Refactor operation doc pages

Extended the use of the :option: directive to all currently written
operation doc pages. In the process several more default parameters were
added for easy inclusion in other doc pages.

Expanded the info tables for each operation with "domain", "alias",
"input type" and "output type".

Corrected various errors along the way. Mainly formatting errors and
references to invalid parameters such as +sym in hatano.

1 files changed, 112 insertions, 47 deletions

diff --git a/docs/source/operations/transformations/helmert.rst b/docs/source/operations/transformations/helmert.rst
index 941fc676..a3e43c13 100644
--- a/docs/source/operations/transformations/helmert.rst
+++ b/docs/source/operations/transformations/helmert.rst
@@ -4,59 +4,26 @@
 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
 14-parameter kinematic counterparts.
 
+
++-----------------+-------------------------------------------------------------------+
+| **Alias**       | helmert                                                           |
++-----------------+-------------------------------------------------------------------+
++ **Domain**      | 2D, 3D and 4D                                                     |
 +-----------------+-------------------------------------------------------------------+
 | **Input type**  | Cartesian coordinates (spatial), decimalyears (temporal).         |
 +-----------------+-------------------------------------------------------------------+
 | **Output type** | Cartesian coordinates (spatial), decimalyears (temporal).         |
 +-----------------+-------------------------------------------------------------------+
-| **Options**                                                                         |
-+----------------+--------------------------------------------------------------------+
-| `x`            | Translation of the x-axis [m]. *Optional*.                         |
-+----------------+--------------------------------------------------------------------+
-| `y`            | Translation of the y-axis [m]. *Optional*.                         |
-+----------------+--------------------------------------------------------------------+
-| `z`            | Translation of the z-axis. [m]. *Optional*.                        |
-+----------------+--------------------------------------------------------------------+
-| `s`            | Scale factor [ppm]. *Optional*.                                    |
-+----------------+--------------------------------------------------------------------+
-| `rx`           | X-axis rotation in the 3D Helmert [arc seconds]. *Optional*.       |
-+----------------+--------------------------------------------------------------------+
-| `ry`           | Y-axis rotation in the 3D Helmert [arc seconds]. *Optional*.       |
-+----------------+--------------------------------------------------------------------+
-| `rz`           | Z-axis rotation in the 3D Helmert [arc seconds]. *Optional*.       |
-+----------------+--------------------------------------------------------------------+
-| `theta`        | Rotation angle in the 2D Helmert. [arc seconds]. *Optional*.       |
-+----------------+--------------------------------------------------------------------+
-| `dx`           | Translation rate of the x-axis. [m/year]. *Optional*.              |
-+----------------+--------------------------------------------------------------------+
-| `dy`           | Translation rate of the y-axis. [m/year]. *Optional*.              |
-+----------------+--------------------------------------------------------------------+
-| `dz`           | Translation rate of the z-axis. [m/year]. *Optional*.              |
-+----------------+--------------------------------------------------------------------+
-| `ds`           | Scale rate factor [ppm/year]. *Optional*.                          |
-+----------------+--------------------------------------------------------------------+
-| `drx`          | Rotation rate of the x-axis [arc seconds/year]. *Optional*.        |
-+----------------+--------------------------------------------------------------------+
-| `dry`          | Rotation rate of the y-axis [arc seconds/year]. *Optional*.        |
-+----------------+--------------------------------------------------------------------+
-| `drz`          | Rotation rate of the y-axis [arc seconds/year]. *Optional*.        |
-+----------------+--------------------------------------------------------------------+
-| `t_epoch`      | Central epoch of transformation. [decimalyear]. Only used in       |
-|                | spatiotemporal transformations. *Optional*.                        |
-+----------------+--------------------------------------------------------------------+
-| `t_obs`        | Observation time of coordinate(s). Mostly useful in 2D and 3D      |
-|                | transformations. [decimalyear]. *Optional*.                        |
-|                | If not specified, will be get from the t component of 4D input     |
-|                | points.                                                            |
-+----------------+--------------------------------------------------------------------+
-| `exact`        | Use exact transformation equations. *Optional*.                    |
-+----------------+--------------------------------------------------------------------+
-| `transpose`    | Transpose rotation matrix. *Optional*.                             |
-+----------------+--------------------------------------------------------------------+
+| **Input type**  | Cartesian coordinates                                             |
++-----------------+-------------------------------------------------------------------+
+| **Output type** | Cartesian coordinates                                             |
++-----------------+-------------------------------------------------------------------+
 
 The Helmert transform, in all it's various incarnations, is used to perform reference
 frame shifts. The transformation operates in cartesian space. It can be used to transform
@@ -107,6 +74,102 @@ Transformation from `ITRF2000@2017.0`  to `ITRF93@2017.0` using 15 parameters:
          drx=-0.00011 dry=-0.00019 drz=0.00007
          epoch=1988.0 tobs=2017.0    transpose
 
+Parameters
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+.. note::
+
+    All parameters are optional but at least one should be used, otherwise the
+    operation will return the coordinates unchanged.
+
+.. option:: +x=<value>
+
+    Translation of the x-axis given in meters.
+
+.. option:: +y=<value>
+
+    Translation of the x-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 in the 3D Helmert given arc seconds.
+
+
+.. option:: +ry=<value>
+
+    Y-axis rotation in the 3D Helmert given in arc seconds.
+
+.. option:: +rz=<value>
+
+    Z-axis rotation in the 3D Helmert given in arc seconds.
+
+
+.. option:: +theta=<value>
+
+    Rotation angle in the 2D Helmert given in arc seconds.
+
+.. option:: +dx=<value>
+
+    Translation rate of the x-axis given in m/year.
+
+.. option:: +dy=<value>
+
+    Translation rate of the y-axis given in m/year.
+
+.. option:: +dz=<value>
+
+    Translation rate of the z-axis given in m/year.
+
+.. option:: +ds=<value>
+
+    Scale rate factor given in ppm/year.
+
+.. option:: +drx=<value>
+
+    Rotation rate of the x-axis given in arc seconds/year.
+
+.. option:: +dry=<value>
+
+    Rotation rate of the y-axis given in arc seconds/year.
+
+.. option:: +drz=<value>
+
+    Rotation rate of the y-axis given in arc seconds/year.
+
+.. option:: +t_epoch=<value>
+
+    Central epoch of transformation given in decimalyear. Only used
+    spatiotemporal transformations.
+
+.. option:: +t_obs=<value>
+
+    Observation time of coordinate(s) given in decicalyear. Mostly useful
+    in 2D and 3D transformations where the observation time is not passed
+    as part of the input coordinate. Can be used to override the observation
+    time from the input coordinate.
+
+.. option:: +exact
+
+    Use exact transformation equations.
+
+    See :eq:`rot_exact`
+
+.. option:: +transpose
+
+    Transpose rotation matrix and follow the **Cordinate Frame** rotation
+    convention. If :option:`+transpose` is not added the **Position Vector**
+    rotation convention is used.
+
+
+
 Mathematical description
 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
@@ -244,6 +307,7 @@ The three rotation matrices can be combined in one:
         \mathbf{R} = \mathbf{R_X} \mathbf{R_Y} \mathbf{R_Y}
     \end{align}
 
+
 For :math:`\mathbf{R}`, this yields:
 
 .. math::
@@ -302,9 +366,10 @@ using the approximated rotation matrix:
 
 If the rotation matrix is transposed the transformation is effectively reversed.
 This is cause for some confusion since there is no correct way of defining the
-rotation matrix. Two conventions exists and they seem to be equally popular.
-In PROJ the rotation matrix can be transposed by adding the `transpose` flag in
-the transformation setup.
+rotation matrix. Two conventions exists and they seem to be equally popular. PROJ
+uses the **Position Vector** rotation convention. The rotation matrix can be transposed by
+adding the :option:`+transpose` flag in the transformation setup which makes PROJ
+follow the **Coordinate Frame** rotation convention.
 
 Applying :eq:`propagation` we get the kinematic version of the approximated
 3D Helmert: