Molodensky-Badekas transform

New in version 6.0.0.

The Molodensky-Badekas transformation changes coordinates from one reference frame to another by means of a 10-parameter shift.

Note

It should not be confused with the Molodensky transform transform which operates directly in the geodetic coordinates. Molodensky-Badekas can rather be seen as a variation of Helmert transform

Alias molobadekas
Domain 3D
Input type Cartesian coordinates
Output type Cartesian coordinates

The Molodensky-Badekas transformation is a variation of the Helmert transform where the rotational terms are not directly applied to the ECEF coordinates, but on cartesian coordinates relative to a reference point (usually close to Earth surface, and to the area of use of the transformation). When px = py = pz = 0, this is equivalent to a 7-parameter Helmert transformation.

Example

Transforming coordinates from La Canoa to REGVEN:

proj=molobadekas convention=coordinate_frame
       x=-270.933 y=115.599 z=-360.226 rx=-5.266 ry=-1.238 rz=2.381
       s=-5.109 px=2464351.59 py=-5783466.61 pz=974809.81

Parameters

Note

All parameters (except convention) are optional but at least one should be used, otherwise the operation will return the coordinates unchanged.

+convention=coordinate_frame/position_vector

Indicates the convention to express the rotational terms when a 3D-Helmert / 7-parameter more transform is involved.

The two conventions are equally popular and a frequent source of confusion. The coordinate frame convention is also described as an clockwise rotation of the coordinate frame. It corresponds to EPSG method code 1034 (in the geocentric domain) or 9636 (in the geographic domain) The position vector convention is also described as an anticlockwise (counter-clockwise) rotation of the coordinate frame. It corresponds to as EPSG method code 1061 (in the geocentric domain) or 1063 (in the geographic domain).

The result obtained with parameters specified in a given convention can be obtained in the other convention by negating the rotational parameters (rx, ry, rz)

+x=<value>

Translation of the x-axis given in meters.

+y=<value>

Translation of the y-axis given in meters.

+z=<value>

Translation of the z-axis given in meters.

+s=<value>

Scale factor given in ppm.

+rx=<value>

X-axis rotation given arc seconds.

+ry=<value>

Y-axis rotation given in arc seconds.

+rz=<value>

Z-axis rotation given in arc seconds.

+px=<value>

Coordinate along the x-axis of the reference point given in meters.

+py=<value>

Coordinate along the y-axis of the reference point given in meters.

+pz=<value>

Coordinate along the z-axis of the reference point given in meters.

Mathematical description

In the Position Vector convention, we define \(R_x = radians \left( rx \right)\), \(R_z = radians \left( ry \right)\) and \(R_z = radians \left( rz \right)\)

In the Coordinate Frame convention, \(R_x = - radians \left( rx \right)\), \(R_z = - radians \left( ry \right)\) and \(R_z = - radians \left( rz \right)\)

(1)\[\begin{split}\begin{align} \begin{bmatrix} X \\ Y \\ Z \\ \end{bmatrix}^{output} = \begin{bmatrix} T_x + P_x\\ T_y + P_y\\ T_z + P_z\\ \end{bmatrix} + \left(1 + s \times 10^{-6}\right) \begin{bmatrix} 1 & -R_z & R_y \\ Rz & 1 & -R_x \\ -Ry & R_x & 1 \\ \end{bmatrix} \begin{bmatrix} X^{input} - P_x\\ Y^{input} - P_y\\ Z^{input} - P_z\\ \end{bmatrix} \end{align}\end{split}\]