{{Short description|Transformation method within a three-dimensional space}} {{Use dmy dates|date=February 2020}} [[File:Helmert (seven-parameter) transformation.svg|thumb|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 method of [[parametrization (geometry)|parametrization]] of [[similarity (geometry)|geometric similarities]], which is used in [[geodesy]] to compute [[datum transformation|transformation]]s between [[datum (geodesy)|datums]]. The Helmert transformation is also called '''seven-parameter transformation'''.

==Definition== The Helmert transformation can be expressed as:

: <math>X_T = C + \mu R X \, </math>

where

* {{mvar|X}} is the initial point represented by a [[coordinate vector]] * {{math|''X''<sub>''T''</sub>}} is the point-transformed vector

The [[parameter]]s are:

* {{mvar|C}} – [[Translation (geometry)|translation vector]], whose components are translations along the coordinate axes * {{mvar|μ}} – [[scale factor]], which must be divided by 1,000,000 and added to 1 when expressed in [[parts per million]] * {{mvar|R}} – [[rotation matrix]], which is decomposed into three rotations {{math|''r''<sub>''x''</sub>}}, {{math|''r''<sub>''y''</sub>}}, {{math|''r''<sub>''z''</sub>}} around the three [[coordinate axes]], whose angles are small in the geodesic context.

=== Variations === 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.

===Restrictions=== The Helmert transformation only uses one scale factor, so it is not suitable for: * The manipulation of measured drawings and [[Photography|photographs]] * The comparison of paper deformations while [[Image scanner|scanning]] old plans and maps. In these cases, a more general [[affine transformation]] is preferable.

==Application== {{main|Datum transformation}} 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 [[Global Positioning System|GPS]].

For example, starting with the [[Gauss–Krüger coordinate system|Gauss–Krüger coordinate]], {{mvar|x}} and {{mvar|y}}, plus the height, {{mvar|h}}, are converted into 3D values in steps: # Undo the [[map projection]]: calculation of the ellipsoidal latitude, longitude and height ({{mvar|W}}, {{mvar|L}}, {{mvar|H}}) # Convert from [[geodetic coordinates]] to [[geocentric coordinates]]: Calculation of {{mvar|x}}, {{mvar|y}} and {{mvar|z}} relative to the [[reference ellipsoid]] of surveying # 7-parameter transformation (where {{mvar|x}}, {{mvar|y}} and {{mvar|z}} almost always change by a few hundred metres at most, and distances by a few mm per km). # 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 <math>\mu = 1 + s</math> (with a value near 1) and the addition of the three translations, {{math|''c''<sub>''x''</sub>}}, {{math|''c''<sub>''y''</sub>}}, {{math|''c''<sub>''z''</sub>}}.

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):<ref>{{Cite web |title=Equations Used for Datum Transformations |url=https://www.linz.govt.nz/data/geodetic-system/coordinate-conversion/geodetic-datum-conversions/equations-used-datum |access-date=2022-06-30 |website=Toitū Te Whenua Land Information New Zealand |language=en-NZ}}</ref>

:<math>\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 </math>

or for each single parameter of the coordinate:

:<math>\begin{align} 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{align} </math>

For the reverse transformation, each element is multiplied by &minus;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 [[least squares adjustment|adjusted]] using the method of [[least squares]] – that is, eliminated in a statistically plausible manner.

=== Standard parameters === :''Note: the rotation angles given in the table are in [[arcsecond|arcseconds]] and must be converted to [[radian]]s before use in the calculation.''

{| class="wikitable" ! [[EPSG Geodetic Parameter Dataset|EPSG Code]] ! Region ! Source datum ! Target datum !Accuracy ([[metre]]) ! {{math|''c''<sub>''x''</sub>}} (metre) ! {{math|''c''<sub>''y''</sub>}} (metre) ! {{math|''c''<sub>''z''</sub>}} (metre) ! {{mvar|s}} ([[Parts per million|ppm]]) ! {{math|''r''<sub>''x''</sub>}} ([[arcsecond]]) ! {{math|''r''<sub>''y''</sub>}} ([[arcsecond]]) ! {{math|''r''<sub>''z''</sub>}} ([[arcsecond]]) |- | [https://epsg.org/transformation_8048/GDA94-to-GDA2020-1.html 8048] | Australia | [[Geocentric Datum of Australia 1994|GDA94]] ([https://epsg.org/crs_4283/GDA94.html EPSG:4283]) | GDA2020 ([https://epsg.org/crs_7844/GDA2020.html EPSG:7844]) |0.01 | 0.06155 | -0.01087 | -0.04019 | -0.009994 | -0.0394924 | -0.0327221 | -0.0328979 |- |[https://epsg.org/transformation_9690/WGS-84-to-GDA2020-3.html 9690] |Australia |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |GDA2020 ([https://epsg.org/crs_7844/GDA2020.html EPSG:7844]) |3 |0.06155 | -0.01087 | -0.04019 | -0.009994 | -0.0394924 | -0.0327221 | -0.0328979 |- |[https://epsg.org/transformation_1618/MGI-to-WGS-84-3.html 1618] |Austria |MGI ([https://epsg.org/crs_4312/MGI.html EPSG:4312]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |1.5 |577.326 |90.129 |463.919 |2.4232 |5.137 |1.474 |5.297 |- |[https://epsg.org/transformation_1776/DHDN-to-ETRS89-2.html 1776] |Germany (West) |DHDN ([https://epsg.org/crs_4314/DHDN.html EPSG:4314]) |ETRS89 ([https://epsg.org/crs_4258/ETRS89.html EPSG:4258]) |3 |598.1 |73.7 |418.2 |6.7 |0.202 |0.045 | -2.455 |- |[https://epsg.org/transformation_1777/DHDN-to-WGS-84-2.html 1777] |Germany (West) |DHDN ([https://epsg.org/crs_4314/DHDN.html EPSG:4314]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |3 |598.1 |73.7 |418.2 |6.7 |0.202 |0.045 | -2.455 |- |[https://epsg.org/transformation_15869/DHDN-to-WGS-84-3.html 15869] |Germany (East) |DHDN ([https://epsg.org/crs_4314/DHDN.html EPSG:4314]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |2 |612.4 |77 |440.2 |2.55 | -0.054 |0.057 | -2.797 |- |[https://epsg.org/transformation_1641/TM65-to-WGS-84-2.html 1641] |Ireland |TM65 ([https://epsg.org/crs_4299/TM65.html EPSG:4299]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |1 |482.5 | -130.6 |564.6 |8.15 | -1.042 | -0.214 | -0.631 |- |[https://epsg.org/transformation_1953/TM75-to-ETRS89-2.html 1953] |Ireland |TM75 ([https://epsg.org/crs_4300/TM75.html EPSG:4300]) |ETRS89 ([https://epsg.org/crs_4258/ETRS89.html EPSG:4258]) |1 |482.5 | -130.6 |564.6 |8.15 | -1.042 | -0.214 | -0.631 |- |[https://epsg.org/transformation_1954/TM75-to-WGS-84-2.html 1954] |Ireland |TM75 ([https://epsg.org/crs_4300/TM75.html EPSG:4300]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |1 |482.5 | -130.6 |564.6 |8.15 | -1.042 | -0.214 | -0.631 |- |[https://epsg.org/transformation_8689/MGI-1901-to-Slovenia-1996-12.html 8689] |Slovenia |MGI 1901 ([https://epsg.org/crs_3906/MGI-1901.html EPSG:3906]) |Slovenia 1996 ([https://epsg.org/crs_4765/Slovenia-1996.html EPSG:4765]) |1 |476.08 |125.947 |417.81 |9.896638 | -4.610862 | -2.388137 |11.942335 |- |[https://epsg.org/transformation_1314/OSGB36-to-WGS-84-6.html 1314] |United Kingdom |[[Ordnance Survey National Grid|OSGB36]] ([https://epsg.org/crs_4277/OSGB36.html EPSG:4247]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |2 |446.448 | -125.157 |542.06 | -20.489 |0.15 |0.247 |0.842 |- |[https://epsg.org/transformation_1315/OSGB36-to-ED50-1.html 1315] |United Kingdom |[[Ordnance Survey National Grid|OSGB36]] ([https://epsg.org/crs_4277/OSGB36.html EPSG:4247]) |[[ED50]] ([https://epsg.org/crs_4230/ED50.html EPSG:4230]) |2 |535.948 | -31.357 |665.16 | -21.689 |0.15 |0.247 |0.998 |- |[https://epsg.org/transformation_1901/NAD83-HARN-to-WGS-84-3.html 1901] |United States |[[North American Datum|NAD83(HARN)]] ([https://epsg.org/crs_4152/NAD83-HARN.html EPSG:4152]) |[[World Geodetic System|WGS84]] ([https://epsg.org/crs_4326/WGS-84.html EPSG:4326]) |1 | -0.991 |1.9072 |0.5129 |0 |{{val|1.25033e-7}} |{{val|4.6785e-8}} |{{val|5.6529e-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"<ref>On geodetic transformations, Bo-Gunnar Reit, 2009 https://www.lantmateriet.se/contentassets/4a728c7e9f0145569edd5eb81fececa7/rapport_reit_eng.pdf</ref>). The translations {{math|''c''<sub>''x''</sub>}}, {{math|''c''<sub>''y''</sub>}}, {{math|''c''<sub>''z''</sub>}} are sometimes described as {{math|''t''<sub>''x''</sub>}}, {{math|''t''<sub>''y''</sub>}}, {{math|''t''<sub>''z''</sub>}}, or {{mvar|dx}}, {{mvar|dy}}, {{mvar|dz}}. The rotations ''r''<sub>''x''</sub>, ''r''<sub>''y''</sub>, and ''r''<sub>''z''</sub> are sometimes also described as <math>\omega</math>, <math>\phi</math> and <math>\kappa</math>.{{who|date=February 2020}} 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 {{val|7|u=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 {{val|1|ul=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 [[conformal map projection]]s near an arbitrary point, the Helmert transformation parameters can be calculated exactly from the [[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== {{Reflist}}

==External links== * [https://proj.org/operations/transformations/helmert.html Helmert transform] in [[PROJ]] coordinate transformation software * [https://www.physics.dundee.ac.uk/gawatson/helmertrev.pdf Computing Helmert Transformations] {{Webarchive|url=https://web.archive.org/web/20220913121643/https://www.physics.dundee.ac.uk/gawatson/helmertrev.pdf |date=13 September 2022 }}

{{DEFAULTSORT:Helmert Transformation}} [[Category:Geodesy]] [[Category:Transformation (function)]]