Datum transformation by Gauss projection of the ellipsoid onto the sphere and bilinear function of complex variable: An application to ETRS89-ED50

In this paper a simple method for accurate transformation of geodetic coordinates is presented. The process uses the regional network of points allocated in both geodetic systems researching only the two-dimensional problem (longitude and latitude). Basically, transformation of any point P consists to determine first the three network vertexes nearest to P such as this point is located inside the triangle. Computing both geodetic systems separately, these three vertexes are projected from the ellipsoid onto the Gauss sphere tangent along a parallel. Given that both geodetic variables are isometric on the sphere, and since only three network vertexes have been considered, it can find of closed way the bilinear transformation of complex variable which applies the triangle in study of a sphere in the other. Starting from here, the P image is obtained projecting this point on the sphere corresponding to their geodetic system, applying it later on the other sphere, and computing finally the inverse transformation which projects this sphere on the second ellipsoid. This procedure achieves higher accuracy than matrix spatial methods of seven-parameters, and it could be considered of the same order that methods based on distortion analysis. However, besides being simpler, the presented process provides better results there where regional reference network is not very dense.