The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a method of parametrization of geometric similarities, which is used in geodesy to compute transformations between datums. The Helmert transformation is also called seven-parameter transformation.
Definition The Helmert transformation can be expressed as:
X T = C + μ R X {\displaystyle X_{T}=C+\mu RX\,}
where
X is the initial point represented by a coordinate vector XT is the point-transformed vector The parameters are:
C – translation vector, whose components are translations along the coordinate axes μ – scale factor, which must be divided by 1,000,000 and added to 1 when expressed in parts per million R – rotation matrix, which is decomposed into three rotations rx, ry, rz 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 photographs The comparison of paper deformations while scanning old plans and maps. In these cases, a more general affine transformation is preferable.
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. For example, starting with the Gauss–Krüger coordinate, x and y, plus the height, h, are converted into 3D values in steps:
Undo the map projection: calculation of the ellipsoidal latitude, longitude and height (W, L, H) Convert from geodetic coordinates to geocentric coordinates: Calculation of x, y and z relative to the reference ellipsoid of surveying 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). 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 μ = 1 + s {\displaystyle \mu =1+s} (with a value near 1) and the addition of the three translations, cx, cy, cz. 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):
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