EPSG:1061

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.

To eliminate high correlation between the translations and rotations in the derivation of parameter values for the Helmert transformation methods (coordinate operation method codes 1032 and 1033), instead of the rotations being derived about the geocentric coordinate reference system origin they may be derived at a location within the points used in the determination. Three additional parameters, the coordinates of the rotation point, are then required. The formula is:

   (Xt)          (  1      -rZ     +rY)     (Xs - Xp)     (Xp)     (tX)
   (Yt)  =  M *  ( +rZ      1      -rX)  *  (Ys - Yp)  +  (Yp)  +  (tY)
   (Zt)          ( -rY     +rX      1 )     (Zs - Zp)     (Zp)     (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 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 point's vector.  The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target 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.

(Xp, Yp, Zp): Coordinates of the point about which the coordinate reference frame is rotated, given in the source Cartesian coordinate reference system. 

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).

Reversibility.
The Molodensky-Badekas (PV) transformation in a strict mathematical sense is not reversible, i.e. in principle the same parameter values cannot be used to execute the reverse transformation. This is because the evaluation point coordinates are in the forward direction source coordinate reference system and the rotations have been derived about this point. They should not be applied about the point having the same coordinate values in the target coordinate reference system, as is required for the reverse transformation. However, in practical application there are exceptions when applied to the approximation of small differences between the geometry of a set of points in two different coordinate reference systems. The typical vector difference in coordinate values is in the order of 6*10^1 to 6*10^2 metres, whereas the evaluation point on or near the surface of the earth is 6.3*10^6 metres from the origin of the coordinate systems at the Earth's centre. This difference of four or five orders of magnitude allows the transformation in practice to be considered reversible. Note that in the reverse transformation, only the signs of the translations and rotation parameter values are reversed; the coordinates of the evaluation point remain unchanged.