Helmert transformation

It can be expressed as: where The parameters are: A special case is the two-dimensional Helmert transformation.

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 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] or for each single parameter of the coordinate: For the reverse transformation, each element is multiplied by −1.

To bring them into agreement, the small inconsistencies (usually only a few cm) are adjusted using the method of least squares – that is, eliminated in a statistically plausible manner.

The translations cx, cy, cz are sometimes described as tx, ty, tz, or dx, dy, dz.

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.

Depending upon the actual location, the "locally best aligned ellipsoid" has been used for surveying and mapping purposes.

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.

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.