IGN document NTG_88.pdf, "Grille de parametres de transformation de coordonnees". http://www.ign.fr
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. In France the national mapping agency (IGN) has promolgated a transformation between the old geographic 2D coordinate reference system NTF and the modern 3-dimensional system RGF93 which uses geocentric translations interpolated from a grid file. This is a two-step operation which first requires interpolation of a gridded dataset in the geographic 2D domain to obtain the geocentric translations, then the transformation is made using the Geocentric translations method. It is described in IGN document NTG-88. The method has also been used in New Caledonia. In summary: • The grid file nodes are referenced to the modern (RGF93 in France) geographic 2D CRS (the Interpolation CRS). • Within the grid file the sense of the parameter values is from old to modern (for France, NTF to RGF93), that is the offsets are additive when the old CRS (in France, NTF) is the source CRS and the modern CRS (in France, RGF93) is the target CRS. For the forward transformation, old CRS to modern CRS, because the interpolation CRS is not the source CRS, in principle iteration is required. The source CRS coordinates are used for the initial bi-linear interpolation of the grid to obtain preliminary geocentric translation offsets. These are applied to calculate provisional target CRS coordinates: X't = Xs + tX Y't = Ys + tY Z't = Zs + tZ This process is iterated until the changes in target coordinates are insignificant. An approximation to an accuracy of 1cm may be made if a suitable 'standard transformation' is available. This avoids iteration. The steps for the forward transformation then are: • Convert source CRS coordinates to interpolation CRS coordinates by using the standard transformation. • Using these, interpolate within the grid file to obtain the three geocentric translations (tX, tY, tZ) applicable at the point. • Using the Geocentric translation (geog 2D domain) method, transform coordinates of the point in the source CRS to the target CRS: Xt = Xs + tX Yt = Ys + tY Zt = Zs + tZ For the reverse transformation from modern CRS (in France, RGF93) to old CRS (in France, NTF) where the modern CRS is the Interpolation CRS, the steps are: • Using the modern (RGF93) geographic coordinates, interpolate within the grid file to obtain the three geocentric translations (tX, tY, tZ) applicable at the point. • Transform modern CRS (RGF93) coordinates to old CRS (NTF) coordinates, taking into account the sense of the geocentric translation parameter values: X(old) = X(modern)+(– tX) which for France may be expressed as X(NTF) = X(RGF93)+(– tX) Y(old) = Y(modern)+(– tY) which for France may be expressed as Y(NTF) = Y(RGF93)+(– tY) Z(old) = Z(modern)+(– tZ) which for France may be expressed as Z(NTF) = Z(RGF93)+(– tZ)
Forward transformation from NTF to RGF93 for a point in France with NTF coordinates of: latitude (NTF) = 48°50'40.2441"N, longitude (NTF) = 2°25'32.4187"E (of Greenwich), This transformation avoids iteration in the forward direction by first using NTF to ETRS89 (1) (EPSG coordinate transformation code 1651) as the 'standard transformation'. This uses the Geocentric translations (geog2D domain) method, code 9603, with parameter values of tX = –168 m tY = –60 m tZ = +320 m and gives approximate RGF93 coordinates of latitude (RGF93') = 48°50'39.9967"N = 48.84444352° longitude (RGF93') = 2°25'29.8273"E = 2.42495203° Then using these approximate RGF93 coordinates, bilinear interpolation of the relevant four grid node values 48.9°N, 2.4°E: tX = –168.275 tY = –58.606 tZ = 320.189 48.9°N, 2.5°E: tX = –168.253 tY = –58.554 tZ = 320.165 48.8°N, 2.4°E: tX = –168.252 tY = –58.630 tZ = 320.170 48.8°N, 2.5°E: tX = –168.204 tY = –58.594 tZ = 320.125 gives interpolated tX = –168.253m, tY = –58.609m and tZ = 320.170m. Then using the Geocentric translations (geog2D domain) method, NTF coordinates latitude = 48°50'40.2441"N, longitude = 2°25'32.4187"E (of Greenwich) and an assumed ellipsoid height of 0.000m convert to geocentric Cartesian values X(NTF) = 4201905.725 m Y(NTF) = 177998.072 m Z(NTF) = 4778904.260 m Applying the interpolated geocentric translations as an additive correction: X(RGF93) = 4201905.725 + (–168.253) = 4201737.472 m Y(RGF93) = 177998.072 + ( –58.609) = 177939.463 m Z(RGF93) = 4778904.260 + ( 320.189) = 4779224.430 m then latitude (RGF93) = 48°50'40.0050"N longitude (RGF93) = 2°25'29.8960"E. For the reverse calculation from RGF93 to NTF of the same point: latitude (RGF93) = 48°50'40.0050"N longitude (RGF93) = 2°25'29.8960"E first use these RGF93 coordinates to interpolate within the grid to give geocentric translations of tX = –168.253 m, tY = –58.609 m and tX = 320.170 m. Then using the Geocentric translations (geog2D domain) method, the RGF93 geographical coordinates (with assumed ellipsoidal height of 0.000m) are converted to geocentric Cartesian coordinates and the interpolated geocentric translations applied with reverse sign: X(NTF) = 4201709.097 – (–168.253) = 4201877.350 m Y(NTF) = 177938.261 – ( –58.609) = 177996.870 m Z(NTF) = 4779191.937 – ( 320.170) = 4778871.767 m then latitude (NTF) = 48°50'40.244"N, longitude (NTF) = 2°25'32.419"E (of Greenwich).