position plotted on the chart can be the starting point of new updated great circle, or saying more closely, great ellipse or geodesic trajectory.
It is an important question whether in the ECDIS time Mercator projection is still essential for marine navigation. Do we really need it? And what about loxodrome? Let start navigation based on geodesics. It is high time to forget the rhumb line navigation and great circle navigation, too. But first we need clear established methods, algorithms and formulae for sailing calculations. But it is already indicating the real revolution in navigation – total revolution. We will be forced to make the revision of such fundamental notions as the course, the heading and the bearing.
And another very important question: do you really know what kind of algorithms and formulae are used in your GPS receiver and your ECS/ECDIS systems for calculations mentioned above? We are almost sure your answer is negative. So, we have got a problem – a serious problem.
REVIEW OF RECENTLY PUBLISHED PAPERS
We surveyed last reports and research results in the field of navigational calculations’ methods applied in marine navigation that deserve to be collected together in [Weintrit, Kopacz, 2011]. Some of these results have often been rediscovered as lemmas to other results. Since 1950 till 2010 many professional magazines and journals published some papers on the great ellipse and on the spheroidal Earth. The following particular problems were discussed among the others: practical rhumb line calculations on the spheroid [Bennet, 1996], geodesic inverse problem, direct and inverse solutions for the great elliptic and line on the reference ellipsoid, loxodromic navigation [CarltonWippern, 1992], formulas for the solution of direct and inverse problems on reference ellipsoids using pocket calculators, geometry of loxodrome on the ellipsoid, geometry of geodesics, geodesic line on the surface of a spheroid [Bourbon, 1990], great circle equation, novel approach to great circle sailing [Chen, Hsu & Chang, 2004], vector function of traveling distance for great circle navigation, great circle navigation with vectorial methods [Nastro & Tancredi, 2010], vector solution for great circle navigation [Earle, 2005], vector solution for navigation on a great ellipse [Earle, 2000], navigation on a great ellipse, great ellipse solution for distances and headings to steer between waypoints [Walwyn, 1999], great ellipse on the surface of the spheroid [Williams, 1996], vector solutions for azimuth [Earle, 2008], sphere to spheroid comparisons [Earle, 2006], great circle versus rhumb line crosstrack distance at midlongitude [Hickley, 1987], modification of sailing calculations, practical sailing formulas for rhumb line tracks on an oblate Earth, distance between two widely separated points on the surface of the Earth, traveling on the curve Earth, new meridian arc formulas for sailing calculations in GIS [Pallikaris, Tsoulos & Paradissis, 2009], new calculations algorithms for GIS navigational systmes and receivers, improved algorithms for sailing calculations [Pallikaris, Tsoulos & Paradissis, 2010], new algorithm for great elliptic sailing (GES) [Pallikaris & Latsas, 2009], shortest paths, sailing in everdecreasing circles [Prince & Williams, 1995], long geodesics on the ellipsoid, spheroidal sailing and the middle latitude [Sadler, 1956], general noniterative solution of the inverse and direct geodetic problems, comparison of spherical and ellipsoidal measures [Tobler, 1964], navigating on the spheroid [Tyrrell, 1955; Williams, 2002], direct and inverse solutions of geodesics on the ellipsoid with application of nested equations, loxodromic distances on the terrestrial spheroid [Williams, 1950], Mercator’s rhumb lines: a multivariable application of arc length, navigating along geodesic paths on the surface of a spheroid [Williams & Phythian, 1989], shortest distance between two nearly antipodean points on the surface of a spheroid, shortest spheroidal distance [Zukas, 1994], navigating on a spheroid.
CONCLUSIONS
This article is written with a variety of readers in mind, ranging from practising navigators to theoretical analysts. It was also our goal to present the current and uniform approaches to sailing calculations highlighting recent developments. Much insight may be gained by considering the examples that have recently proliferated in the literature reviewed above. We present our approach to the subject and place special emphasis on the geometrical base from a general point of view. Of particular interest are geodesic lines, in particular great ellipse calculations. The geometry of modelling structures implies the calculus essentially, in particular the mathematical formulae in the algorithms applied in the navigational electronic device and systems. Thus, is the spherical or spheroidal model the best fit in the local approximations of the Earth surface? We show that generally in navigation the essential calculating procedure refers to the distance and angle measurement what may be transferred to more general geometrical structures, for instance metric spaces, Riemannian manifolds. The authors point out that the locally modelling structure has a different “shape” and thus the different curvature and the flow of geodesics. That affects the calculus provided on it. The algorithm applied for navigational purposes, in particular ECDIS should inform the user on actually used mathematical model and its limitations. The question we also ask affects the range and point in applying the loxodrome sailing in case the ECDIS equipped with the great circle (great ellipse) approximation algorithms of given accuracy replaces the traditional nautical charts based on Mercator projection. The shortest distance (geodesics) depends on the type of metric we use on the considered surface in general navigation. The geodesics can look different even on the same plane if different metrics are taken into consideration. Let us observe for instance the diameter of the parallel of latitude on a conical circle which does not pass its centre. That differs from both the plane and spherical case. Our intuition insists on the way of thinking to look at the diameter as a part of geodesic of the researched surface crossing the centre of a circle. However the diameter depends on the applied metric, thus the shape of the circles researched in the metric spaces depends on the position of the centre and the radius. It is also important to know how the distance between two points on considered structure is determined, where the centre of the circle is positioned and how the diameter passes. Changing the metric causes the differences in the obtained distances. For example π as a number is constant and has the same value in each geometry it is used in calculations. However π as a ratio of the circumference to its diameter can achieve different values, in particular π [Kopacz, 2010]. The navigation based on geodesic lines and connected software of the ship’s devices (electronic chart, positioning and steering systems) gives a strong argument to research and use geodesicbased methods for calculations instead of the loxodromic trajectories in general. The theory is developing as well what may be found in the books on geometry and topology. This motivates us to discuss the subject and research the components of the algorithm of calculations for navigational purposes.
REFERENCES
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CarltonWippern, K. 1992. On Loxodromic Navigation, The Journal of Navigation, Vol. 45, No. 2., p.292297.
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