Application of CAS to geodesy: a ‘live’ approach

1. The Dixon resultant package has to be loaded: <<Resultant`Dixon` Then write the polynomial system directly p = x2 + y2 – 1; q = x2 + z2 – 1; r = y2 + z2 – 1; The resultant monomial can be received with one command (eliminating y, z from the polynomial system and using ,  auxiliary variables) dr = DixonResultant[{p ,q ,r},{y, z},{, }] And finally the solutions are Solve[dr ==0,x]//Union Figure 3. Calculating rotation matrix in Mathematica Figure 4. 3D intersection Application of CAS to geodesy: a ‘live’ approach P. Zaletnyik1, B. Paláncz2, J.L. Awange3, E.W. Grafarend4 1,2 Budapest University of Technology and Economics - Hungary 3 Curtin University of Technology - Australia, 4 Stuttgart University - Germany Introduction Nowadays, computer algebra systems (CAS) offer the possibility of 'live' interaction to users. This is in contrast to the widely held believe by most scientists, even today, that CAS language is a programming language. CAS can be used like 'live' mathematics for creating, proving as well as evaluating algorithms and expressions in numeric or symbolic form. Most nonlinear geodetic computational problems, e.g., finding initial values for iterative algorithms, avoiding ill-conditioned numerical problems, or finding effective global or local minimums are immaterial when CAS is properly employed. Live application of CAS To solve the nonlinear geodetic problems the Mathematicasoftware can be used effectively. One powerful tool for these problems is the Dixon resultant, which is implemented into Mathematica. To solve the previous threevariate polynomial is very easy with this CAS system. Nonlinear problems in geodesy Ranging by GPS Another tipically nonlinear problem which can be solved symbolically with CAS is position determination using Global Positioning System. The distance of the receiver from i-th satellite is related to the unknown position of the receiver (x1, x2, x3) and the receiver clock bias (x4). For i=1…4: Employing Dixon resultant one can get three linear equations for variables {x1, x2, x3} with parameter x4 eliminating the other two variables. Substituting these results into the last nonlinear equation (f4), then we get a quadratic equation for x4, which can be solved easily. CAS- Computer Algebra System CAS – Computer Algebra Systems integrate the modern numeric and symbolic mathematical algorithms as built-in functions. Their major advantage over traditional programming languages that they are interactive. Besides symbolic and numeric computations, we have also strong visualization capabilities. For solving mathematical problems with CAS, we do not need to study the theory of the algorithms, we just can use them in a simple way. This simplicity allows us to concentrate on the essential task and ignore peripheral matters. The most frequently used CAS systems are: Mathematica, Maple, MuPADand Macsyma. With these systems, symbolic manipulations can be done like symbolic simplifications, differentiation, integration, matrix operations, solution of polynomial systems of equations, polynomial factorization, greatest common divisor etc. Figure 5. Point positioning using Global Navigation Satellite System 3D Helmert and affine transformation The 3D Helmert transformation uses three translation parameters (X0,Y0,Z0), three rotation parameters (a, b,c) in the R rotation matrix (see Fig. 3) and a scale parameter (s) to transform one set of coordinates in a given system into another coordinate system. The 9-parameter affine transformation is the generalization of the 7-parameter similarity transformation model, where 3 different scales represented by a diagonal scale-matrixare used instead of one scale factor. For the determination of the 7 or 9 parameters of the transformation we needat least 3 points with known coordinates in both coordinate systems. The 3D Helmert transformation can be solved symbolically using Dixon resultant and selecting 7 equations from the 9 equation which belongs to the 3 known points. In case of affine transformation a semi-symbolic solution can be given which can provide reduction in computing time comparing with numerical solution. The advantage of using a CAS system is evident not only for solving this problem, but for solving for example matrix calculations symbolically which are also very frequent in geodesy. For example in the coordinate transformation problem (which we will examine later) the rotation matrix (R) need to be calculated with the skew-symmetric matrix (S). To calculate R symbolic matrix operations are needed such as extraction, multiplication and inverse determination. where I3 is a 3x3 identity matrix. The symbolic result can be obtainedusing one Mathematica command (see Figure 3). Solving polynomial system of equations Solution of nonlinear systems of equations is an indispensable task in all geosciences. The basic concept of thesolution of a multivariate nonlinear system of polynomial equations, when the number of equations (n) is equal to the number of variables (m), is to reduce the number of the variables from m to 1. After this reduction, the received monomial can be solved easily mostly numerically. Conclusion CAS proved to be a very effective, user friendly tool for solving nonlinear geodetic problems, like ranging with GPS, intersection, 3D Helmert and affine transformations. The user can formulate his or her problems interactively, in the usual mathematical way and solve them with transparent built-in functions implemented on the bases of algorithms representing the state of art of mathematics without programming knowledge. Contact: Piroska ZaletnyikEmail: piri@agt.bme.hu Let us solve the next threevariate polynomial system, by eliminating two variables (y, z) from the equations). Nonlinear problems in geodesy -Intersection In intersection method angularobservation are considered from known points to the unknown point. Arranging the equations to zero and eliminating x2, x3 with Dixon resultant (as in the previous section) we get a monomial for x1 which can be solved numerically. Variable x2, x3 can be determinded similarly. Figure 1. Solutions of system The received monomial: Acknowledgements The first author wishes to thank to the Hungarian Eötvös Fellowship for supporting her visit at the Department of Geodesy and Geoinformatics of the University of Stuttgart (Germany) where this work has been accomplished. The roots of this monomial are shown in Figure 2. Figure 2. The x oordinates of the solution