New definitions of isometric latitude and the Mercator projection of the ellipsoid
Vol 7, Issue 2, 2024
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Abstract
The article discusses the interrelationships of the loxodrome or rhumb line, isometric latitude, and the Mercator projection of the rotational ellipsoid. It is shown that by applying the isometric latitude, a very simple equation of the rhumb line on the ellipsoid is obtained. The consequence of this is that the isometric latitude can be defined using the generalized geodetic longitude and not only using the geodetic latitude, as was usual until now. Since the image of the rhumb line in the plane of the Mercator projection is a straight line, the isometric latitude can also be defined using this projection. Finally, a new definition of the normal aspect of the Mercator projection of the ellipsoid is given. It is a normal aspect cylindrical projection in which the images of the rhumb line on the ellipsoid are straight lines in the plane of projection that, together with the images of the meridians in the projection, form equal angles as the rhumb line forms with the meridians on the ellipsoid. The article provides essential knowledge to all those who are interested in the use of maps in navigation. It will be useful for teachers and students studying cartography and GIS, maritime, or applied mathematics. The author uses mathematical methods, especially differential geometry. The assumption is that the readers are no strangers to mathematical cartography.
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1. Lapaine M. New Definitions of the Isometric Latitude and the Mercator Projection. Revue Internationale de Géomatique. 2024; 33(1): 155-165. doi: 10.32604/rig.2024.052258
2. Snyder JP. Map projections: A working manual. Professional Paper. Geological Survey Professional Paper. 1987. doi: 10.3133/pp1395
3. Bugaevsky LM, Snyder JP. Map Projections: A Reference Manual. London, Bristol: Taylor & Francis; 1995.
4. Deakin RE. 3-D coordinate transformations. Surveying and land information systems. 1998; 58(4): 223−234. doi: 10.1017/S0373463315000181
5. Bian SF, Li HP. Mathematical Analysis in Cartography by Means of Computer Algebra System. Cartography - A Tool for Spatial Analysis. Published online August 17, 2012. doi: 10.5772/50159
6. Bermejo-Solera M, Otero J. Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude. Journal of Geodesy. 2008; 83(1): 1-12. doi: 10.1007/s00190-008-0224-y
7. Li H, Bian S, Chen L. The direct calculating formulae for transformations between authalic latitude function and isometric latitude. Geomatics and Information Science of Wuhan University. 2011; 36(7): 843−846.
8. Thomas PD. Conformal projections in geodesy and cartography (United States. Coast and Geodetic Survey. Special publication). National Technical Information Service; 1977.
9. Kawase K. Concise derivation of extensive coordinate conversion formulae in the Gauss-Krüger projection. Bulletin of the Geospatial information authority of Japan. 2013; 60: 1−6.
10. Abee M. The Spread of the Mercator Projection in Western European and United States Cartography. The International Journal for Geographic Information and Geovisualization. 2021; 56(2): 151-165. doi: 10.3138/cart-2019-0024
11. Lapaine M, Frančula N. Web Mercator Projection – One of Cylindrical Projections of an Ellipsoid to a Plane. Kartografija i geoinformacije. 2021; 20(35): 30-47. doi: 10.32909/kg.20.35.2
12. Pápay G. Mercator’s geometric method in the construction of his projection of 1569 (German). KN - Journal of Cartography and Geographic Information. 2022; 72(4): 261-267. doi: 10.1007/s42489-022-00115-5
13. Kerkovits K. Secant Cylinders Are Evil—A Case Study on the Standard Lines of the Universal Transverse Mercator and Universal Polar Stereographic Projections. ISPRS International Journal of Geo-Information. 2024; 13(2): 56. doi: 10.3390/ijgi13020056
14. Lenart AS. Orthodromes and Loxodromes in Marine Navigation. Journal of Navigation. 2016; 70(2): 432-439. doi: 10.1017/s0373463316000552
15. Gao M, Shi GY. Ship-handling behavior pattern recognition using AIS sub-trajectory clustering analysis based on the T-SNE and spectral clustering algorithms. Ocean Engineering. 2020; 205: 106919. doi: 10.1016/j.oceaneng.2020.106919
16. Nicolai R. The map projection of portolan charts. Nieuw archief voor wiskunde. 2021; 5(1): 33−41.
17. Jiao C, Wan X, Li H, et al. Dynamic Projection Method of Electronic Navigational Charts for Polar Navigation. Journal of Marine Science and Engineering. 2024; 12(4): 577. doi: 10.3390/jmse12040577
18. Alexander J. Loxodromes: A Rhumb Way to Go. Mathematics Magazine. 2004; 77(5): 349-356. doi: 10.1080/0025570x.2004.11953279
19. Kos S, Filjar R, Hess M. Differential Equation of the Loxodrome on a Rotational Surface. Available online: https://www.ion.org/publications/abstract.cfm?articleID=8379 (accessed on 9 March 2024).
20. Kos S, Vranić D, Zec D. Differential Equation of a Loxodrome on a Sphere. Journal of Navigation. 1999; 52(3): 418-420. doi: 10.1017/s0373463399008395
21. Elhashash A. Existence, Uniqueness and Angle Computation for the Loxodrome on an Ellipsoid of Revolution. Journal of Geometry and Symmetry in Physics; 2012. doi: 10.7546/jgsp-13-2009-75-88
22. Petrović M. Differential Equation of a Loxodrome on the Spheroid. Naše more. 2007; 54(3-4): 87-89.
23. Petrović M. Orthodrome-Loxodrome Correlation by the Middle Latitude Rule. Journal of Navigation. 2013; 67(3): 539-543. doi: 10.1017/s037346331300074x
24. Weintrit A, Kopcz P. A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General. In: Methods and Algorithms in Navigation. CRC Press; 2014.
25. Babaarslan M, Yayli Y. Differential Equation of the Loxodrome on a Helicoidal Surface. Journal of Navigation. 2015; 68(5): 962-970. doi: 10.1017/s0373463315000181
26. Kovalchuk V, Mladenov IM. λ-Spheres as a New Reference Model for Geoid: Explicit Solutions of the Direct and Inverse Problems for Loxodromes (Rhumb Lines). Mathematics. 2022; 10(18): 3356. doi: 10.3390/math10183356
27. Lambrinos N, Repanidou I, Intzidou G. A Didactical Approach of Rhumb Line vs. Great Circle in Web Mercator Projection for 6th Grade Pupils. Didactic Strategies and Resources for Innovative Geography Teaching. Published online June 10, 2022: 168-202. doi: 10.4018/978-1-7998-9598-5.ch008
28. Tseng WK, Earle MA, Guo JL. Direct and Inverse Solutions with Geodetic Latitude in Terms of Longitude for Rhumb Line Sailing. Journal of Navigation. 2012; 65(3): 549-559. doi: 10.1017/s0373463312000148
29. Viličić M, Lapaine M. Loxodrome and Isometric Latitude. Cartography and Geoinformation. 2024; 23(41).
30. Lapaine M. Mappings in the theory of map projections (Croatian) [PhD thesis]. University of Zagreb, Faculty of Geodesy; 1996.
31. Heck B. Calculation methods and evaluation models for land surveying (German). Herbert Wichmann Verlag, Karlsruhe; 1987.
32. Lapaine M. Isometric latitude and rhumb line (Croatian). Geodetski list. 1993; 1: 5−14.
DOI: https://doi.org/10.24294/jgc.v7i2.6694
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