Helmert transformation

Definition

A Helmert transformation is a map $T \colon \R^3 \to \R^3$ of the form $T(X) = C + (1 + s) R X.$ Here $C \in \R^3$ is a translation vector and $s > -1$ is a real number, the scale change. The matrix $R \in SO(3)$ is a rotation matrix. In geodesy $R$ is given by three small angles $r_x, r_y, r_z$ , in radians, and replaced by its linear approximation $R \approx \begin{pmatrix} 1 & -r_z & r_y \\ r_z & 1 & -r_x \\ -r_y & r_x & 1 \end{pmatrix}.$ The seven numbers $C_x, C_y, C_z, s, r_x, r_y, r_z$ are its parameters.

Remarks

Example: $C = (100, 0, 0)$ m, $s = 0$ , $R = I$ moves every point $100$ m along the $x$ -axis; it preserves all distances, as $s = 0$ . Non-example: the map $X \mapsto \operatorname{diag}(1, 1, 1 + 10^{-6}) X$ is not a Helmert transformation, since it scales the axes by different factors. A Helmert transformation is a similarity transformation: it multiplies all distances by $1 + s$ and preserves angles. Two sign conventions for the angles are in use, the position vector and the coordinate frame conventions; they differ by the sign of $r_x, r_y, r_z$ . It is applied to Cartesian coordinates; geodetic coordinates $(\varphi, \lambda, h)$ are converted to Cartesian form first.

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Tag	`2N7G6ZZA69`
URI	`m2://2N7G6ZZA69`
Kind	definition
Writing	W0 stub
MSC	86A30, 51N20
Defines	Helmert transformation
Uses	Similarity transformation, Rotation, Geodetic datum, Geodetic coordinates
Changed	3 November 2026

Helmert transformation

Definition

Remarks

Reading path

Further reading

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