May 2010 | 15 tion the navigator is interested only in the inverse RLS problem that solves for the “course to steer” and the “sailing distance” from one location to another. In contemporary navigational systems (ECDIS, ECS) the accurate solu- tion of the direct RLS is required for the portrayal of the RLS paths on the displayed electronic chart. The RLS paths (loxodromes) are depicted on the electronic chart through the calculation of the coordinates of intermediate points between the points of departure and destination for specific distance inter- vals and for a given steering course. Inverse RLS problem This section of the algorithm consists of three parts: Part 1: Calculation of Meridian distance The calculation of the length of the arc of the meridian is a prerequisite not only for RLS calculations, but also for “geodesic sailing” and “great elliptic sailing”. The fundamental equation for the calculation of the length of the meridian arc on the ellipsoid (figure 1), is: (1) Where R M is the radius of curvature of the meridian. Replacing the well known value of R M in (1) we obtain (2): (2) Formula 2 can be transformed into an elliptic integral of the second type, which cannot be evaluated in a “closed” form. The calculation can be performed either by numerical integration methods, such as Simpson’s rule, or by the binomial expansion of the denominator to rapidly converging series, retention of a few F rom the early days of the develop- ment of basic navigational software built into satellite navigational receivers, it has been noted that for the sake of sim- plicity and a number of other reasons, this navigational software is often based on methods of limited accuracy [1]. It is surprising that even nowadays the use of navigational software is still used in a loose manner, sometimes ignoring basic principles and adopting oversimplified as- sumptions and errors such as the wrong mixture of spherical and ellipsoidal calcula- tions in different steps of the solution of a particular sailing problem [2]. The lack of official standardization on both the “accuracy required” and the equivalent “methods employed”, in conjunction to the “black box solutions” provided by GNSS navigational receivers and navigational systems (ECDIS and ECS) suggest the necessity of a thorough examination of the issue of sailing calculations for navigational systems and GNSS receivers [3]. New formulas have been derived for both Rhumb Line Sailing (RLS) on the ellipsoid and Great Elliptic Sailing (GES). The RLS formulas result from the analysis of the geometry of the loxodrome on the ellipsoid and are simpler and faster than those traditionally used for RLS on the ellipsoid, which are normally based on the Mercator projection formulas. The proposed new algorithm for Great Elliptic Sailing (GES), provides extremely high accuracies comparable to those obtained by the computations of geodesics. Numerical tests show that discrepan- cies in the computed distances between “geodesic” and “great elliptic arc” are practically negligible for navigation [4]. Rhumbline Sailing (RLS) The proposed formulas for rhumbline sailing calculations on the ellipsoid solve both the direct and the inverse RLS problem. In traditional naviga- NAVIGATION Improved algorithms for sailing calculations This paper presents new algorithms for rhumbline sailing (RLS) and great elliptic sailing (GES) calculations for route planning and portrayal of navigational paths on Electronic Chart Systems Figure 1: The meridian arc Fig-2: The geometry of the loxodrome (rhumbline) on the ellipsoid Athanasios Pallikaris Associate Professor Hellenic Naval Academy palikaris@snd.edu.gr Lysandros Tsoulos Professor, National Technical University of Athens lysandro@central. ntua.gr Demitris Paradissis Professor, National Technical University of Athens dempar@central.ntua.gr P 1 KP 2 : infintesimal right triangle with hypotenuse ds on the rhumbline