Editing Geographic coordinate conversion (section)

=== Molodensky-Badekas transformation ===

To eliminate the coupling between the rotations and translations of the Helmert transform, three additional parameters can be introduced to give a new XYZ center of rotation closer to coordinates being transformed. This ten-parameter model is called the ''Molodensky-Badekas transformation'' and should not be confused with the more basic Molodensky transform.{{r|OGP7_2|page1=133-134}}

Like the Helmert transform, using the Molodensky-Badekas transform is a three-step process:
# Convert from geodetic coordinates to ECEF coordinates for datum <math>A</math>
# Apply the Molodensky-Badekas transform, with the appropriate <math>A\to B</math> transform parameters, to transform from datum <math>A</math> ECEF coordinates to datum <math>B</math> ECEF coordinates
# Convert from ECEF coordinates to geodetic coordinates for datum <math>B</math>

The transform has the form<ref name=MB_NGA>{{cite web|title=Molodensky-Badekas (7+3) Transformations|url=http://earth-info.nga.mil/GandG/coordsys/datums/molodensky.html|publisher=National Geospatial Intelligence Agency (NGA)|access-date=5 March 2014|archive-date=19 July 2013|archive-url=https://web.archive.org/web/20130719151529/http://earth-info.nga.mil/GandG/coordsys/datums/molodensky.html|url-status=live}}</ref>

: <math>
  \begin{bmatrix} X_B \\ Y_B \\ Z_B \end{bmatrix} =
  \begin{bmatrix} X_A \\ Y_A \\ Z_A \end{bmatrix} +
    \begin{bmatrix} \Delta X_A \\ \Delta Y_A \\ \Delta Z_A \end{bmatrix} +
    \begin{bmatrix}
         1 & -r_z &  r_y \\
       r_z &    1 & -r_x \\
      -r_y &  r_x &    1
    \end{bmatrix}
    \begin{bmatrix} X_A - X^0_A \\ Y_A - Y^0_A \\ Z_A - Z^0_A \end{bmatrix} +
    \Delta S \begin{bmatrix} X_A - X^0_A \\ Y_A - Y^0_A \\ Z_A - Z^0_A \end{bmatrix}.
</math>

where <math>\left(X^0_A,\, Y^0_A,\, Z^0_A\right)</math> is the origin for the rotation and scaling transforms and <math>\Delta S</math> is the scaling factor.

The Molodensky-Badekas transform is used to transform local geodetic datums to a global geodetic datum, such as WGS 84. Unlike the Helmert transform, the Molodensky-Badekas transform is not reversible due to the rotational origin being associated with the original datum.{{r|OGP7_2|page1=134}}