ALGEBRAIC GEODESY AND GEOINFORMATICS
CONTENTS INTRODUCTION MOTIVATION MODERN CHALLENGES FACING THE CHALLENGES CONCLUDING REMARKS PART I ALGEBRAIC SYMBOLIC AND NUMERIC METHODS BASICS OF RING THEORY SOME APPLICATIONS TO GEODESY AND GEOINFORMATICS NUMBERS FROM OPERATIONAL PERSPECTIVE NUMBER RINGS BASICS OF POLYNOMIAL THEORY POLYNOMIAL EQUATIONS POLYNOMIAL RINGS POLYNOMIAL OBJECTS AS RINGS OPERATIONS “ADDITION” AND “MULTIPLICATION” FACTORING POLYNOMIALS POLYNOMIAL ROOTS MINIMAL POLYNOMIALS UNIVARIATE POLYNOMIALS WITH REAL COE?CIENTS QUADRATIC POLYNOMIALS CUBIC POLYNOMIALS QUARTIC POLYNOMIALS METHODS FOR INVESTIGATING ROOTS LOGARITHMIC AND CONTOUR PLOTS ON COMPLEX PLANE ISOGRAPH SIMULATOR APPLICATION OF INVERSE SERIES COMPUTATION OF ZEROS OF POLYNOMIAL SYSTEMS GROEBNER BASIS THE ORIGIN BASICS OF GROEBNER BASIS BUCHBERGER ALGORITHM MATHEMATICA COMPUTATION OF GROEBNER BASIS MAPLE COMPUTATION OF GROEBNER BASIS POLYNOMIAL RESULTANTS RESULTANTS: AN ALTERNATIVE TO GROEBNER BASIS SYLVESTER RESULTANTS MULTIPOLYNOMIAL RESULTANTS F MACAULAY FORMULATION B STURMFELS’ FORMULATION THE DIXON RESULTANT BASIC CONCEPTS FORMULATION OF THE DIXON RESULTANT DIXON’S GENERALIZATION OF THE CAYLEY-B´EZOUT METHOD IMPROVED DIXON RESULTANT - KAPUR-SAXENA-YANG METHOD HEURISTIC METHODS TO ACCELERATE THE DIXON RESULTANT EARLY DISCOVERY OF FACTORS: THE EDF METHOD LINEAR HOMOTPY INTRODUCTORY REMARKS BACKGROUND TO HOMOTOPY DE?NITION AND BASIC CONCEPTS SOLVING NONLINEAR EQUATIONS VIA HOMOTOPY TRACING HOMOTOPY PATH AS INITIAL VALUE PROBLEM TYPES OF LINEAR HOMOTOPY FIXED POINT HOMOTOPY NEWTON HOMOTOPY START SYSTEM FOR POLYNOMIAL SYSTEMS SOLUTIONS OF OVERDETERMINED SYSTEMS ESTIMATING GEODETIC AND GEOINFORMATICS UNKNOWNS ALGEBRAIC LEAST SQUARE SOLUTION (ALESS) TRANSFORMING OVERDETERMINED SYSTEMS TO DETERMINED SOLVING THE DETERMINED SYSTEM GAUSS-JACOBI COMBINATORIAL ALGORITHM COMBINATORIAL APPROACH: THE ORIGIN LINEAR AND NONLINEAR GAUSS-MARKOV MODELS GAUSS-JACOBI COMBINATORIAL FORMULATION COMBINATORIAL SOLUTION OF NONLINEAR GAUSS-MARKOV MODEL CONSTRUCTION OF MINIMAL COMBINATORIAL SUBSETS OPTIMIZATION OF COMBINATORIAL SOLUTIONS THE GAUSS-JACOBI COMBINATORIAL ALGORITHM EXTENDED NEWTON-RAPHSON METHOD THE STANDARD NEWTON-RAPHSON APPROACH EXAMPLES OF LIMITATIONS OF THE STANDARD APPROACH OVERDETERMINED POLYNOMIAL SYSTEMS OVERDETERMINED NON-POLYNOMIAL SYSTEM DETERMINED POLYNOMIAL SYSTEM UNDERDETERMINED POLYNOMIAL SYSTEM EXTENDING THE NEWTON-RAPHSON APPROACH USING PSEUDOINVERSE APPLICATIONS OF THE EXTENDED NEWTON-RAPHSON METHOD PROCRUSTES SOLUTION PROCRUSTES: ORIGIN AND APPLICATIONS PROCRUSTES AND THE MAGIC BED MULTIDIMENSIONAL SCALING APPLICATIONS OF PROCRUSTES IN MEDICINE GENE RECOGNITION IDENTI?CATION OF MALARIA PARASITES PARTIAL PROCRUSTES SOLUTION CONVENTIONAL FORMULATION PARTIAL DERIVATIVE FORMULATION THE GENERAL PROCRUSTES SOLUTION EXTENDED GENERAL PROCRUSTES SOLUTION MILD ANISOTROPY IN SCALING STRONG ANISOTROPY IN SCALLING WEIGHTED PROCRUSTES TRANSFORMATION PART II APPLICATIONS TO GEODESY AND GEOINFORMATICS LPS-GNSS ORIENTATIONS AND VERTICAL DE?ECTIONS 10-1 INTRODUCTORY REMARKS 10-2 POSITIONING SYSTEMS 10-3 GLOBAL POSITIONING SYSTEM (GPS) 10-4 LOCAL POSITIONING SYSTEMS (LPS) 10-41 LOCAL DATUM CHOICE IN AN LPS 3-D NETWORK 10-42 RELATIONSHIP BETWEEN GLOBAL AND LOCAL LEVEL REFERENCE FRAMES 10-43 OBSERVATION EQUATIONS 10-5 THREE-DIMENSIONAL ORIENTATION PROBLEM 10-51 PROCRUSTES SOLUTION OF THE ORIENTATION PROBLEM 10-52 DETERMINATION OF VERTICAL DE?ECTION 10-6 EXAMPLE: TEST NETWORK STUTTGART CENTRAL 10-7 CONCLUDING REMARKS CARTESIAN TO ELLIPSOIDAL MAPPING 11-1 INTRODUCTORY REMARKS 11-2 MAPPING TOPOGRAPHICAL POINTS ONTO REFERENCE ELLIPSOID 11-3 MAPPING GEOMETRY 11-4 MINIMUM DISTANCE MAPPING A,A,B 11-5 CONCLUDING REMARKS POSITIONING BY RANGING 12-1 APPLICATIONS OF DISTANCES 12-2 RANGING BY GLOBAL NAVIGATION SATELLITE SYSTEM (GNSS) 12-21 THE PSEUDO-RANGING FOUR-POINTS PROBLEM STURMFELS’ APPROACH GROEBNER BASIS APPROACH 12-22 RANGING TO MORE THAN FOUR GPS SATELLITES EXTENDED NEWTON-RAPHSON SOLUTION HOMOTOPY SOLUTION OF GPS N-POINT PROBLEM LEAST SQUARES VERSUS GAUSS-JACOBI COMBINATORIAL 12-3 RANGING BY LOCAL POSITIONING SYSTEMS (LPS) 12-31 PLANAR RANGING CONVENTIONAL APPROACH SYLVESTER RESULTANTS APPROACH REDUCED GROEBNER BASIS APPROACH PLANAR RANGING TO MORE THAN TWO KNOWN STATIONS ALESS SOLUTION OF OVERDETERMINED PLANAR RANGING PROBLEM 12-32 THREE-DIMENSIONAL RANGING CLOSED FORM THREE-DIMENSIONAL RANGING CONVENTIONAL APPROACHES SOLUTION BY ELIMINATION APPROACH-2 POLYNOMIAL RESULTANTS APPROACH N-POINT THREE-DIMENSIONAL RANGING ALESS SOLUTION EXTENDED NEWTON-RAPHSON’S SOLUTION 12-4 CONCLUDING REMARKS POSITIONING BY RESECTION METHODS 13-1 RESECTION PROBLEM AND ITS IMPORTANCE 13-2 GEODETIC RESECTION 13-21 PLANAR RESECTION CONVENTIONAL ANALYTICAL SOLUTION STURMFELS’ RESULTANT APPROACH 13-22 THREE-DIMENSIONAL RESECTION EXACT SOLUTION SOLUTION OF GRUNERT’S DISTANCE EQUATIONS GROEBNER BASIS SOLUTION OF GRUNERT’S EQUATIONS POLYNOMIAL RESULTANTS’ SOLUTION OF GRUNERT’S DISTANCE EQUATIONS LINEAR HOMOTOPY SOLUTION GRAFAREND-LOHSE-SCHA?RIN APPROACH 3D-RESECTION TO MORE THAN THREE KNOWN STATIONS 13-3 PHOTOGRAMMETRIC RESECTION 13-31 GRAFAREND-SHAN M¨OBIUS PHOTOGRAMMETRIC RESECTION 13-32 ALGEBRAIC PHOTOGRAMMETRIC RESECTION 13-4 CONCLUDING REMARKS POSITIONING BY INTERSECTION METHODS 14-1 INTERSECTION PROBLEM AND ITS IMPORTANCE 14-2 GEODETIC INTERSECTION 14-21 PLANAR INTERSECTION CONVENTIONAL SOLUTION REDUCED GROEBNER BASIS SOLUTION 14-22 THREE-DIMENSIONAL INTERSECTION CLOSED FORM SOLUTION REDUCED GROEBNER BASIS SOLUTION STURMFELS’ RESULTANTS SOLUTION INTERSECTION TO MORE THAN THREE KNOWN STATIONS 14-3 PHOTOGRAMMETRIC INTERSECTION GRAFAREND-SHAN M¨OBIUS APPROACH COMMUTATIVE ALGEBRAIC APPROACHES 14-4 CONCLUDING REMARKS GNSS ENVIRONMENTAL MONITORING 15-1 SATELLITE ENVIRONMENTAL MONITORING 15-2 GNSS REMOTE SENSING 15-21 SPACE BORNE GNSS METEOROLOGY 15-22 GROUND BASED GNSS METEOROLOGY 15-3 REFRACTION (BENDING) ANGLES 15-31 TRANSFORMATION OF TRIGONOMETRIC EQUATIONS TO ALGEBRAIC 15-32 ALGEBRAIC DETERMINATION OF BENDING ANGLES APPLICATION OF GROEBNER BASIS SYLVESTER RESULTANTS SOLUTION 15-4 ALGEBRAIC ANALYSIS OF SOME CHAMP DATA 15-5 CONCLUDING REMARKS ALGEBRAIC DIAGNOSIS OF OUTLIERS 16-1 OUTLIERS IN OBSERVATION SAMPLES 16-2 ALGEBRAIC DIAGNOSIS OF OUTLIERS 16-21 OUTLIER DIAGNOSIS IN PLANAR RANGING 16-22 DIAGNOSIS OF MULTIPATH ERROR IN GNSS POSITIONING 16-3 CONCLUDING REMARKS DATUM TRANSFORMATION PROBLEMS 17-1 THE 7-PARAMETER DATUM TRANSFORMATION AND ITS IMPORTANCE 17-2 ALGEBRAIC SOLUTION OF THE 7-PARAMETER TRANSFORMATION PROBLEM 17-21 GROEBNER BASIS TRANSFORMATION 17-22 DIXON RESULTANT SOLUTION 17-23 GAUSS-JACOBI COMBINATORIAL TRANSFORMATION 17-3 THE 9-PARAMETER (A?NE) DATUM TRANSFORMATION 17-4 ALGEBRAIC SOLUTION OF THE 9-PARAMETER TRANSFORMATION 17-41 THE 3-POINT A?NE TRANSFORMATION PROBLEM SIMPLI?CATIONS FOR THE SYMBOLIC SOLUTION SYMBOLIC SOLUTION WITH DIXON RESULTANT SYMBOLIC SOLUTION WITH REDUCED GROEBNER BASIS 17-42 THE N-POINTS PROBLEM ALESS APPROACH TO OVERDETERMINED CASES HOMOTOPY SOLUTION OF THE ALESS-DETERMINED MODEL 17-43 PROCRUSTES SOLUTION 17-5 CONCLUDING REMARK