A Solution of the Korn-Lichtenstein Equations of Conformal Mapping which Directly Generates Ellipsoidal Gauß-Krüger Conformal Coordinates or the Transverse Mercator Projection
The differential equations which generate a general conformai mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn-Lichtenstein equations subject to the integrability conditions of type vectorial Laplace-Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and normal ellipsoidal latitude. The related coefficient constraints are collected and the constraints to the general solution of the Korn-Lichtenstein equations which directly generates Gauß-Krüger conformai coordinates as well as the Universal Transverse Mercator Projection (UTM) in one step, avoiding any intermediate isometric coordinate representation, are presented. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. The detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form. In addition, a fresh derivation of the Korn-Lichtenstein equations of conformai mapping for a (pseudo-) Riemann manifold of arbitrary dimension extending initial results for higher-dimensional manifolds of Riemann type is presented.