Algebraic geodesy and geoinformatics [extra materials extras.springer.com

The book presents modern and efficient methods for solving Geodetic and Geoinformatics algebraic problems. Numerous examples are illustrated with Mathematica using the computer algebra techniques of Ring, Polynomials, Groebner basis, Resultants (including Dixon resultants), Gauss-Jacobi combinatorial and Procrustes algorithms, as well as homotopy methods. While these problems are traditionally solved by approximate methods, this book presents alternative algebraic techniques based on computer algebra tools. ¬ This new approach meets such modern challenges as resection by laser techniques, solution of orientation in Robotics, transformation and bundle block adjustment in Geoinformatics, densification of Engineering networks, analytical solution for GNSS-meteorology and many other problems. For Mathematicians, the book provides some practical examples of the application of abstract algebra and multidimensional scaling. While preparing and teaching ‘Introduction to Geodesy I and II’ to undergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taught required some skills in algebra, and in particular, computer algebra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we have attempted to put together basic concepts of abstract algebra which underpin the techniques for solving algebraic problems. Algebraic computational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds, the concepts and techniques presented herein are nonetheless applicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robotics apply resection and intersection techniques which require algebraic solutions. Solution of nonlinear systems of equations is an indispensable task in almost all geosciences such as geodesy, geoinformatics, geophysics (just to mention but a few) as well as robotics. These equations which require exact solutions underpin the operations of ranging, resection, intersection and other techniques that are normally used. Examples of problems that require exact solutions include; • three-dimensional resection problem for determining positions and orientation of sensors, e. g. , camera, theodolites, robots, scanners etc. The book presents modern and efficient methods for solving Geodetic and Geoinformatics algebraic problems illustrated with Mathematica using computer algebra techniques of Ring, Polynomials, Groebner basis, Resultants including Dixon resultant, Gauss-Jacobi combinatorial and Procrustes algorithms as well as homotopy methods. These problems are traditionally solved by approximate methods. This book presents alternative algebraic techniques based on computer algebra tools. This new approach meets such modern challenges as resection by laser techniques, solution of orientation in Robotics, transformation and bundle block adjustment in Geoinformatics, densification of Engineering networks, analytical solution for GPS-meteorology and many other problems. For Mathematicians, the book provides some practical examples of application of abstract algebra and multidimensional scaling. Front Matter....Pages i-xviii Introduction....Pages 1-5 Front Matter....Pages 7-7 Basics of ring theory....Pages 9-16 Basics of polynomial theory....Pages 17-32 Groebner basis....Pages 33-47 Polynomial resultants....Pages 49-62 Linear homotpy....Pages 63-77 Solutions of Overdetermined Systems....Pages 79-99 Extended Newton-Raphson method....Pages 101-110 Procrustes solution....Pages 111-135 Front Matter....Pages 137-137 LPS-GNSS orientations and vertical deflections....Pages 139-153 Cartesian to ellipsoidal mapping....Pages 155-171 Positioning by ranging....Pages 173-216 Positioning by resection methods....Pages 217-248 Positioning by intersection methods....Pages 249-263 GNSS environmental monitoring....Pages 265-287 Algebraic diagnosis of outliers....Pages 289-301 Datum transformation problems....Pages 303-338 Back Matter....Pages 339-377