Computing Helmert transformations

The Helmert transformation is used in geodesy. It transforms a set of points into another by rotation, scaling and translation. When both sets of points are given, then least squares can be used to solve the inverse problem of determining the parameters. In particular, the parameters of the so-called seven-parameter transformation can be obtained by standard methods. In this note, it is shown how a Gauss–Newton method in the rotation parameters alone can easily be implemented to determine the parameters of the nine-parameter transformation (when different scale factors for the variables are needed).