Helmert Transformation

The Helmert transformation formula as we know it:

\[ \begin{equation} \begin{bmatrix} x \\ y \\ z \end{bmatrix}^B % = % \begin{bmatrix} t_x \\ t_y \\ t_z \end{bmatrix} % + (1+s\cdot10^{-6}) % \begin{bmatrix} 1 & -r_{z} & r_{y} \\ r_{z} & 1 & -r_{x} \\ -r_{y} & r_{x} & 1 \end{bmatrix} % \begin{bmatrix} x \\ y \\ z \end{bmatrix}^A \end{equation} \]

can also be expressed as

\[ \begin{equation} P_{B} = T + c \bm{R}P_{A} \end{equation} \]

where $P_{A}$ and $P_{B}$ are the input and output coordinate vectors, $T$ is the translation vector, $\bm R$ is the rotation matrix and the scale factor $c$ is expressed as

\[ \begin{equation} c = 1+s\cdot10^{-6} \end{equation} \]

Chaining transformations

It is quite common to perform several Helmert transformations in a row, e.g.

\[ \begin{equation} P_B = T_1 + c_1\bm{R_1}P_A \end{equation} \]

\[ \begin{equation} P_C = T + c_2\bm{R_2}P_B \end{equation} \]

where the coordinate $P_A$ is transformed with the parameters of the first step ($T_1$, $c_1$ & $\bm{R_1}$) and then the parameters of the second step ($T_2$, $c_2$ & $\bm{R_2}$).

This is computationally heavy and can be avoided by substituting one into the other:

\[ \begin{equation} P_C = T_2 + c_2\bm{R_2} \left( T_1 + c_1 \bm{R_1} P_A \right) \end{equation} \]

$$ \begin{equation} P_C = T_2 + c_2\bm{R_2}T_1 + c_2c_1 \bm{R_2}\bm{R_1} P_A \end{equation} $$ which can be simplified using the following expressions

\[ \begin{equation} T_3 = T_2 + c_2\bm{R_2}T_1 \end{equation} \]

\[ \begin{equation} c_3 = c_2 \cdot c_1 \end{equation} \]

\[ \begin{equation} R_3 = \bm{R_2}\bm{R_1} \end{equation} \]

And with a bit of substitution we get

\[ \begin{equation} P_C = T_3 + c_3 \bm{R_3} P_A \end{equation} \]

which is on the same form as the Helmert transformation shown in the beginning.

A chained Helmert transformation can be sped up significantly by first determining $T_3$, $c_3$ and $\bm R_3$ and using them as parameters in the original Helmert transformation formula.

Shown here is only the 7-parameter version of the Helmert transformation but of course this obviously also extends to the 14-parameter version and combinations of the two.