Helmert transformation

The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917; also called a seven-parameter transformation) is a transformation method within a three-dimensional space. It is frequently used in geodesy to produce distortion-free transformations from one datum to another using:


 * $$X_T=C+\mu RX \, $$

where


 * XT is the transformed vector
 * X is the initial vector

The parameters are:


 * $$C$$ &mdash; translation vector. Contains the three translations along the coordinate axes
 * $$\mu$$ &mdash; scale factor, which is unitless, and as it is usually expressed in ppm, it must be divided by 1,000,000.
 * $$R$$ &mdash; rotation matrix. Consists of three axes (small rotations around the coordinate axes) $$r_x$$, $$r_y$$, $$r_z$$. The rotation matrix is an orthogonal matrix. The rotation is given in radians.

Thus, the Helmert transformation is a similarity mapping. It is a special implementation of Galilean transformation, along with affine transformations and projection transformations. However, the latter two distort the lengths of line segments.

Calculating the parameters
If the transformation parameters are unknown, they can be calculated with reference points (that is, points whose coordinates are known before and after the transformation. Since a total of seven parameters (three translations, one scale, three rotations) have to be determined, at least two points and one coordinate of a third point (for example, the Z-coordinate) must be known. This gives a system of linear equations with seven equations and seven unknowns, which can be solved.

In practice, it is best to use more points. Through this correspondence, more accuracy is obtained, and a statistical assessment of the results becomes possible. In this case, the calculation is adjusted with the Gaussian least squares method.

A numerical value for the accuracy of the transformation parameters is obtained by calculating the values at the reference points, and weighting the results relative to the centroid of the points.

While the method is mathematically rigorous, it is entirely dependent on the rigour of the parameters that are used. In practice, these parameters are computed from the inclusion of at least three known points in the networks. However the accuracy of these will affect the following transformation parameters, as these points will contain observation errors. Therefore a "real-world" transformation will only be a best estimate and should contain a statistical measure of its quality.

It is not always necessary to use the seven parameter transformation, sometimes it is sufficient to use the five parameter transformation, composed of three translations, one longitude rotation and one change of scale.

Two-dimensional case
A special case is the two-dimensional Helmert transformation. Here, only four parameters are needed (two translations, one scaling, one rotation). These can be determined from two known points; if more points are available then checks can be made.

Application
The Helmert transformation is used, among other things, in geodesy to transform the coordinates of the point from one coordinate system into another. Using it, it becomes possible to convert regional surveying points into the WGS84 locations used by GPS.

In the process, the Gauss–Krüger coordinate, x and y, plus the height, h, are converted into 3D values in steps:
 * 1) Calculation of the ellipsoidal width, length and height (W, L, H)
 * 2) Calculation of X, Y and Z relative to the reference ellipsoid of surveying
 * 3) 7-parameter transformation (where X, Y and Z change almost evenly, a few hundred metres at most, and the distances change a few mm per km).
 * 4) Because of this, terrestrially measured positions can be compared with GPS data; these can then be brought into the surveying as new points &mdash; transformed in the opposite order.

The third step consists of the application of a rotation matrix, multiplication with the scale factor $$\mu=1+s$$ (with a value near 1) and the addition of the three translations, $$c_x$$, $$c_y$$, $$c_z$$.

The coordinates of a reference system B are derived from reference system A by the following formula :


 * $$\begin{bmatrix} X \\ Y \\ Z \end{bmatrix}^B = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix} + (1 + s\times10^{-6}) \cdot \begin{bmatrix} 1&-r_z&r_y \\ r_z&1&-r_x \\ -r_y & r_x & 1 \end{bmatrix} \cdot \begin{bmatrix} X \\ Y \\ Z \end{bmatrix}^A $$

or for each single parameter of the coordinate:



\begin{matrix} X_B=c_x+(1+s\times10^{-6})\cdot (X_A-r_z\cdot Y_A+r_y\cdot Z_A)\\ Y_B=c_y+(1+s\times10^{-6})\cdot (r_z\cdot X_A+Y_A-r_x\cdot Z_A)\\ Z_B=c_z+(1+s\times10^{-6})\cdot (-r_y\cdot X_A+r_x\cdot Y_A+Z_A).\\ \end{matrix} $$

For the reverse transformation, each element is multiplied by -1.

The seven parameters are determined for each region with three or more "identical points" of both systems. To bring them into agreement, the small inconsistencies (usually only a few cm) are adjusted using the method of least squares - that is, eliminated in a statistically plausible manner.

Standard parameters
Note that the rotation angles given in the table are in seconds and must be converted to radians before use in the calculation.

These are standard parameter sets for the 7-parameter transformation (or data transformation) between two ellipsoids. For a transformation in the opposite direction, the signs of all the parameters must be changed. The translations $$c_x$$, $$c_y$$, $$c_z$$, are sometimes described as $$t_x$$, $$t_y$$,$$t_z$$, or $$dx$$, $$dy$$, $$dz$$. The rotations $$x$$, $$y$$ and $$z$$ are sometimes also described as $$\kappa$$, $$\phi$$ and $$\omega$$. In the United Kingdom the prime interest is the transformation between the OSGB36 datum used by the Ordnance survey for Grid References on its Landranger and Explorer maps to the WGS84 implementation used by GPS technology. The Gauss–Krüger coordinate system used in Germany normally refers to the Bessel ellipsoid. A further datum of interest was the ED50 datum based on the Hayford ellipsoid. It is used by NATO for military maps and in future for most EU countries.

The earth does not have a perfect ellipsoidal shape, but is described as a geoid. Instead, the geoid of the earth is described by many ellipsoids. Depending upon the actual location, the "locally best aligned ellipsoid" has been used for surveying and mapping purposes. The standard parameter set gives an accuracy of about 7 m for a OSGB36/WGS84 transformation. This is not precise enough for surveying, and the Ordnance Survey supplements these results by using a lookup table of further translations in order to reach 1 cm accuracy.

Restrictions
The Helmert transformation only uses one scale factor, so it is not suitable for: In these cases, use another affine transformation.
 * The manipulation of measured drawings and photographs
 * The comparison of paper deformations while scanning old plans and maps.