EPSG

EPSG guidance note #7-2, https://epsg.org/guidance-notes.html

2022-01-03

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.

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