Vertical Offset and Slope


Data source: EPSG

Information source: OGP

Revision date: 2017-10-16

Remarks: This transformation allows calculation of height in the target system by applying the parameter values to the height value of the point in the source system.


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 Europe, national height systems are related to the pan-European height system through three transformation parameters and the formula:

Ht = Hs + A + [IncLat * rhoO * (Lat – LatO)] + [IncLong * nuO * (Long
– LongO) * cos(Lat)]

Ht = gravity-related height value in the target vertical coordinate reference system.
Hs = gravity-related height value in the source vertical coordinate reference system.
A is the value of the vertical offset to be applied.
IncLat is the value in radians of the inclination parameter in the latitude domain, i.e. in the plane of the meridian, derived at an evaluation point with coordinates of LatO, LongO.
IncLon is the value of the inclination parameter in the longitude domain, i.e. perpendicular to the plane of the meridian.
rhoO is the radius of curvature of the meridian at latitude LatO, where rhoO = a(1 – e^2)/(1 – e^2 * sin^2(LatO))^1.5
nuO is the radius of curvature on the prime vertical (i.e. perpendicular to the meridian) at latitude LatO, where nuO = a /(1 – e^2 * sin^2(LatO))^0.5
LatO , LongO are the coordinates of the evaluation point in the 'Horizontal CRS' coordinate reference system, in radians. The ID of the 'Horizontal CRS' is a defining parameter.
Lat , Long are the horizontal coordinates of the point in the same coordinate reference system as the evaluation point (the 'Horizontal CRS'), in radians.

The horizontal location of the point must always be given in 'Horizontal CRS' terms. If given in another CRS the coordinates must first be transformed to 'Horizontal CRS' values. The method is reversible.


For coordinate transformation LN02 to EVRF2000 (1)

Ordinate 1 of evaluation point: 46deg 55min N = 0.818850307  radians
Ordinate 2 of evaluation point: 8deg 11min E of Greenwich = 0.142826110  radians
Transformation Parameters:
A = -0.245m 
IncLat = -0.210" = -0.000001018  radians
IncLong = -0.032" = -0.000000155  radians

Consider a point having a gravity-related height in the LN02 system (Hs) of 473.0m and with horizontal coordinates in the ETRS89 geographical coordinate reference system (CRS code 4258) of:
ETRS89 latitude: 47deg 20 min N = 0.826122513  radians
ETRS89 longitude: 9 deg 40min E of Greenwich = 0.168715161 radians

Then rhoO = 6369526.88 m 
IncLat term = -0.047 m 
nuO = 6389555.64 m  
incLong term = -0.017 m
whence EVRF2000 height (Ht) = 473.0 +(-0.245) + (-0.047) + (-0.017) = 472.690 m.
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