Groebner Basis in Geodesy and Geoinformatics

Abstract

In geodesy and geoinformatics, most problems are nonlinear in nature and often require the solution of systems of polynomial equations. Before 2002, solutions of such systems of polynomial equations, especially of higher degree remained a bottleneck, with iterative solutions being the preferred approach. With the entry of Groebner basis as algebraic solution to nonlinear systems of equations in geodesy and geoinformatics in the pioneering work “Gröbner bases, multipolynomial resultants and the Gauss Jacobi combinatorial algorithms : adjustment of nonlinear GPS/LPS observations", the playing field changed. Most of the hitherto unsolved nonlinear problems, e.g., coordinate transformation problems, global navigation satellite systems (GNSS)'s pseudoranges, resection-intersection problems in photogrammetry, and most recently, plane fitting in point clouds in laser scanning have been solved. A comprehensive overview of such applications are captured in the first and second editions of our book Algebraic Geodesy and Geoinformatics published by Springer. In the coming third edition, an updated summary of the newest techniques and methods of combination of Groebner basis with symbolic as well as numeric methods will be treated. To quench the appetite of the reader, this presentation considers an illustrative example of a two-dimension coordinate transformation problem solved through the combination of symbolic regression and Groebner basis.