Coordinate Frame rotation (geocentric domain)


Data source: EPSG

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2019-02-21

Remarks: This method is a specific case of the Molodensky-Badekas (CF) method (code 1034) in which the evaluation point is at the geocentre with coordinate values of zero. Note the analogy with the Position Vector method (code 1033) but beware of the differences!


<<<<<This text is continued from the description of the Position Vector transformation formula>>>>>

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Although being common practice particularly in the European E&P industry, the Position Vector transformation sign convention is not universally accepted.  A variation on this formula is also used, particularly in the USA E&P industry.  That formula is based on the same definition of translation and scale parameters, but a different definition of the rotation parameters.  The associated convention is known as the "Coordinate Frame rotation" convention. 
The formula is:

   (Xt)          (  1      +rZ      -rY)    (Xs)     (tX)
   (Yt)  =  M *  ( -rZ       1      +rX)  * (Ys)  +  (tY)
   (Zt)          ( +rY     -rX       1 )    (Zs)     (tZ)

and the parameters are defined as:

(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.

(rX, rY, rZ): Rotations to be applied to the coordinate reference frame.  The sign convention is such that a positive rotation of the frame about an axis is defined as a clockwise rotation of the coordinate reference frame when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis, that is a positive rotation about the Z-axis only from source coordinate reference system to target coordinate reference system will result in a smaller longitude value for the point in the target coordinate reference system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).

In the absence of rotations the two formulas are identical; the difference is solely in the rotations. The name of the second method reflects this.

Note that the same rotation that is defined as positive in the first method is consequently negative in the second and vice versa.  It is therefore crucial that the convention underlying the definition of the rotation parameters is clearly understood and is communicated when exchanging datum transformation parameters, so that the parameters may be associated with the correct coordinate transformation method (algorithm).


The same example as for the Position Vector transformation (coordinate operation method 1033) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:

Transformation parameters for the Coordinate Frame rotation convention:
tX (m) = 0.000 
tY (m) = 0.000 
tZ (m) = +4.5 
rX (") = 0.000
rY (") = 0.000
rZ (") = -0.554 = -0.000002685868 radians
dS (ppm) = +0.219

M = 1 + dS = 1.000000219

Please note that only the rotation has changed sign as compared to the Position Vector transformation.
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