# Helmert Transformation

The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917; also called a seven-parameter transformation) is a transformation method within a three-dimensional space. It is frequently used in geodesy to produce distortion-free transformations from one datum to another using:

where

• XT is the transformed vector
• X is the initial vector

The parameters are:

• — translation vector. Contains the three translations along the coordinate axes
• — scale factor, which is unitless, and as it is usually expressed in ppm, it must be divided by 1,000,000.
• rotation matrix. Consists of three axes (small rotations around the coordinate axes), . The rotation matrix is an orthogonal matrix. The rotation is given in radians.

Thus, the Helmert transformation is a similarity transformation.

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