Helmert 3-Parameter Transformation

The 3-parameter Helmert transformation is a simple translation in the three principal directions of an earth-centered, earth-fixed coordinate system.

Type

type: helmert_3param

Options

Option	Type	Required	Description
`x`	`float`	No	Translation in X direction (meters)
`y`	`float`	No	Translation in Y direction (meters)
`z`	`float`	No	Translation in Z direction (meters)

Example

operators:
- name: Apply offset
  type: helmert_3param
  x: 100.0
  y: 50.0
  z: -25.0

Mathematical Background

Forward Transformation

A coordinate in vector \(\mathbf{V}_a\) is transformed into vector \(\mathbf{V}_b\) using:

\[ \mathbf{V}_b = \mathbf{T} + \mathbf{V}_a \]

Where \(\mathbf{T} = (x, y, z)^T\) is the translation vector:

\[ \mathbf{T} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \]

Inverse Transformation

The inverse transformation is:

\[ \mathbf{V}_a = \mathbf{V}_b - \mathbf{T} \]

Parameter Estimation

The translation parameters are estimated as the weighted mean difference between source and target coordinates:

The mean translation vector is computed as:

\[ \mathbf{T} = \bar{\mathbf{V}}_{target} - \bar{\mathbf{V}}_{source} \]

Where the weighted means are:

\[ \bar{\mathbf{V}}_{source} = \frac{\sum_{i=1}^{n} w_i \mathbf{V}_{source,i}} {\sum_{i=1}^{n} w_i} , \quad \bar{\mathbf{V}}_{target} = \frac{\sum_{i=1}^{n} w_i \mathbf{V}_{target,i}} {\sum_{i=1}^{n} w_i} \]

For each coordinate component \((x, y, z)\):

\[ \bar{v}_{source} = \frac{\sum_{i=1}^{n} w_i v_{source,i}} {\sum_{i=1}^{n} w_i} \]

The weight \(w_i\) is taken from the corresponding coordinate's weight in the weight matrix.