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 = EF + rho * sin (lon – lonO)¶N = NF – rhoF + rho * cos (lon – lonO)¶where¶mF = cos latF  / (1 – e^2 sin^2(latF))^0.5¶tF  = tan (p/4 + latF/2) / {[(1 + e sin(latF)) / (1 – e sin(latF))]^(e/2)}¶t  = tan (p/4 + lat/2) / {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)}¶rhoF = a mF¶rho = rhoF *  t / tF¶¶For the north pole case, mF, *F, * and E are found as for the south pole case but¶tF = tan (p/4 – latF/2) * {[(1 + e sin(latF)) / (1 – e sin(latF))]^(e/2)}¶t = tan (p/4 – lat/2) * {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)}¶N = NF + rhoF – [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 for the south pole case¶rho' = [(E-EF)^2  + (N – NF + rhoF)^2] ^0.5¶t'   =  rho' *  tF / rhoF¶chi  = 2 atan(t' ) – pi/2¶and where mF and tF are as for the forward conversion¶¶For reverse conversion north pole case, mF, tF and rhoF are found as for the north pole case of the forward conversion, and¶rho' = [(E-EF)^2  + (N – NF – rhoF)^2]^0.5¶t' is found as for the south pole case of the reverse conversion =  rho' *  tF / rhoF¶chi  =  pi/2 - 2 atan t'¶¶Then for for both north and south pole cases ¶if E = EF, lon = lonO¶else for the south pole case¶lon = lonO + atan [(E – EF) / (N – NF + rhoF)]¶and for the north pole case¶lon = lonO + atan [(E – EF) / –(N – NF – rhoF)] = lonO + atan [(E – EF) / (NF + rhoF – N)]