EPSG

EPSG guidance note #7-2, http://www.epsg.org

2018-08-29

Latitude of natural origin must be either 90 degrees or -90 degrees (or equivalent in alternative angle unit).

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 forward conversion from latitude and longitude, for the south pole case E = FE + rho * sin(lon – lonO) N = FN + rho * cos(lon – lonO) where t = tan(pi/4 + lat/2) / {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)} rho = 2*a*ko*t / {[(1+e)^(1+e) (1–e)^(1–e)]^0.5} For the north pole case, rho and E are found as for the south pole case but t = tan(pi/4 – lat/2) * {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)} N = FN – rho * cos(lon – lonO) For the reverse conversion from easting and northing to latitude and longitude, lat = chi + (e^2/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2 chi) + (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4 chi) + (7e^6/120 + 81e^8/1120) sin(6 chi) + (4279e^8/161280) sin(8 chi) where rho' = [(E-FE)^2 + (N – FN)^2]^0.5 t' =rho' {[(1+e)^(1+e) * (1– e)^(1-e)]^0.5} / (2 a ko) and for the south pole case chi = 2 atan(t' ) – pi/2 but for the north pole case chi = pi/2 - 2 atan(t') Then for for both north and south cases if E = FE, lon = lonO else for the south pole case lon = lonO + atan2[(E – FE),(N – FN)] and for the north pole case lon = lonO + atan2[(E – FE),(FN – N)] (see GN7-2 implementation notes in preface for atan2 convention)

For Projected Coordinate Reference System: WGS 84 / UPS North Parameters: Ellipsoid: WGS 84 a = 6378137.0 metre 1/f = 298.2572236 then e = 0.081819191 Latitude of natural origin (latO): 90°00'00.000"N =1.570796327 rad Longitude of origin (longO): 0°00'00.000"E=0.0 rad Scale factor at natural origin (ko): 0.994 False easting (FE) 2000000.00 metre False northing (FN) 2000000.00 metre Forward calculation for: Latitude (lat) =73°N =1.274090354 rad Longitude (lon) =44°E =0.767944871 rad t = 0.150412808 rho = 1900814.564 whence E = 3320416.75 m N = 632668.43 m Reverse calculation for the same Easting and Northing (3320416.75 E, 632668.43 N) first gives: rho' = 1900814.566 t' = 0.150412808 chi = 1.2722090 Then Latitude (lat) = 73°00'00.000"N Longitude (lon) = 44°00'00.000"E