The Optimal Universal Transverse Mercator Projection

Abstract

The Korn-Lichtenstein partial differential equations subject to an integrability condition of Laplace-Beltrami type which govern conformal mapping are reviewed. They are completed by an extensive review of deformation measures (Cauchy-Green deformation tensor, Euler-Lagrange deformation tensor, simultaneous diagonalization of a pair of symmetric matrices) extending the Tissot deformation portrait. W.r.t. one system of isometric parameters which cover a surface (oriented two-dimensional Riemann manifold) the d’Alembert-Euler equations (Cauchy-Riemann equations) subject to an integrability condition of Laplace-Beltrami type are solved in real analysis by various systems of functions (fundamental solution: 2d-polynomial, separation of variables) plus a properly chosen boundary value problem, namely the equidistant mapping of one parameter line. Finally the optimal transverse Mercator projection is outlined by solving a boundary value problem of the d’Alembert-Euler equations (Cauchy-Riemann equations) of a biaxial ellipsoid (ellipsoid of revolution) where a dilatation factor of a central meridian is to be determined. It is proven that for a non-symmetric and a symmetric UTM strip the total areal distortion approaches zero once the total departure from an isometry is minimized. According to the “Geodetic Reference System 1980” for a strip [-l _E ,+l _E ] × [B _S ,B _N ] = [-3.5°,+3.5°] × [80°S,84°N] - the standard UTM strip - an optimal dilatation factor is p = 0.999,578, while for a strip [-2°, +2°] × [80°S, 84°N] - the standard Gauβ-Krüger strip - an optimal dilatation factor is p = 0.999,864. The paper is being published in manuscripta geodaetica.