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Simple and highly accurate formulas for the computation of Transverse

Bermejo-Solera, Mercedes and Otero Juez, Jesús (2009) Simple and highly accurate formulas for the computation of Transverse. Journal of Geodesy, 83 (1). pp. 1-12. ISSN 0949-7714

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A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude lambda 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-Kruger 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 +/- 20A degrees 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 Poincar, method of the small parameter.

Item Type:Article
Uncontrolled Keywords:Geometrical geodesy; Map projections; Cartography; Conformal mapping; Perturbation theory
Subjects:Sciences > Mathematics > Geodesy
ID Code:17284

Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York.

Bermejo M, Otero J (2005) Minimum conformal mapping distortion according to Chebyshev’s principle: case-study over peninsular Spain. JGeod 79(1–3):124–134. doi:10.1007/s00190-005-0450-5

Bugayevskiy LM, Snyder JP (1995) Map projections: a reference manual. Taylor & Francis, London.

DefenseMapping Agency (1989) TheUniversal Grids: UniversalTransverse Mercator and Universal Polar Stereographic. DMA Technical Manual 8358.2, Fairfax.

Dorrer E (2003) From elliptic arc length to Gauss-Krüger coordinates by analytical continuation. In:Grafarend E, KrummFW, Schwarze VS (eds)Geodesy—the challenge of the 3rd millennium. Springer, Berlin, pp 293–298.

Dozier J (1980) Improved algorithm for calculation of UTM coordinates and geodetic coordinates. NOAA Technical Report NESS81,NationalOceanicandAtmosphericAdministration,WashingtoEnríquez 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.

Gradshteyn IS, Ryzhik IM (1980) Table of integrals, series, and products. Academic Press, New York.

Grafarend E, Syffus R (1998) The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the transverse Mercator Projection. J Geod 72(5):282–293. doi:10.1007/s001900050167.

Hotine M (1946) The orthomorphic projection of the spheroid. Empire Surv Rev 8(62):300–311 ntergovernmental Committee on Surveying and Mapping (2002) Geocentric DaTum of Australia: technical manual, Version 2.2., ISBN 0-9579951-0-5.

Johnson WP (2002) The curious history of Faà di Bruno’s formula. Am. Math Monthly 109(3):217–234.

Krüger L (1912) Konforme Abbildung des Erdellipsoids in der Ebene. Druck und Verlag von B.G. Teubner, Leipzig.

KuzminaRP (2000) Asymptotic methods for ordinary differential equations. Kluwer, Dordrecht.

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. 8 1972.

Lee LP (1962) The Transverse Mercator projection of the entire spheroid. Empire Surv Rev 16(123):208–217.

Lee LP (1976) Conformal projections based on elliptic functions. Cartographica, Monograph 16, suplement 1 to Canadian Cartographer, 13, 128 p.

Maplesoft (2005) Maple 10 user manual. Maplesoft, a division of Waterloo Maple Inc., Canada.

Moritz H (1980) Geodetic reference system 1980. Bull Geod

62(3):348–358 Redfearn JCB (1948) Transverse Mercator Formulae. Empire Surv Rev 9(69):318–322.

Struik DJ (1950) Lectures on classical differential geometry. Addison-Wesley, Cambridge.

The Mathworks Inc. (2006a) MATLAB Math. The MathWorks, Inc.,


The Mathworks Inc. (2006b) Mapping toolbox for use with MATLAB. The MathWorks, Inc., Natick.

Thompson EH (1975) A note on conformal map projections. Surv Rev 23(175):17–28

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