ISSN 2083-6473
ISSN 2083-6481 (electronic version)




Associate Editor
Tomasz Neumann

Published by
TransNav, Faculty of Navigation
Gdynia Maritime University
3, John Paul II Avenue
81-345 Gdynia, POLAND
www http://www.transnav.eu
e-mail transnav@am.gdynia.pl
A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General
1 Gdynia Maritime University, Gdynia, Poland
ABSTRACT: We survey last reports and research results in the field of navigational calculations' methods applied in marine navigation that deserve to be collected together. Some of these results have often been rediscovered as lemmas to other results. We present our approach to the subject and place special emphasis on the geometrical base from a general point of view. The geometry of approximated structures implies the calculus essentially, in particular the mathematical formulae in the algorithms applied in the navigational electronic devices and systems. The question we ask affects the range and point in applying the loxodrome (rhumb line) 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. We also cover the subject on approximating models for navigational purposes. Moreover, 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 geodesic-based methods for calculations instead of the loxodromic trajectories in general.
Admiralty Manual of Navigation, 1987, Volume 1, General navigation, Coastal Navigation and Pilotage, Ministry of Defence (Navy), London, The Stationery Office.
Bennett, G.G. 1996. Practical rhumb line calculations on the spheroid, The Journal of Navigation, Vol. 49, No. 1.
Bowditch, N. 2002. American Practical Navigator, Pub. No. 9, The Bicentennial Edition, National Imagery and Mapping Agency.
Bowring, B.R. 1983. The Geodesic Inverse Problem, Bulletin Geodesique, Vol. 57, p. 109 (Correction, Vol. 58, p. 543).
Bowring, B.R. 1984. The direct and inverse solutions for the great elliptic and line on the reference ellipsoid, Bulletin Geodesique, 58, p. 101-108.
Bowring, B.. 1985. The geometry of Loxodrome on the Ellipsoid, The Canadian Surveyor, Vol. 39, No. 3.
Busemann, H. 1955. The geometry of geodesics, Elsevier Academic Press, Washington.
Bourbon, R. 1990. Geodesic line on the surface of a spheroid, The Journal of Navigation, Vol. 43, No. 1.
Carlton-Wippern, K. 1992. On Loxodromic Navigation, The Journal of Navigation, Vol. 45, No. 2., p.292-297.
Chen, C.L., Hsu, T.P., Chang, J.R. 2004. A Novel Approach to Great Circle sailing: The Great Circle Equation, The Journal of Navigation, Vol. 57, No. 2, p. 311-325.
Cipra, B. 1993. What’s Happening in the Mathematical Sciences, Vol. I. RI: American Mathematical Society, Providence.
Earle, M.A. 2000. Vector Solution for Navigation on a Great Ellipse, The Journal of Navigation, Vol. 53, No. 3., p. 473-481.
Earle, M.A. 2005. Vector Solution for Great Circle Navigation, The Journal of Navigation, Vol. 58, No. 3, p. 451-457.
Earle, M.A. 2006. Sphere to spheroid comparisons, The Journal of Navigation, Vol. 59, No. 3, p. 491-496.
Earle, M.A. 2008. Vector Solutions for Azimuth, The Journal of Navigation, Vol. 61, p. 537-545.
Funar, L., Gadgil, S. 2001. Topological geodesics and virtual rigidity, Algebraic & Geometric Topology, Vol. 1, pp. 369-380.
Goldberg, D. 1991. What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys, March.
Gradshteyn, I.S., & Ryzhik, I.M.. 2000. Tables of Integrals, Series and Products, 6th ed. CA Academic Press, San Diego.
Hickley, P. 1987. Great Circle Versus Rhumb Line Cross-track Distance at Mid-Longitude. The Journal of Navigation, Vol.
57, No. 2, p. 320-325 (Forum), May.
Hiraiwa, T. 1987. Proposal on the modification of sailing calculations. The Journal of Navigation, Vol. 40, 138 (Forum).
Hohenkerk, C., & Yallop B.D. 2004. NavPac and Compact Data 2006 – 2010 Astro-Navigation Methods and Software for the PC. TSO, London.
Kaplan, G.H. 1995. Practical Sailing Formulas For Rhumb-Line Tracks on an Oblate Earth, Navigation, Vol. 42, No. 2, pp. 313-326, Summer 1995;
Kelley, J. L. 1955. General Topology. New York: Van Nostrand.
Knippers, R. 2009. Geometric aspects of mapping, International Institute for Geo-Information Science and Earth Observation, Enschede, http://kartoweb.itc.nl/geometrics.
Kopacz, P. 2006. On notion of foundations of navigation in maritime education, 7th Annual General Assembly and Conference AGA-7, The International Association of Maritime Universities IAMU, Dalian, China.
Kopacz, P. 2007. On Zermelo navigation in topological structures, 9th International Workshop for Young Mathematicians “Topology”, pp. 87–95, Institute of Mathematics, Jagiellonian University, Cracow.
Kopacz, P. 2010. Czy pi jest constans?, „Delta” - Matematyka - Fizyka – Astronomia – Informatyka, No 12/2010, Instytut Matematyki, Uniwersytet Warszawski, Warszawa (in Polish).
Lambert, W.D. 1942. The Distance Between Two Widely Separated Points on the Surface of the Earth, The Journal of the Washingtin Academy of Sciences, Vol. 32, No. 5, p. 125, May.
Lyusternik, L.A. 1964. Shortest Paths, Pergamon Press.
Meade, B.K. 1981. Comments on Formulas for the Solution of Direct and Inverse Problems on Reference Ellipsoids using Pocket Calculators, Surveying and Mapping, Vol. 41, March.
Miller, A.R., Moskowitz, I.S., Simmen J. 1991. Traveling on the Curve Earth, The Journal of The Institute of Navigation, Vol. 38.
Nastro, V., Tancredi U. 2010. Great Circle Navigation with Vectorial Methods, Journal of Navigation, Vol.63, Issue 3.
Nord, J., Muller, E. 1996. Mercator’s rhumb lines: A multivariable application of arc length, College Mathematics Journal, No. 27, p. 384–387.
Pallikaris, A., Tsoulos, L., Paradissis, D. 2009a. New meridian arc formulas for sailing calculations in GIS, International Hydrographic Review.
Pallikaris, A., Tsoulos, L., Paradissis, D. 2009b. New calculations algorithms for GIS navigational systmes and receivers, Proceeidngs of the European Navigational Conference ENC - GNSS, Naples, Italy.
Pallikaris, A., Tsoulos L., Paradissis D. 2010. Improved algorithms for sailing calculations. Coordinates, Vol. VI, Issue 5, May, p. 15-18.
Pallikaris, A., Latsas, G. 2009. New algorithm for great elliptic sailing (GES), The Journal of Navigation, vol. 62, p. 493-507.
Prince, R., Williams, R. 1995. Sailing in ever-decreasing circles, The Journal of Navigation, Vol. 48, No. 2, p. 307-313.
Rainsford, H.F. 1953. Long Geodesics on the Ellipsoid, Bulletin Geodesique, Vol. 30, p. 369-392.
Rainsford, H.F. 1955. Long Geodesics on the Ellipsoid, Bulletin Geodesique, Vol. 37.
Sadler, D.H. 1956. Spheroidal Sailing and the Middle Latitude, The Journal of Navigation, Vol. 9, issue 4, p. 371-377.
Schechter, M. 2007. Which Way Is Jerusalem? Navigating on a Spheroid. The College Mathematics Journal, Vol. 38, No. 2, March.
Silverman, R. 2002. Modern Calculus and Analytic Geometry. Courier Dover Publications, Dover.
Sinnott, R. 1984. Virtues of the Haversine. Sky and Telescope. Vol. 68, No. 2.
Snyder, J.P. 1987. Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395.
Sodano, E.M. 1965. General Non-iterative Solution of the Inverse and Direct Geodetic Problems, Bulletin Geodesique, Vol. 75.
Tobler, W. 1964. A comparison of spherical and ellipsoidal measures, The Professional Geographer, Vol. XVI, No 4, p. 9-12.
Torge, W. 2001. Geodesy. 3rd edition. Walter de Gruyter, Berlin, New York.
Tseng, W.K., Lee, H.S. 2007. The vector function of traveling distance for great circle navigation, The Journal of Navigation, Vol. 60, p. 158-164.
Tseng, W.K., Lee, H.S. 2010. Navigation on a Great Ellipse, Journal of Marine Science and Technology, Vol. 18, No. 3, p. 369-375.
Tyrrell, A.J.R. 1955. Navigating on the Spheroid, The Journal of Navigation, vol. 8, 366 (Forum).
Vincenty, T. 1975. Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations. Survey Review, Vol. XXII, No. 176, p. 88-93.
Vincenty, T. 1976. Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations – additional formulas. Survey Review Vol. XXIII, No. 180, p. 294.
Walwyn, P.R. 1999. The Great ellipse solution for distances and headings to steer between waypoints, The Journal of Navigation, Vol. 52, p. 421-424.
Weintrit, A. 2009. The Electronic Chart Display and Information System (ECDIS). An Operational Handbook. A Balke-ma Book. CRC Press, Taylor & Francis Group, Boca Raton – London - New York - Leiden, pp. 1101.
Weintrit, A., Kopacz, P. 2004. Safety Contours on Electronic Navigational Charts. 5th International Symposium ‘Information on Ships’ ISIS 2004, organized by German Institute of Navigation and German Society for Maritime Technology, Hamburg.
Weisstein, E. Great Circle, MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/GreatCircle.html;
Willard, S. 1970. General Topology, Addison-Wesley. Re-printed by Dover Publications, New York, 2004 (Dover edition).
Williams, J.E.D. 1950. Loxodromic Distances on the Terrestrial Spheroid, The Journal of Navigation, Vol. 3, No. 2, p. 133-140.
Williams, E. 2002. Navigation on the spheroidal Earth, http://williams.best.vwh.net/ellipsoid/ellipsoid.html.
Williams, R. 1996. The Great Ellipse on the surface of the spheroid, The Journal of Navigation, Vol. 49, No. 2, p. 229-234.
Williams, R. 1998. Geometry of Navigation, Horwood series in mathematics and applications, Horwood Publishing, Chichester,
Williams, R., Phythian, J.E. 1989. Navigating Along Geodesic Paths on the Surface of a Spheroid, The Journal of Navigation, Vol. 42, p. 129-136.
Williams, R., Phythian, J.E. 1992. The shortest distance between two nearly antipodean points on the surface of a spheroid, The Journal of Navigation, Vol. 45, p. 114.
Wippern, K.C.C. 1988. Surface Navigation and Geodesy, A Parametric Approach, ASFSPA-CECOM Technical Note, March, with addendums;
Wylie, C.R., Barrett, L.C. 1982. Advanced Engineering Mathematics,McGraw-Hill, p. 834.
Zukas, T. 1994. Shortest Spheroidal Distance, The Journal of Navigation, Vol. 47, p. 107-108.
Citation note:
Weintrit A., Kopacz P.: A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 5, No. 4, pp. 507-517, 2011
Authors in other databases:
Piotr Kopacz:

Other publications of authors:

A. Weintrit, R. Wawruch, C. Specht, L. Gucma, Z. Pietrzykowski
L. Murawski, S. Opoka, K. Majewska, M. Mieloszyk, W. Ostachowicz, A. Weintrit

File downloaded 3103 times

Important: TransNav.eu cookie usage
The TransNav.eu website uses certain cookies. A cookie is a text-only string of information that the TransNav.EU website transfers to the cookie file of the browser on your computer. Cookies allow the TransNav.eu website to perform properly and remember your browsing history. Cookies also help a website to arrange content to match your preferred interests more quickly. Cookies alone cannot be used to identify you.
Akceptuję pliki cookies z tej strony