Helmert-Transformation

The Helmert transformation (after Friedrich Robert Helmert, 1843-1917 ), also known as 7- parameter transformation is a transformation of coordinates for three-dimensional Cartesian coordinates, which is often used in geodesy for distortion-free conversion from one to another, also three-dimensional system:

• Transformed vector ...
• Output vector ...

The seven parameters are:

• ... Displacement vector. Contains the three displacements along the coordinate axes
• Scale factor ...
• ... Rotation matrix. Consists of three rotation angles (slight rotations about the coordinate axes ) rx, ry, rz. The rotation matrix is an orthogonal matrix

Thus, the Helmert transformation is a similarity transformation. It is a specialization of the Galilean transformations, including affine transformations include, among others, and projective; However, the latter distort the route lengths.

Calculation of the parameters

When the transformation parameters are unknown, they may have identical points (that is, points whose coordinates are known before and after transformation ) is calculated. Since a total of 7 parameters ( 3 shifts, scale 1, 3 rotations ) are to be determined, at least 2 points and a third point, a coordinate must be known (eg, the z- coordinate). This creates a system of equations with seven equations and as many unknowns, which can be solved.

In practice, you will be anxious to use more points. Through this determination is obtained first, a check on the accuracy of points used and second, the possibility of a statistical evaluation of the results. The calculation is performed in this case with a compensation bill after the Gaussian method of least squares.

To obtain numerically favorable values ​​for the calculation of the transformation parameters, the coordinate calculations of differences in relation to the mean value of the given points is performed.

The two-dimensional case

A special case is the two-dimensional Helmert transformation for plane coordinate systems with low surface area. It is used, inter alia, in geodesy, when a small-scale survey network with overdetermination is connected to the national coordinate system, or in astrometry for easy plate reduction in well maßhältigen photographic plates. The transformation corresponds to a rotation and expansion with parallel displacement in any direction.

It requires only 4 instead of 7 transformation parameters, namely two shifts, one scale factor and one rotation. The calculation of these four parameters requires two identical points in the two coordinate systems; are given more points, again carried out an adjustment.

Application

The Helmert transformation is used in geodesy, among other things, to transform coordinates of points from one coordinate system to another. Thus, for example, the conversion of points of the regional land surveying in the WGS84 used for GPS positioning is possible.

Here, y are the Gauss-Krueger - coordinates x, gradually converted plus the height H in 3D values:

This will terrestrially measured positions with GPS data comparable; latter - can be introduced as new points in the National Survey - transformed in reverse order.

The third step ( the Helmert transformation ) is the application of a rotation matrix, the multiplication by a scale factor ( μ is close to the value 1) and the addition of a shift C.

Since the sub-operations of this transformation all result in only small changes, the coordinates of a reference system B may be derived by the following formula from the reference system A:

The angle of rotation, and are to be used at their value in radians.

Or for each component:

All parameters are multiplied by -1 for the inverse transformation.

The seven parameters are determined for each region (surveying Separately, state, etc. ) with 3 or more " identical points " of both systems. In case of determining the small inconsistencies (usually only a few cm ) offset by adjustment by the least squares method - that is, remove the statistically plausible way.

Default parameter sets

In the examples it is standard parameter sets for the 7- parameter transformation (or datum transformation ) between two ellipsoids. For the transformation in the opposite direction, the sign must be changed for all parameters. The angle of rotation, and are sometimes referred to as κ, φ and ω. The datum transformation from WGS84 by Bessel is interesting in that the GPS technology refers to the WGS84 ellipsoid, which is widespread in Germany Gauss -Krüger coordinate system in general, however, the Bessel ellipsoid.

Since the Earth has not a perfect ellipsoid shape, but is described as a geoid, not enough for a datum transformation with measurement accuracy of the standard set of parameters. Geoidform the earth is instead defined by a plurality of ellipsoids. Depending on the actual location of the parameters are " locally bestangleichenden ellipsoids " used. These values ​​may differ greatly from the standard values ​​, however, result in the transformation calculation usually only changes the result in the centimeter range.

Limitations

Since the Helmert transformation has only one scale factor, it can not be used as a similarity transformation:

• The equalization of the measurement images or photos, here is a projective transformation or the photogrammetric rectification apply;
• The adjustment of a paper delay when scanning old plans and maps. In these cases, an affine transformation is used.