[Proj] Meaning of 7-parameter transformation rotations

[Oscar van Vlijmen]

> From: Martin Vermeer <martin.vermeer-hut.fi>

> I find in the document gen_parms.html the remark
> "The seven parameter case uses delta_x, delta_y, delta_z, Rx - rotation
> X, Ry -  rotation  Y, Rz - rotation Z, M_BF - Scaling. The three
> translation parameters are in meters as in the three parameter case. The
> rotational parameters are not in physical units. They are something like the
> sine of the rotational angle times the ellipoid axis length, but I don't know
> the exact details of how to derive these from a physical description of
> the rotation."

In the 7-parameter Helmert transformation the rotation parameters are indeed
in arcseconds according to geodetic practice.
In principle the transformation should be done with mathematical accuracy
and in that case in the rotation matrix combinations of sines and cosines of
angles are found.
Since the translations are relatively small, the rotation angles are small,
the scaling factor is small and a (spherical) Helmert transformation on an
ellipsoidal earth introduces small errors anyway, the sines/cosines in the
rotation matrix are approximated. sin(angle) is nearly equal to angle
(expressed in rad), cos(angle) is nearly 1 and sin(angle)*sin(angle) is
about 0.
See EPSG coordinate operation method code 9606 and ISO 19111.
This approximation has several interesting practical consequences. The order
(around geocentric X-Y-Z axes) of the rotations is no longer important, the
inverse transformation can be done by negating all parameters, switch from
coordinate frame to position vector rotation can be done by negating the
angles only.