EPSG

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

2018-08-29

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. First calculate the scale factor at natural origin: for the south pole case tF = tan (pi/4 + latF/2) / {[(1 + e sin(latF)) / (1 – e sin(latF))]^(e/2)} but for the north pole case tF = tan (pi/4 - latF/2) * {[(1 + e sin(latF)) / (1 – e sin(latF))]^(e/2)} then for both cases mF = cos(latF) / (1 – e^2 sin^2(latF))^0.5 ko = mF {[(1+e)^(1+e) (1–e)^(1–e)]0.5} / (2 tF) The forward and reverse conversions then follow the formulae for the Polar Stereographic (variant A) method: 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 implementation notes in preface for atan2 convention)

For Projected Coordinate Reference System: WGS 84 / Australian Antarctic Polar Stereographic Parameters: Ellipsoid: WGS 84 a = 6378137.0 metres 1/f = 298.2572236 then e = 0.081819191 Latitude of standard parallel (latF): 71°00'00.000"S = -1.239183769 rad Longitude of origin (lonO): 70°00'00.000"E = 1.221730476 rad False easting (FE): 6000000.00 metres False northing (FN): 6000000.00 metres Forward calculation for: Latitude (lat) = 75°00'00.000"S = -1.308996939 rad Longitude(lon) = 120°00'00.000"E = 2.094395102 rad tF = 0.168407325 mF = 0.326546781 ko = 0.97276901 t = 0.132508348 pho = 1638783.238 whence E = 7255380.79 m N = 7053389.56 m Reverse calculation for the same Easting and Northing (7255380.79 E, 7053389.56 N) first gives: tF = 0.168407325 mF = 0.326546781 and ko = 0.97276901 then rho' = 1638783.236 t' = 0.132508347 chi = -1.3073146 Then Latitude (lat) = 75°00'00.000"S Longitude (lon) = 120°00'00.000"E