The application of commutative algebra to geodesy: two examples

Abstract

Algebra, in particular commutative algebra, is applied here to provide a general unified solution to nonlinear systems of equations encountered in geodesy. Starting with the “Abelian group”, the “polynomial ring” is defined and used to form generators of ideals. By applying Buchberger or polynomial resultant algorithms, these generators are reduced to simple structures often comprising a univariate polynomial in one of the unknowns. The advantage of the proposed unified approach is that it provides exact solutions to geodetic nonlinear systems of equations without the traditional requirements of linearization, iterations or approximate starting values. The commutative algebraic approach therefore alleviates the need for isolated exact solutions to various geodetic nonlinear systems of equations. The procedure is applied to GPS meteorology to compute refraction angles, and Helmert’s one-to-one mapping of topographical points onto the reference ellipsoid.