Journal of Geodesy

, Volume 83, Issue 1, pp 1–12 | Cite as

Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude

Original Article


A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude λ and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values of the TM coordinates are given by conveniently truncated Gauss–Krüger series expansions, we use the maximum norm of the absolute error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals 0.21  mm, whereas for the region of the ellipsoid bounded by the meridians  ±20° the accuracy is equal to 0.3  mm. Our approach is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43  mm. The small parameter of the power series expansion is the square of the eccentricity of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincaré method of the small parameter.


Geometrical geodesy Map projections Cartography Conformal mapping Perturbation theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

190_2008_224_MOESM1_ESM.txt (1 kb)
ESM 1 (TXT 2 kb)
190_2008_224_MOESM2_ESM.txt (1 kb)
ESM 2 (TXT 2 kb)


  1. Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New YorkGoogle Scholar
  2. Bermejo M, Otero J (2005) Minimum conformal mapping distortion according to Chebyshev’s principle: case-study over peninsular Spain. J Geod 79(1–3):124–134. doi:10.1007/s00190-005-0450-5 CrossRefGoogle Scholar
  3. Bugayevskiy LM, Snyder JP (1995) Map projections: a reference manual. Taylor & Francis, LondonGoogle Scholar
  4. Defense Mapping Agency (1989) The Universal Grids: Universal Transverse Mercator and Universal Polar Stereographic. DMA Technical Manual 8358.2, FairfaxGoogle Scholar
  5. Dorrer E (2003) From elliptic arc length to Gauss-Krüger coordinates by analytical continuation. In: Grafarend E, Krumm FW, Schwarze VS (eds) Geodesy—the challenge of the 3rd millennium. Springer, Berlin, pp 293–298Google Scholar
  6. Dozier J (1980) Improved algorithm for calculation of UTM coordinates and geodetic coordinates. NOAA Technical Report NESS 81, National Oceanic and Atmospheric Administration, WashingtonGoogle Scholar
  7. Enríquez C (2004) Accuracy of the coefficient expansion of the Transverse Mercator Projection. Int J Geogr Inf Sci 18(6):559–576. doi:10.1080/13658810410001701996 CrossRefGoogle Scholar
  8. Gradshteyn IS, Ryzhik IM (1980) Table of integrals, series, and products. Academic Press, New YorkGoogle Scholar
  9. Grafarend E, Syffus R (1998) The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gau–Krüger conformal coordinates or the transverse Mercator Projection. J Geod 72(5):282–293. doi:10.1007/s001900050167 CrossRefGoogle Scholar
  10. Hotine M (1946) The orthomorphic projection of the spheroid. Empire Surv Rev 8(62): 300–311Google Scholar
  11. Intergovernmental Committee on Surveying and Mapping (2002) Geocentric DaTum of Australia: technical manual, Version 2.2., ISBN 0-9579951-0-5
  12. Johnson WP (2002) The curious history of Faà di Bruno’s formula. Am Math Monthly 109(3): 217–234CrossRefGoogle Scholar
  13. Krüger L (1912) Konforme Abbildung des Erdellipsoids in der Ebene. Druck und Verlag von B.G. Teubner, LeipzigGoogle Scholar
  14. Kuzmina RP (2000) Asymptotic methods for ordinary differential equations. Kluwer, DordrechtGoogle Scholar
  15. Lambert JH (1772) Anmerkungen und Zusätze zur Entwerfung der Land und Himmelskarten, Berlin. English translation by W.R. Tobler: Notes and comments on the composition of terrestrial and celestial maps, University of Michigan, 1972, Geographical Publication n.~8Google Scholar
  16. Lee LP (1962) The Transverse Mercator projection of the entire spheroid. Empire Surv Rev 16(123): 208–217Google Scholar
  17. Lee LP (1976) Conformal projections based on elliptic functions. Cartographica, Monograph 16, suplement 1 to Canadian Cartographer, 13, 128 pGoogle Scholar
  18. Maplesoft (2005) Maple 10 user manual. Maplesoft, a division of Waterloo Maple Inc., Canada.
  19. Moritz H (1980) Geodetic reference system 1980. Bull Geod 62(3): 348–358CrossRefGoogle Scholar
  20. Redfearn JCB (1948) Transverse Mercator Formulae. Empire Surv Rev 9(69): 318–322Google Scholar
  21. Struik DJ (1950) Lectures on classical differential geometry. Addison-Wesley, CambridgeGoogle Scholar
  22. The Mathworks Inc. (2006a) MATLAB Math. The MathWorks, Inc., Natick.
  23. The Mathworks Inc. (2006b) Mapping toolbox for use with MATLAB. The MathWorks, Inc., Natick.
  24. Thompson EH (1975) A note on conformal map projections. Surv Rev 23(175): 17–28Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de FísicaCentro Universitario de Mérida (Ingeniería Técnica en Topografía), Universidad de ExtremaduraMéridaSpain
  2. 2.Instituto de Astronomía y Geodesia (UCM-CSIC), Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain

Personalised recommendations