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Molodensky-Badekas transform

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+

New in version 6.0.0.

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+

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

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+

Note

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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

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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.

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+

Example

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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
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Parameters

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Note

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

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++convention=coordinate_frame/position_vector
+

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

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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).

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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)

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++x=<value>
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Translation of the x-axis given in meters.

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+ +
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++y=<value>
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Translation of the y-axis given in meters.

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+ +
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++z=<value>
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Translation of the z-axis given in meters.

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++s=<value>
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Scale factor given in ppm.

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+ +
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++rx=<value>
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X-axis rotation given arc seconds.

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++ry=<value>
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Y-axis rotation given in arc seconds.

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+ +
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++rz=<value>
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Z-axis rotation given in arc seconds.

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+ +
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++px=<value>
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Coordinate along the x-axis of the reference point given in meters.

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++py=<value>
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Coordinate along the y-axis of the reference point given in meters.

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++pz=<value>
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Coordinate along the z-axis of the reference point given in meters.

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Mathematical description

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