Simple and highly accurate formulas for the computation of Transverse

Abstract

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.