Merge pull request #1161 from rouault/molobadekas

Implement Molodensky-Badekas transform (fixes #1160)

docs/source/operations/transformations/helmert.rst

@@ -7,7 +7,7 @@ Helmert transform
 .. versionadded:: 5.0.0
 
 The Helmert transformation changes coordinates from one reference frame to
-anoether by means of 3-, 4-and 7-parameter shifts, or one of their 6-, 8- and
+another by means of 3-, 4-and 7-parameter shifts, or one of their 6-, 8- and
 14-parameter kinematic counterparts.
 
 
@@ -25,7 +25,7 @@ anoether by means of 3-, 4-and 7-parameter shifts, or one of their 6-, 8- and
 | **Output type** | Cartesian coordinates                                             |
 +-----------------+-------------------------------------------------------------------+
 
-The Helmert transform, in all it's various incarnations, is used to perform reference
+The Helmert transform, in all its various incarnations, is used to perform reference
 frame shifts. The transformation operates in cartesian space. It can be used to transform
 planar coordinates from one datum to another, transform 3D cartesian
 coordinates from one static reference frame to another or it can be used to do fully
@@ -117,7 +117,7 @@ Parameters
 
 .. option:: +y=<value>
 
-    Translation of the x-axis given in meters.
+    Translation of the y-axis given in meters.
 
 .. option:: +z=<value>
 

docs/source/operations/transformations/index.rst

@@ -16,5 +16,6 @@ systems are based on different datums.
    helmert
    horner
    molodensky
+   molobadekas
    hgridshift
    vgridshift

docs/source/operations/transformations/molobadekas.rst

@@ -0,0 +1,147 @@
+.. _molobadekas:
+
+================================================================================
+Molodensky-Badekas transform
+================================================================================
+
+.. versionadded:: 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 :ref:`Molodensky` transform which
+    operates directly in the geodetic coordinates. Molodensky-Badekas can rather
+    be seen as a variation of :ref:`Helmert`
+
++-----------------+-------------------------------------------------------------------+
+| **Alias**       | molobadekas                                                       |
++-----------------+-------------------------------------------------------------------+
+| **Domain**      | 3D                                                                |
++-----------------+-------------------------------------------------------------------+
+| **Input type**  | Cartesian coordinates                                             |
++-----------------+-------------------------------------------------------------------+
+| **Output type** | Cartesian coordinates                                             |
++-----------------+-------------------------------------------------------------------+
+
+The Molodensky-Badekas transformation is a variation of the :ref:`Helmert` 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.
+
+.. option:: +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``)
+
+.. option:: +x=<value>
+
+    Translation of the x-axis given in meters.
+
+.. option:: +y=<value>
+
+    Translation of the y-axis given in meters.
+
+.. option:: +z=<value>
+
+    Translation of the z-axis given in meters.
+
+.. option:: +s=<value>
+
+    Scale factor given in ppm.
+
+.. option:: +rx=<value>
+
+    X-axis rotation given arc seconds.
+
+.. option:: +ry=<value>
+
+    Y-axis rotation given in arc seconds.
+
+.. option:: +rz=<value>
+
+    Z-axis rotation given in arc seconds.
+
+.. option:: +px=<value>
+
+    Coordinate along the x-axis of the reference point given in meters.
+
+.. option:: +py=<value>
+
+    Coordinate along the y-axis of the reference point given in meters.
+
+.. option:: +pz=<value>
+
+    Coordinate along the z-axis of the reference point given in meters.
+
+Mathematical description
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+
+In the *Position Vector* convention, we define :math:`R_x = radians \left( rx \right)`,
+:math:`R_z = radians \left( ry \right)` and :math:`R_z = radians \left( rz \right)`
+
+In the *Coordinate Frame* convention, :math:`R_x = - radians \left( rx \right)`,
+:math:`R_z = - radians \left( ry \right)` and :math:`R_z = - radians \left( rz \right)`
+
+.. math::
+    :label: 10param
+
+    \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}