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The Optimal Universal Transverse Mercator Projection

  • E. Grafarend
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 114)

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.

Keywords

Fundamental Solution Conformal Mapping Integrability Condition Deformation Measure Parameter Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • E. Grafarend
    • 1
  1. 1.Geodätisches InstitutUniversität StuttgartStuttgartGermany

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