# Helmert transformation

The transformation from a reference frame 1 to a reference frame 2 can be described with three shifts Δx, Δy, Δz, three angles of rotation Rx, Ry, Rz and a scale factor μ

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

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

• ${\ displaystyle X_ {T}}$ ... transformed vector
• ${\ displaystyle X}$ ... output vector

The seven parameters are:

• ${\ displaystyle C}$... displacement vector. Contains the three shifts along the coordinate axes
• ${\ displaystyle \ mu}$... scale factor
• ${\ displaystyle R}$... rotation matrix. Consists of three angles of rotation ( rotations around the coordinate axes ) r x , r y , r z . The rotation matrix is ​​an orthogonal matrix .

The Helmert transformation is thus a similarity transformation . It is a specialization of the Galileo transformations , which include affine and projective transformations ; the latter, however, distort the length of the route.

## Calculation of the parameters

If the transformation parameters are unknown, they can be calculated using identical points (i.e. points whose coordinates are known before and after the transformation). Since a total of 7 parameters (3 shifts, 1 scale, 3 rotations) have to be determined, at least 2 points and one coordinate of a 3rd point (e.g. the z coordinate) must be known. This creates a system of equations with seven equations and just as many unknowns that can be solved.

In practice, more points are usually used. This overdetermination gives you, firstly, a check on the correctness of the points used and, secondly, the possibility of a statistical assessment of the result. In this case, the calculation is carried out using a compensation calculation using the Gaussian method of least squares .

In order to obtain numerically favorable values ​​for the calculation of the transformation parameters, the calculations are carried out with coordinate differences based on the mean value of the given points.

## Two-dimensional case

A special case is the two-dimensional Helmert transformation for plane coordinate systems . It is used u. a. in geodesy , when a small-scale surveying network with overdetermination is connected to the national coordinate system, or in astrometry for simple plate reduction with well-dimensional photo plates. The transformation corresponds to a rotational stretching with parallel displacement in any direction.

Instead of 7, it only requires 4 transformation parameters , namely 2 shifts, 1 scale factor and 1 rotation. The calculation of these 4 parameters requires two identical points in the two coordinate systems; if more points are given, an adjustment is made again.

## application

The Helmert transformation is used in geodesy , among other things , to transform coordinates of points from one coordinate system to another. So z. B. the conversion of points of the regional land survey into the WGS84 used for GPS -locations possible.

The Gauss-Krüger coordinates plus the height are converted into 3D values ​​step by step: ${\ displaystyle x, y}$${\ displaystyle H}$

1. Calculation of the ellipsoidal latitude, longitude and height ( )${\ displaystyle B, L, H}$
2. Calculation of the reference ellipsoid of the national survey${\ displaystyle X, Y, Z}$
3. 7-parameter transformation (which changes almost uniformly by a maximum of a few hundred meters and the distances by a few mm per km).${\ displaystyle X, Y, Z}$
4. Inverse transformation into ellipsoidal latitude, longitude and height

This enables terrestrially measured positions to be compared with GPS data; the latter can - transformed in the reverse order - be introduced as new points in the national survey.

The third step (the Helmert transformation) consists in the application of a rotation matrix , the multiplication with a scale factor (µ is close to the value 1) and the addition of a shift . ${\ displaystyle \ mu = 1 + m / 10 ^ {6}}$${\ displaystyle C}$

Since the sub-operations of this transformation all cause only small changes, the coordinates of a reference system can be derived from the reference system using the following formula : ${\ displaystyle B}$${\ displaystyle A}$

${\ displaystyle {\ begin {bmatrix} X \\ Y \\ Z \ end {bmatrix}} ^ {B} = {\ begin {bmatrix} c_ {x} \\ c_ {y} \\ c_ {z} \ end {bmatrix}} + \ mu \ cdot {\ begin {bmatrix} 1 & r_ {z} & - r_ {y} \\ - r_ {z} & 1 & r_ {x} \\ r_ {y} & - r_ {x} & 1 \ end {bmatrix}} \ cdot {\ begin {bmatrix} X \\ Y \\ Z \ end {bmatrix}} ^ {A}}$

where the angles of rotation , and are to be used with their value in radians. ${\ displaystyle r_ {x}}$${\ displaystyle r_ {y}}$${\ displaystyle r_ {z}}$

Or for each individual component:

{\ displaystyle {\ begin {aligned} & X_ {B} = c_ {x} + \ mu \ cdot (X_ {A} + r_ {z} \ cdot Y_ {A} -r_ {y} \ cdot Z_ {A} ) \\ & Y_ {B} = c_ {y} + \ mu \ cdot (-r_ {z} \ cdot X_ {A} + Y_ {A} + r_ {x} \ cdot Z_ {A}) \\ & Z_ { B} = c_ {z} + \ mu \ cdot (r_ {y} \ cdot X_ {A} -r_ {x} \ cdot Y_ {A} + Z_ {A}) \\\ end {aligned}}}

For the reverse transformation, all parameters are multiplied by −1.

The 7 parameters are determined for the respective region (separate survey , federal state, etc.) with 3 or more "identical points" of both systems. In the event of overdetermination, the small contradictions (usually only a few cm) are balanced out using the least squares method - that is, eliminated in the most statistically plausible way.

## Standard parameter sets

 area Starting system Target system c x (meters) c y (meters) c z (meters) m ( ppm ) r x ( arcsecond ) r y ( arcsecond ) r z ( arcsecond ) England , Scotland , Wales WGS84 OSGB36 −446.448 125.157 −542.06 20.4894 0.1502 0.247 0.8421 Ireland Ireland 1965 −482.53 130.596 −564.557 −8.15 −1.042 −0.214 −0.631 Germany DHDN / Potsdam 2001 −598.1 −73.7 −418.2 −6.7 0.202 0.045 −2.455 Pulkowo S42 / 83 2001 −24.9 126.4 93.2 −1.01 −0.063 −0.247 −0.041 Austria (BEV) MGI −577.326 −90.129 −463.919 −2.423 5.137 1.474 5.297 Switzerland LV95 −674.374 −15,056 −405.346 0 0 0 0 United States Clarke 1866 8th −160 −176 0 0 0 0

The examples are 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 angles of rotation , and are sometimes referred to as κ, φ and ω. The datum transformation from WGS84 to Bessel is interesting for Central Europe insofar as GPS technology refers to the WGS84 ellipsoid, whereas the Gauß-Krüger coordinate system used in Germany and Austria is usually based on the Bessel ellipsoid. ${\ displaystyle r_ {x}}$${\ displaystyle r_ {y}}$${\ displaystyle r_ {z}}$

Since the earth does not have a perfect ellipsoid shape, but is described as a geoid , the standard parameter set is not sufficient for a datum transformation with measuring accuracy. Instead, the geoid shape of the earth is described by a multitude of ellipsoids. Depending on the actual location, the parameters of the "locally best-matching ellipsoid" are used. These values ​​can deviate significantly from the standard values ​​and usually lead to significant changes in the result in the transformation calculation.

## restrictions

Since the Helmert transformation only knows one scale factor, it cannot be used as a similarity transformation for: