Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude |
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| Authors: | Mercedes Bermejo-Solera Jesús Otero |
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| Institution: | (1) Departamento de Física, Centro Universitario de Mérida (Ingeniería Técnica en Topografía), Universidad de Extremadura, Mérida, 06800, Spain;(2) Instituto de Astronomía y Geodesia (UCM-CSIC), Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain |
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| Abstract: | 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.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. |
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| Keywords: | Geometrical geodesy Map projections Cartography Conformal mapping Perturbation theory |
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