MERCATOR PROJECTION ON THE SPHERE, A DEDUCTION WITHOUT MATHEMATICAL GAP

Isaac Ramos; Andréa de Seixas; Silvio Jacks dos Anjos Garnés; Lucas Gonzales Lima Pereira Calado

doi:10.4215/rm2025.e24014

Isaac Ramos Federal University of Pampa, Itaqui (RS), Brazil http://orcid.org/0009-0009-9657-4060
Andréa de Seixas Federal University from Pernambuco, Recife (PE), Brazil http://orcid.org/0000-0002-5879-4902
Silvio Jacks dos Anjos Garnés Federal University from Pernambuco, Recife (PE), Brazil http://orcid.org/0000-0002-0098-6645
Lucas Gonzales Lima Pereira Calado Federal University of Pernambuco, Recife (PE), Brazil http://orcid.org/0000-0002-2456-8288

DOI: https://doi.org/10.4215/rm2025.e24014

Abstract

Map projection is the mathematical process of converting the Earth's surface, considered as a sphere or an ellipsoid, into a map. This conversion is performed by projecting the Earth's points onto a surface, which can be a plane, a cone, or a cylinder. Its basic objective is to develop a mathematical basis for creating maps, essential in areas such as cartography, geodesy, and navigation. It would be ideal if all maps were isometric, but for large areas, the curvature of the Earth makes it impossible, causing distortions. For the reasons above, the mathematics behind map projection is complex, but it is important to understand it. Among the most varied types, the Mercator projection, created by Gerard Mercator in 1569, is a conformal cylindrical projection, widely used in navigation, as it represents the rhumb lines on the map as straight lines, but, despite preserving angles, it generates other distortions. The objective of this article is to present a complete mathematical derivation of the Mercator projection on the sphere, avoiding simplifications and omissions as much as possible. As an application, the deduced equations will be used to implement a visualization of the continents in Python.

Keywords: Mathematical Cartography, Mapping, Cylindrical Conformal Projection.

Author Biographies

Isaac Ramos, Federal University of Pampa, Itaqui (RS), Brazil

Master in Geodetic Sciences and Geoinformation Technologies. Federal University of Pernambuco, UFPE, Brazil.

Andréa de Seixas, Federal University from Pernambuco, Recife (PE), Brazil

PhD in Geodetic Engineering (Doctor of Technical Sciences) from the Technical University of Vienna, Austria (2001). She is currently Associate Professor IV at the Federal University of Pernambuco. She has experience in the area of Geosciences, with emphasis on Geodesy, working mainly on the following topics: Topographic and Geodetic Surveys, Cadastral Surveys, Reference Systems, Geodetic Instrumentation, Geodetic Engineering, Deformation Measurement, Settlement Control and Monitoring, Optical 3D Measurement Techniques. She developed activities focused on Judicial Expertise designated by a Federal Judge, regarding the Definition of Intermunicipal Limits (2009 to 2012) and Coastal Dynamics at the Mouth of the São Francisco River and Adjacencies (2013 to 2016). She is a member of the Postgraduate Program in Geodetic Sciences and Geoinformation Technologies at the Federal University of Pernambuco (she served as Vice Coordinator of Postgraduate Studies from 2005 to 2008 and 2015 to 2016 and as Coordinator of Postgraduate Studies from 2009 to 2010 and 2017 to 2021). She teaches in the Undergraduate Courses in Cartographic and Surveying Engineering, Civil Engineering, Architecture and Urbanism and in the Postgraduate Course in Geodetic Sciences and Geoinformation Technologies. She works in the training of human resources, supervises monitoring, scientific initiations, and completion work for undergraduate and postgraduate courses. Awarded the Order of Cartographic Merit - OMC in the rank of Officer, ceremony commemorating the Day of the Cartographic Engineer in 2016.

Silvio Jacks dos Anjos Garnés, Federal University from Pernambuco, Recife (PE), Brazil

PhD in Geodetic Sciences from the Federal University of Paraná (2001). He worked from 1994 to 2008 at UNIDERP in the courses of Surveying Engineering, Civil Engineering and Agronomy. For seven years he coordinated the geoprocessing laboratory at UNIDERP and was a professor in the postgraduate programs in Environment and Regional Development and Agroindustrial Production and Management, in the chairs of geoprocessing and precision agriculture. Since the end of 2008 he has been a professor at the Federal University of Pernambuco, in the Department of Cartographic Engineering of the Cartographic and Surveying Engineering Course and of the postgraduate program in Geodetic Sciences and Geoinformation Technologies, working with greater emphasis on Geodesy, mainly in the following themes: GNSS positioning, physical geodesy, observation adjustment, tectonic movements, high-precision geodetic networks. He has worked in technical registration and land regularization of federal areas with the SPU from 2012 to 2016. He is the leader of the CNPq Research Group on Land Regularization. He coordinates LAASTRO - UFPE's Astronomy Laboratory with social projects in the area. He has worked in judicial expertise and stands out as a programmer, being the author of the AstGeoTop and CRDF (Digital Certificate of Land Regularization) software. Since 2016, he has been developing Regularization work with the Ministry of Regional Development and since 2021 has been working in partnership between UFPE and the Legal Housing Program of TJPE and CGJPE.

Lucas Gonzales Lima Pereira Calado, Federal University of Pernambuco, Recife (PE), Brazil

PhD in Cartographic Sciences from FCT/Unesp - Presidente Prudente Campus. My main research areas focus on the application of Geodesy to: i) understand sea level variations, ii) estimate the vertical displacement of the Earth's crust and iii) apply satellite positioning. I am the founder and coordinator of the Study Group on Geodesy for Sea Level Monitoring (GEOMAR). I am a permanent member of the Postgraduate Program in Geodetic Sciences and Geoinformation Technologies - UFPE. Institutional affiliation: Department of Cartographic Engineering, Center for Technology and Geosciences.

References

BOWYER, Ronald Edwin; GERMAN, George Arthur. A Guide to Map Projections. John Murray (Publishers) Ltd. Norwich: 1959.

BUGAYEVSKIY, Lev Moseevich; SNYDER, John Parr. Map Projections, a Reference Manual. Taylor & Francis Ltd. London, 1995.

DEAKIN, Rodney Edwin. Map Projection Theory. RMIT University, 2003.

DIMITRIJEVIĆ, Aleksandar; MILOSAVLJEVIĆ, Aleksandar; RANČIĆ, Dejan. Efficient Distortion Mitigation and Partition Reduction in Mapping Global Geodata: Dual Orthogonal Equidistant Cylindrical Projection Approach. ISPRS Int. J. Geo-Inf. 2023, 12, 289.

FRANČULA, Nedjeljko. Kartografske projekcije, skripta, Geodetski fakultet Sveučilišta u Zagrebu, 2004.

KENNEDY, Melita; KOPP, Steve. Understanding Map Projections. Environmental Systems Research Institute, Inc.New York: 2001.

KESSLER, Fritz; BATTERSBY, Sarah. Working with Map Projections: A Guide to Their Selection. CRC Press. Boca Raton: 2019.

KIMERLING, Jon; MUEHRCKE, Phillip; MUEHRCKE, Juliana. Map use: Reading, Analysis, and Interpretation, Fifth Edition. JP Publications. Madison: 2005.

KRAKIWSKY, Edward. Conformal Map Projections in Geodesy. Technical Report n° 217. Department of Geodesy and Geomatics Engineering, University of New Brunswick. Fredericton: 1973.

LAPAINE, Miljenko. Companions of the Mercator Projection. Geoadria n°. 1, Vol. 24, 2019, p-p 7-21.

LAPAINE, Miljenko. New Definitions of the Isometric Latitude and the Mercator Projection. Revue Internationale de Géomatique, v. 33, p. 155-165, 2024.

LAPAINE, Miljenko; DIVJAK, Ana Kuveždić. Famous People and Map Projections. In: Choosing a Map Projection. Lecture Notes in Geoinformation and Cartography Subseries: Publications of the International Cartographic Association (ICA). Editors: Miljenko Lapaine E. Lynn Usery. Springer International Publishing, 2017.

LEE, Laurence Patrick. The transverse Mercator projection of the spheroid, Survey Review, 8 (Part 58), pp.142–152, 1945.

MALING, Derek Hylton. Coordinate Systems and Map Projections, second edition. Pergamon Press. Oxford: 1992.

MONMONIER, Mark. Rhumb lines and map wars : a social history of the Mercator projection. The University of Chicago Press, Chicago: 2004.

OSBORNE, Peter. The Mercator Projections. Edinburgh, 2013.

PEARSON, Frederick. Map ProjectionsTheory and Applications (1st ed.). CRC Press, Boca Raton: 1990.

REDFEARN, John Charles Burford. Transverse Mercator formulae, Survey Review, 9 (Part 69), pp. 318–322, 1948.

RICHARDUS, Peter; ADLER, Ron. Map projections for geodesists, cartographers and geographers. North-Holland publishing company. Amsterdam: 1972.

SANTOS, Adeildo Antão dos. Representações Cartográficas. Editora universitária da Universidade Federal de Pernambuco. Recife: 1985.

SMETANOVÁ, Dana; VARGOVÁ, Michaela; BIBA, Vladislav; HINTERLEITNER, Irena. Mercator’s Projection – a Breakthrough in Maritime Navigation. NAŠE MORE, 2016, 63 (3 Special Issue), 182-184.

SNYDER, John Parr. Map Projections, a Working Manual. United States Geological Survey Professional Paper 1395. United States Government Printing Office. Washington: 1987.

SNYDER, John Parr. Map Projections. In: Encyclopedia of Planetary Sciences, ed. James H. Shirley. Chapman & Hall. London: 1997.

TOBLER, Waldo. A new companion for Mercator. Cartography and Geographic Information Science, 2017, 45(3), 284–285.

TOBLER, Waldo. A Classification of Map Projections. Annals of the Association of American Geographers, 1962, vol 52: p-p 167-175.

VERMEER, Martin; RASILA, Antti Map of the World: An Introduction to Mathematical Geodesy. CRC Press. Boca Raton: 2020.