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 geodesy to produce datum transformations between datums. The Helmert transformation is also called a seven-parameter transformation and is a similarity transformation. It can be expressed as: where XT is the transformed vector X is the initial vector The parameters are: C – translation vector. Contains the three translations along the coordinate axes μ – scale factor, which is unitless; if it is given in ppm, it must be divided by 1,000,000 and added to 1. R – rotation matrix. Consists of three axes (small rotations around each of the three coordinate axes) rx, ry, rz. 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 old plans and maps. In these cases, a more general affine transformation is preferable. 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 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).