Transverse Mercator 3D

Formula

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.

For the calculation of easting and northing from latitude and longitude, the formulas for the Transverse Mercator projection (method 9807) are followed. The third coordinate, ellipsoidal height (h), is carried across unchanged.

For the reverse calculation to convert easting and northing projected coordinates to latitude and longitude, the reverse formulas for the Transverse Mercator projection (method 9807) are followed. The third coordinate, ellipsoidal height (h), is carried across unchanged.

Example

For Projected Coordinate System LUREF / Luxembourg TM (3D)

Parameters:
Ellipsoid  International 1924  a = 6378388.0 m  1/f = 297.0
then e'^2 = 0.006768170 and e^2 = 0.006722670

Latitude of natural origin (LatO) = 49°50'00"N = 0.865105111 rad
Longitude of natural origin (LonO) = 6°10'00"E = 0.103634480 rad
Scale factor (ko) = 1.0
False Eastings (FE) = 80000.00 m
False Northings (FN) = 100000.00 m

Forward calculation for: 
Latitude = 49°34'00.7381"N = 0.865105111 rad
Longitude = 5°56'16.1460"E = 0.103634480 rad
Ellipsoidal height = 328.061 m

Constants of the projection:
n = 0.001686341
B = 6367654.500
h1 = 0.0008412760
h2 = 0.0000007673
h3 = 1.21291E-09
h4 = 2.48508E-12
QO = 1.0010215805
ξO0 = 0.8664312653
ξO1 = 0.0008302519
ξO2 = -0.0000002308
ξO3 = -1.08789E-09
ξO4 = 1.5005E-12
Mo = 5522420.140
Q = 0.9938515103
β = 0.8617755951
η0 = 0.8617795370
ξ0 = -0.0026005578
η1 = 0.0008314901
ξ1 = 0.0000006658
η2 = -0.0000002308
ξ2 = 0.0000000076
η3 = -1.08789E-09
ξ3 = -8.37324E-12
η4 = 1.42607E-12
ξ4 = -4.23484E-14
η = 0.8626107952
ξ = -0.0025998844

Then
Northing                 X = 70387.372 metres
Easting	                 Y = 63444.834 metres
Ellipsoidal height  h = 328.061 m

Reverse calculation for same easting, northing and ellipsoidal height first gives:

h1' = 0.0008412763
h2' = 0.0000000596
h3' = 1.69485E-10
h4' = 2.20474E-13
Then
ξ' = -0.0025998844
η' = 0.8626107952
ξ1' = 0.0000006728
η1' = 0.0008312765
ξ2' = 0.0000000006
η2' = -0.0000000181
ξ3' = -1.18154E-12
η3' = -1.51639E-10
ξ4' = -3.73855E-15
η4' = 1.27718E-13
ξ0' = -0.0026005578
η0' = 0.8617795370
β' = 0.8617755951
Q' = 0.9938515103
Q" 1st iteration	= 0.9989606180
Q" 2nd iteration	= 0.9989751785
Q" 3rd iteration	= 0.9989752198
Q" 4th iteration	= 0.9989752198

Then
Latitude	                = 49°34'00.7381"N
Longitude              =  5°56'16.1460"E
Ellipsoidal height = 328.061 m

EPSG:1111

Transverse Mercator 3D

Attributes

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Formula

Example