Abstract
The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa–Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.
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Appendix: A numerical example for different solutions of the 3D, 7-parameter transformation
Appendix: A numerical example for different solutions of the 3D, 7-parameter transformation
We are considering the example in Awange and Grafarend (2002), which was also used in Závoti (2013) (Table 2). The origins of the two coordinate systems are given in VGS84 and in a local system. To verify the numerical computations, we have written a MATLAB program, which allows a choice between solutions I, II, and III (see chapter 3, formula (10), (12), (13)) for the scale-factor. Following the determination of the scale-factor, our procedure uses in all three cases our linear model. Thus the rotation and shift parameters of the Bursa–Wolf model are also determined from the linear model. The equivalence of the two solutions II and III has already been proved in Papp (2013) and Závoti (2013).
The rotation angles α, β and γ can be obtained from the R rotation matrix (40) using Eq. (3). It can be seen that there is no need to give a starting value, the equations have not to be expanded, there is no need for iteration, and the procedure can be used for arbitrary rotation angles. The results for the solution of the nonlinear problem using the algorithms of the present work are given in Table 3.
Both methods yield, within computational precision, the following identical values for the Cardan-angles:
α = −0.9984976709[''] β = −0.8936957645[''] γ = 0.9930877298['']
Because solutions II and III yield the same numerical values, these common values are only listed once in the header of Table 3 under the entry Bursa–Wolf.
We note that the quaternions q 1, q 2 and q 3, and the parameters a, b, and c of the skew-symmetric matrix \( {\varvec{C}}^{{\mathbf{'}}} \) are equal only within computational precision. Differences in subsequent digits follow from Eq. (42). Greater differences exist between the scale-factors λ and the shift parameters of the two methods. The differences in the scale-factors are a consequence of the differences between Eqs. (10) and (12), and this may result in the relative differences in the shift parameters. The Bursa–Wolf model makes the measurement errors minimal by an exclusive use of least squares methods, while our model can also be applied not only in least squares cases.
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Závoti, J., Kalmár, J. A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation. Acta Geod Geophys 51, 245–256 (2016). https://doi.org/10.1007/s40328-015-0124-6
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DOI: https://doi.org/10.1007/s40328-015-0124-6