The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the Transverse Mercator Projection

Abstract.

The differential equations which generate a general conformal 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 ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gauß–Krüger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form.