The laws of nature in general, and the relations and laws in geodesy in particular can be expressed in most cases by nonlinear equations which are in general solved by transforming them to linear form and applying iteration. The process of bringing the equations to linear form implies neglections and approximation. In certain cases it is possible to obtain exact, correct solutions for nonlinear problems. In the present work we introduce parameters into the rotation matrix, and using this we derive solutions for the 2D and 3D similarity transformations. This method involves no iteration, and it does not require the transformation of the equations to linear form. The scale parameter is determined in both cases by solving a polynomial equation of second degree. This solution is already known, but our derivation is worth consideration because of its simple nature.
Závoti (2002) presented the mathematical description of the interpolation method especially for modeling the orbit of artificial satellites, which is suitable for approaching only certain 9 points. The task in this form stems from Grafarend and Schaffrin's (1993) study. During the time passed since the elaboration of the method, the generalization of the algorithm became necessary in the case when we have an arbitrary amount of measurement points, which must be approached according to a certain principle. The generalized method was successfully applied for modeling geodynamical processes, for describing the motion of the Earth's poles and for analyzing economical time series.
The closed form solution of 7 parameter 3D transformation.The Gauss-Jacobi combinatorial adjustment is applied to solve the 3D transformation problem with 7 parameters, and it is also demonstrated that the combinatorial algorithm gives the same solution as the conventional linear Gauss-Markov model.
A new, alternative solution of the exterior orientation in photogrammetry is given. The exterior orientation of sensors (e.g. camera-systems) is one of the basic tasks of photogrammetry. The parameters for exterior orientation can be determined from the mathematical equations between the image coordinates and the corresponding object or ground coordinates. The mathematical models for this problem have been available since decades, huge program packages utilize the methods which have proved to be successful in practice. In spite of this we propose in this work an alternative solution which does not use iteration and approximative data. The equations in this work are in coherence with the photogrammetric theory of exterior orientation, the only difference is in the mathematical solution. This kind of mathematical treatment of the problem can be considered as novel, its practical application may come later.
In a basic problem of geodesy the directions from points with known coordinates to an unknown (new) point are measured, and then the resulting angles are used to compute the coordinates of the new point. The relations between angles and lengths lead to a system of nonlinear equations of the form
= 0 (
= 1, 2, 3), where each
is a second degree polynomial of the unknown distances
. Two different direct (non-iterative) solutions are discussed: one is based on the Sylvesterdeterminant of the resultant (this is a new result), the other on the Gröbner-bases. We show that in the general case both methods lead to the same equations in one variable and of fourth degree, but in a special case the equations obtained from Sylvester-determinant are of second degree. As a numerical example, three known points and an unknown point were selected in the city of Sopron. The required space angles were used to make the computations yielding the
X, Y, Z
coordinates of the unknown point.We show that the direct solution of the 2D similarity transformation leads to the same result as applying the Gröbner-bases.
Authors:J. Závoti, J. Somogyi, J. Kalmár, and L. Battha
In the Mathematical Geodesy Division of the Geodetic and Geophysical Research Institute of the Hungarian Academy of Sciences research has been done mainly in two areas: theoretical foundation of the evaluation of geodetic measurements and the practical application of theoretical results. These include interpolation methods, robust estimation, time-series analysis. Results of the research have been applied in areas such as photogrammetry, digital terrain model, polar motion, geodynamics.