Adaptation of Soldner equations for solving geodetic problems using the Gauss coordinate system in the local spherical field – part 1: theoretical principles

Abstract

The present work originates from considerations of spherical trigonometry which, we recall, defines the relations existing between the sides and angles of a spherical triangle. In this regard, Soldner's equations, which are based precisely on spherical trigonometry theorems, relate the rectangular geodesic coordinates of two points of the sphere, when the length of the geodesic line that joins them and the plane azimuth from this format are known. in the first of the two vertices considered. The two fundamental problems of geodesy are therefore solved in the aforementioned coordinate system.

The aforementioned equations are generally used in the Cassini-Soldner representation, where the rectangular geodetic coordinates coincide with the plano-cartographic coordinates, since they allow to solve geodetic problems without having to go through the calculation of the linear deformation modulus, as well as of the reduction to the chord. and of the angular deformation (the aforementioned representation is in fact not compliant).

The modification of the Soldner equations, proposed in this study, allows to apply the same operative modality, when it is necessary to face the same geodetic problems, using the Gauss coordinate system. In the latter case, since the width of the spindles greatly exceeds that of the spherical field, Soldner's formulas can progressively fall into defect, when operating at a considerable distance (greater than 180km) from the central meridian and therefore to maintain the rigor unchanged. of the results it is necessary to decrease the length of the geodesic line with respect to that allowed in the spherical field.

This document is followed by a second part dedicated to numerical applications and complements of what is treated here.