For the correct transformation from grid coordinates of a system to another one, not only the projections and their parameters should be known in both systems but also the datums of them. In ideal case, the two grids are interpreted in the same datum. However, in most cases, they are not.
The three possible ways of the coordinate transformations are shown in Fig. 24. Of course, in the datum is the same, the datum transformations are not needed. However, if the datums are different, we shall choose one of these three ways.
The direct transformation is based on higher-order polynomials, whose several parameters are estimated from the coordinates of base points in both systems. Here this method is not discussed, as – albeit it is the most accurate method – its parameters cannot be supported in most GIS packages.
The second way starts with the usage of the inverse projection equations: we calculate the ellipsoidal coordinates in the first datum from the grid coordinates. In second step, using the abridging Molodensky formulas, we transform the ellipsoidal coordinates from the first datum to the second one. Here we shall know that this transformation can be accomplished by correction grids, too. In the final step, we transform the ellipsoidal coordinates in the second datum to the final grid coordinates, using the direct projection equations of the second grid. The errors of this method are because of the ambiguity of the datum transformation, the distortion between the two geodetic networks; the projection equations can be accepted as exact ones. This method is used by the GPS receivers with one difference: the input data is given in WGS84 ellipsoidal coordinates and the method starts with their transformation to the local datum, omitting the first step, the usage of the inverse projection equations. The method is also supported by all GIS packages.
If we know the Burša-Wolf type datum transformation parameters between the two systems, and our software supports this kind of transformation, it needs geocentric coordinates as input data. So, two more steps are applied: transformation from ellipsoidal coordinates to geocentric one and vice versa. The direct case is part of the trivial trigonometry, however computing the ellipsoidal coordinates from the geocentric one is surprisingly difficult in ellipsoidal case. Its closed formulas, or more precisely, its algorithm, was first given in 1989 (the Borkowski method), and its development is still an important research direction. However, the closed Bowring-formulas can be applied. The horizontal error of this approximation is below 1 centimeter; that’s why this is used in GIS packages. Therefore, the error of this way is caused again by the ambiguity of the datum transformation.
In our practice, it is very rare when we shall make these computations ourselves. They are programmed in our software or GPS device, the only necessary inputs provided by us are the projection and datum parameters, if they are not pre-set in the application. However for us, the specialists, it is worth to know what is inside the ‘black box’.