Helmert transformation

The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a geometric transformation method within a three-dimensional space. It is frequently used in

. The Helmert transformation is also called a seven-parameter transformation and is a .

It can be expressed as:

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

where

• X_T is the transformed vector
• X is the initial vector

The parameters are:

• C – translation vector. Contains the three translations along the coordinate axes
• μ –
  ppm
  , it must be divided by 1,000,000 and added to 1.
• R – rotation matrix. Consists of three axes (small^{[clarification needed]} rotations around each of the three coordinate axes) r_x, r_y, r_z. The rotation matrix is an orthogonal matrix. The angles are given in either degrees or radians.

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.

Sometimes it is sufficient to use the five parameter transformation, composed of three translations, only one rotation about the Z-axis, and one change of scale.

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, a more general affine transformation is preferable.

The Helmert transformation is used, among other things, in

.

For example, starting with the

Gauss–Krüger coordinate

, x and y, plus the height, h, are converted into 3D values in steps:

1. Undo the map projection: calculation of the ellipsoidal latitude, longitude and height (W, L, H)
2. Convert from
   reference ellipsoid
   of surveying
3. 7-parameter transformation (where x, y and z almost always change by a few hundred metres at most, and distances by 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, c_x, c_y, c_z.

The coordinates of a reference system B are derived from reference system A by the following formula (position vector transformation convention and very small rotation angles simplification):^[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:

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

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

– that is, eliminated in a statistically plausible manner.

Standard parameters

Note: the rotation angles given in the table are in

arcseconds and must be converted to radians

before use in the calculation.

EPSG Code	Region	Source datum	Target datum	Accuracy (metre)	cx (metre)	cy (metre)	cz (metre)	s ( ppm )	rx ( arcsecond )	ry ( arcsecond )	rz ( arcsecond )
8048	Australia	GDA94 (EPSG:4283)	GDA2020 (EPSG:7844)	0.01	0.06155	-0.01087	-0.04019	-0.009994	-0.0394924	-0.0327221	-0.0328979
9690	Australia	WGS84 (EPSG:4326)	GDA2020 (EPSG:7844)	3	0.06155	-0.01087	-0.04019	-0.009994	-0.0394924	-0.0327221	-0.0328979
1618	Austria	MGI (EPSG:4312)	WGS84 (EPSG:4326)	1.5	577.326	90.129	463.919	2.4232	5.137	1.474	5.297
1776	Germany (West)	DHDN (EPSG:4314)	ETRS89 (EPSG:4258)	3	598.1	73.7	418.2	6.7	0.202	0.045	-2.455
1777	Germany (West)	DHDN (EPSG:4314)	WGS84 (EPSG:4326)	3	598.1	73.7	418.2	6.7	0.202	0.045	-2.455
15869	Germany (East)	DHDN (EPSG:4314)	WGS84 (EPSG:4326)	2	612.4	77	440.2	2.55	-0.054	0.057	-2.797
1641	Ireland	TM65 (EPSG:4299)	WGS84 (EPSG:4326)	1	482.5	-130.6	564.6	8.15	-1.042	-0.214	-0.631
1953	Ireland	TM75 (EPSG:4300)	ETRS89 (EPSG:4258)	1	482.5	-130.6	564.6	8.15	-1.042	-0.214	-0.631
1954	Ireland	TM75 (EPSG:4300)	WGS84 (EPSG:4326)	1	482.5	-130.6	564.6	8.15	-1.042	-0.214	-0.631
8689	Slovenia	MGI 1901 (EPSG:3906)	Slovenia 1996 (EPSG:4765)	1	476.08	125.947	417.81	9.896638	-4.610862	-2.388137	11.942335
1314	United Kingdom	OSGB36 (EPSG:4247)	WGS84 (EPSG:4326)	2	446.448	-125.157	542.06	-20.489	0.15	0.247	0.842
1315	United Kingdom	OSGB36 (EPSG:4247)	ED50 (EPSG:4230)	2	535.948	-31.357	665.16	-21.689	0.15	0.247	0.998
1901	United States	NAD83(HARN) (EPSG:4152)	WGS84 (EPSG:4326)	1	-0.991	1.9072	0.5129	0	1.25033×10−7	4.6785×10−8	5.6529×10−8

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 geodetic transformations"[2]). The translations c_x, c_y, c_z are sometimes described as t_x, t_y, t_z, or dx, dy, dz. The rotations r_x, r_y, and r_z are sometimes also described as ${\displaystyle \omega }$, ${\displaystyle \phi }$ and ${\displaystyle \kappa }$.^[

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.

Estimating 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 with seven equations and seven unknowns, which can be solved.

For transformations between

Jacobian matrix

of the transformation function.

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 accuracy 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.

See also

• Geographic coordinate conversion
• Procrustes analysis
• Surveying

References

1. ^ "Equations Used for Datum Transformations". Toitū Te Whenua Land Information New Zealand. Retrieved 30 June 2022.
2. ^ On geodetic transformations, Bo-Gunnar Reit, 2009 https://www.lantmateriet.se/contentassets/4a728c7e9f0145569edd5eb81fececa7/rapport_reit_eng.pdf

External links

• Helmert transform in PROJ coordinate transformation software
• Computing Helmert Transformations Archived 13 September 2022 at the Wayback Machine