THE OPTIMAL UNIVERSAL TRANSVERSE MERCATOR PROJECTION

The Korn-Lichtenstein partial differential equations subject to an int egrability condition of Laplace-Beltrami type which govern conformal m apping are reviewed. They are completed by an extensive review of defo rmation measures (Cauchy-Green deformation tenser, Euler-Lagrange defo rmation tenser, simultaneous diagonalization of a pair of symmetric ma trices) extending the Tissot deformation portrait. W.r.t. one system o f isometric parameters which cover a surface (oriented two-dimensional Riemann manifold) the d'Alembert-Euler equations (Cauchy-Riemann equa tions) subject to an integrability condition of Laplace-Beltrami type are solved in real analysis by various systems of functions (fundament al solution: ad-polynomial, separation of variables) plus a properly c hosen boundary value problem, namely the equidistant mapping of one pa rameter line. Finally the optimal universal transverse Mercator projec tion is outlined by solving a boundary value problem of the d'Alembert -Euler equations (Cauchy-Riemann equations) of a biaxial ellipsoid (el lipsoid of revolution) where a dilatation factor of a central meridian is to be determined. It is proven that for a non-symmetric and a symm etric UTM strip the total areal distortion approaches zero once the to tal departure from an isometry is minimized. According to the ''Geodet ic Reference System 1980' for a strip [I-(E), + I-E] x [B-s, B-N] = [- 3.5, + 3.5 degrees] x [80 degrees S, 84 degrees N]- the standard UTM s trip - an optimal dilatation factor is rho = 0.999, 578, while for a s trip [-2 degrees, + 2 degrees] x [80 degrees S, 84 degrees N] - the st andard Gauss-Kruger strip - an optimal dilatation factor is rho = 0.99 9, 864.