EPSG:9830

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 + atan2[(E – EF),(N – NF + rhoF)]
and for the north pole case
lon = lonO + atan2[(E – EF),(NF + rhoF – N)]
(see implementation notes in GN7-2 preface for atan2 convention)