User:Peter Mercator/References

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ADAMS 1921 [1]




BUZENGEIGER Legendre theorem on spherical triangles (to fourth order) [5]

CLARKE Geodesy [6]).

DELAMBRE meridian 1798 [7]

GAUSS Legendre theorem on spherical triangles [8]


GEOTRANS converter [10]


KARNEY transverse Mercator [12]

KRUGER transverse Mercator [13]

LAMBERT transverse Mercator [14]

LEE exact [15]

LEE series [16]

LEGENDRE 1 theorem stated not proved [17]

LEGENDRE 2 theorem proved [18]



NADENIK Legendre theorem survey [21]

NELL Legendre theorem to order 6 [22]


NIST [24]

OSBORNE (Spherical trig page 16 Legendre) [25]

OSBORNE Mercator Projections [26]

OSGB [27]

PEARSON Trig textbook Legendre theorem at para41 page103 [28]

PODER [29]

RAPP [30]


SNYDER flattening [32]

SNYDER workbook [33]




TORGE [37]

TROPKFE Legendre theorem possibly in 1740 [38]

UTM [39]



WGS84 [42]

  1. ^ Adams, Oscar S (1921). Latitude Developments Connected With Geodesy and Cartography, (with tables, including a table for Lambert equal area meridional projection). Special Publication No. 67 of the US Coast and Geodetic Survey. A facsimile of this publication is available from the US National Oceanic and Atmospheric Administration (NOAA) at Warning: Adams uses the nomenclature isometric latitude for the conformal latitude of this article.
  2. ^ The Astronomical Almanac published annually by the National Almanac Office in the United States ( and the United Kingdom (
  3. ^ F. W. Bessel, 1825, ¨Uber die Berechnung der geographischen L¨angen und Breiten aus geod¨atischen Vermessungen, Astron. Nachr., 4(86), 241–254, doi:10.1002/asna.201011352, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852–861 (2010), E-print arXiv:0908.1824,
  4. ^ Borre[1]
  5. ^ Buzengeiger, Karl Heribert Ignatz (1818), [[2] "Vergleichung zweier kleiner Dreiecke von gleichen Seiten, wovon das eine sphärisch, das andere eben ist"], Zeitschrift für Astronomie und verwandte Wissenschaften (v6): 264—270 
  6. ^ Clarke, Alexander Ross (1880), Geodesy, Clarendon Press  Recently republished at Forgotten Books.
  7. ^ Delambre, Jean Baptiste Joseph (1798), [[3] Méthodes analytiques pour la détermination d’un arc du méridien] 
  8. ^ Gauss, Karl Friedrich (1841 page=96), [[4] Elementare Ableitung eines zuerst von Legendre aufgestellten Lehrsatzes der sphärischen Trigonometrie journal=Journal für die reine und angewandte Mathematik (vol 2)] 
  9. ^ Gauss, Karl Friedrich, 1825. "Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" Preisarbeit der Kopenhagener Akademie 1822. Schumacher Astronomische Abhandlungen, Altona, no. 3, p. 5–30. [Reprinted, 1894, Ostwald’s Klassiker der Exakten Wissenschaften, no. 55: Leipzig, Wilhelm Engelmann, p. 57–81, with editing by Albert Wangerin, pp. 97–101. Also in Herausgegeben von der Gesellschaft der Wissenschaften zu Göttingen in Kommission bei Julius Springer in Berlin, 1929, v. 12, pp. 1–9.]
  10. ^ Geotrans, 2010, Geographic translator, version 3.0, URL
  11. ^ Hofmann-Wellenhof, B and Moritz, H (2006 and 2005). 'Physical Geodesy (second edition)' ISBN-103211-33544-7.
  12. ^ Karney, Charles F. F. (2010). Transverse Mercator with an accuracy of a few nanometers. To be published in Computational Physics. Available as a preprint [5] with resource material at [6].
  13. ^ Krüger, L. (1912). Konforme Abbildung des Erdellipsoids in der Ebene. Royal Prussian Geodetic Institute, New Series 52.
  14. ^ Lambert, Johann Heinrich. 1772. Ammerkungen und Zusatze zurder Land und Himmelscharten Entwerfung. In Beyträge zum Gebrauche der Mathematik und deren Anwendung, part 3, section 6) name=wangerin>Albert Wangerin (Editor), 1894. Ostwald's Klassiker der exacten Wissenschaften (54). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the University of Michigan Historical Math Library.
  15. ^ Lee, L.P. (1976). Conformal Projections Based on Elliptic Functions. Supplement No. 1 to Canadian Cartographer, Vol 13. (Designated as Monograph 16). Toronto: Department of Geography, York University. A report of unpublished analytic formulae involving incomplete elliptic integrals obtained by E.H. Thompson in 1945. The article may be purchased from University of Toronto[7]. At the present time (2010) it is necessary to purchase several units in order to obtain the relevant pages: pp 1–14, 92–101 and 107–114.
  16. ^ Lee L P, (1946). Survey Review, Volume 8 (Part 58), pp 142–152. The transverse Mercator projection of the spheroid. (Errata and comments in Volume 8 (Part 61), pp 277–278.NAG WGS84 on the site of National Geodetic Survey
  17. ^ Legendre, Adrien-Marie (1787), Mémoire sur les opérations trigonométriques, dont les résultats dépendent de la figure de la Terre, p. 7-8 (Article  VI[8] 
  18. ^ Legendre, Adrien-Marie (1798), [[9] Méthode pour déterminer la longueur exacte du quart du méridien d’après les observations faites pour la mesure de l’arc compris entre Dunkerque et Barcelone], p. 12-14 (Note III[10])  This article is included in the work of Delambre.
  19. ^ Maxima, 2009, A computer algebra system, version 5.20.1, URL
  20. ^ Maling, Derek Hylton (1992), Coordinate Systems and Map Projections (second ed.), Pergamon Press, ISBN 0080372333 .
  21. ^ Nádeník, Zbyněk, [[11] Legendre theorem on spherical triangles] 
  22. ^ NELL (1874), Zur höherin Geodäsie, p. 324  Section A of this paper proves the Legendre theorem to the sixth order. (Page 329)
  23. ^ Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [12]
  24. ^ Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F. et al., eds. (2010,), NIST Handbook of Mathematical Functions, Cambridge University Press 
  25. ^ Osborne, Peter (2013), Spherical Trigonometry, (Appendix D of The Mercator Projections), p. 16 
  26. ^ Osborne, Peter (2013), The Mercator Projections 
  27. ^ A guide to coordinate systems in Great Britain. This is available as a pdf document at [13]]
  28. ^ Pearson, Henry (1831), A syllabus of plane and spherical trigonometry, Cambridge.  Legendre's theorem is at Article 41, page103 [14]
  29. ^ K. E. Engsager and K. Poder, 2007, A highly accurate world wide algorithm for the transverse Mercator mapping (almost), in Proc. XXIII Intl. Cartographic Conf. (ICC2007), Moscow, p. 2.1.2.
  30. ^ Rapp, Richard H (1991), Geometric Geodesy, Part I, [15] 
  31. ^ Redfearn, J C B (1948). Survey Review, Volume 9 (Part 69), pp 318–322, Transverse Mercator formulae.
  32. ^ Snyder, John P (1993), Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, ISBN 0-226-76747-7 
  33. ^ Snyder, John P. (1987), Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395, United States Government Printing Office, Washington, D.C. 
  34. ^ Stuifbergen, N 2009, Wide zone transverse Mercator projection, Technical Report 262, Canadian Hydrographic Service, URL
  35. ^ Thomas, Paul D (1952). Conformal Projections in Geodesy and Cartography. Washington: U.S. Coast and Geodetic Survey Special Publication 251.
  36. ^ Tobler, Waldo R, Notes and Comments on the Composition of Terrestrial and Celestial Maps, 1972. University of Michigan Press
  37. ^ Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8
  38. ^ Tropfke, Johannes (1903), [[16] Geschichte der Elementar-Mathematik (Volume 2).], Verlag von Veit, p. 295 
  39. ^ J. W. Hager, J.F. Behensky, and B.W. Drew, 1989, The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS), Technical Report TM 8358.2, Defense Mapping Agency, URL publications/tm8358.2/TM8358 2.pdf.
  40. ^ Vincenty (PDF)
  41. ^ Albert Wangerin (Editor), 1894. Ostwald's Klassiker der exacten Wissenschaften (54). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the University of Michigan Historical Math Library.
  42. ^ The WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2 page 3-1.