# Helmert transformation

The transformation from a reference frame 1 to a reference frame 2 can be described with three translations Δx, Δy, Δz, three rotations Rx, Ry, Rz and a scale parameter μ.

The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) 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. The Helmert transformation is also called a seven-parameter transformation and is a similarity transformation.

## Definition

It can be expressed as:

${\displaystyle X_{T}=C+\mu RX\,}$

where

• XT is the transformed vector
• X is the initial vector

The parameters are:

• ${\displaystyle C}$ — translation vector. Contains the three translations along the coordinate axes
• ${\displaystyle \mu }$scale factor, which is unitless, and as it is usually expressed in ppm, it must be divided by 1,000,000.
• ${\displaystyle R}$rotation matrix. Consists of three axes (small rotations around the coordinate axes) ${\displaystyle r_{x}}$, ${\displaystyle r_{y}}$, ${\displaystyle r_{z}}$. The rotation matrix is an orthogonal matrix. The rotation is given in radians.

### Variations

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 rotation (about the Z-axis) and one change of scale.

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.

### Restrictions

The Helmert transformation only uses one scale factor, so it is not suitable for:

• The manipulation of measured drawings and photographs
• The comparison of paper deformations while scanning old plans and maps.

In these cases, use another affine transformation.

## 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 latitude, longitude 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 — transformed in the opposite order.

The third step consists of the application of a rotation matrix, multiplication with the scale factor ${\displaystyle \mu =1+s}$ (with a value near 1) and the addition of the three translations, ${\displaystyle c_{x}}$, ${\displaystyle c_{y}}$, ${\displaystyle c_{z}}$.

The coordinates of a reference system B are derived from reference system A by the following formula:[1]

${\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}^{B}={\begin{bmatrix}c_{x}\\c_{y}\\c_{z}\end{bmatrix}}+(1+s\times 10^{-6})\cdot {\begin{bmatrix}1&-r_{z}&r_{y}\\r_{z}&1&-r_{x}\\-r_{y}&r_{x}&1\end{bmatrix}}\cdot {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}^{A}}$

or for each single parameter of the coordinate:

${\displaystyle {\begin{matrix}X_{B}&=&c_{x}+(1+s\times 10^{-6})\cdot (X_{A}-r_{z}\cdot Y_{A}+r_{y}\cdot Z_{A})\\Y_{B}&=&c_{y}+(1+s\times 10^{-6})\cdot (r_{z}\cdot X_{A}+Y_{A}-r_{x}\cdot Z_{A})\\Z_{B}&=&c_{z}+(1+s\times 10^{-6})\cdot (-r_{y}\cdot X_{A}+r_{x}\cdot Y_{A}+Z_{A}).\\\end{matrix}}}$

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.

Region Start datum Target datum ${\displaystyle c_{x}}$ (Metre) ${\displaystyle c_{y}}$ (Metre) ${\displaystyle c_{z}}$ (Metre) s (ppm) rx (Arcsecond) ry (Arcsecond) rz (Arcsecond)
Slovenia ETRS89 D48 D96 409.545 72.164 486.872 17.919665 −3.085957 −5.469110 11.020289
England, Scotland, Wales WGS84 OSGB36[2] −446.448 125.157 −542.06 20.4894 −0.1502 −0.247 −0.8421
Ireland WGS84 Ireland 1965 −482.53 130.596 −564.557 −8.15 1.042 0.214 0.631
Germany WGS84 DHDN −591.28 −81.35 −396.39 −9.82 1.4770 −0.0736 −1.4580
Germany WGS84 Bessel 1841 −582 −105 −414 −8.3 −1.04 −0.35 3.08
Germany WGS84 Krassovski 1940 −24 123 94 −1.1 −0.02 0.26 0.13
Austria (BEV) WGS84 MGI −577.326 −90.129 −463.920 −2.423 5.137 1.474 5.297
United States WGS84 Clarke 1866 8 −160 −176 0 0 0 0

These are standard parameter sets for the 7-parameter transformation (or data transformation) between two datums. For a transformation in the opposite direction, inverse transformation parameters should be calculated or inverse transformation should be applied (as described in paper "On gedetic transformations"[3]). The translations cx, cy, cz are sometimes described as tx, ty, tz, or dx, dy, dz. The rotations rx, ry, and rz are sometimes also described as ${\displaystyle \omega }$, ${\displaystyle \phi }$ and ${\displaystyle \kappa }$. 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 ED50 (European Datum 1950) based on the Hayford ellipsoid. ED50 was part of the fundamentals of the NATO coordinates up to the 1980s, and many national coordinate systems of Gauss–Krüger are defined by ED50.

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 an 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.

## 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.