EPSG guidance note #7-2, http://www.epsg.org
This transformation is a truncated Taylor series expansion of a transformation between two geographic coordinate systems, modelled as a set of geocentric translations.
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. As an alternative to the computation of the new latitude, longitude and height above ellipsoid in discrete steps through geocentric coordinates, the changes in these geographic coordinates may be derived directly by formulas derived by Molodenski. Abridged versions of these formulas, which are quite satisfactory for three parameter transformations, are as follows: dlat " = [(-dX*sin(lat)*cos(lon)) - (dY*sin(lat)*sin(lon)) + (dZ*cos(lat)) + (((a*Df) + (f*Da))*sin(2*lat))] / (rho * sin(1")) dlon " = (-dX*sin(lon) + dY*cos(lon)) / ((nu*cos(lat)) * sin(1")) dh = (dX*cos(lat)*cos(lon)) + (dY*cos(lat)*sin(lon)) + (dZ*sin(lat)) + ((a*Df + f*Da)*(sin(lat)^2)) - da where the dX, dY and dZ terms are the geocentric translation parameters, and rho and nu are the meridian and prime vertical radii of curvature at the given latitude (lat) on the first ellipsoid, da is the difference in the semi-major axes (a1 - a2) of the first and second ellipsoids and df is the difference in the flattening of the two ellipsoids. The formulas for dlat and dlon indicate changes in latitude and longitude in arc-seconds.
For a North Sea point with coordinates derived by GPS satellite in the WGS84 geographical coordinate reference system, with coordinates of: latitude lat_s =53°48'33.82"N, longitude lon_s = 2°07'46.38"E, and ellipsoidal height h_s = 73.0m, whose coordinates are required in terms of the ED50 geographical coordinate reference system which takes the International 1924 ellipsoid. The three geocentric translations parameter values from WGS84 to ED50 for this North Sea area are given as dX = +84.87m, dY = +96.49m, dZ = +116.95m. Ellipsoid Parameters are: WGS 84 a = 6378137.0 metres 1/f = 298.2572236 International 1924 a = 6378388.0 metres 1/f = 297.0 Then da = 6378388 – 6378137 = 251 df = 0.003367003 - 0.003352811 = 1.41927E-05 whence dlat = 2.543" dlon = 5.097" dh = – 44.909 m ED50 values on the International 1924 ellipsoid are then: latitude lat_t = 53°48'36.563"N, longitude lon_t = 2°07'51.477"E, and ellipsoidal height h_t = 28.091 m.