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.¶¶Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are:¶¶R= sqrt( rho0 * nu0)¶n= {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5¶c=  [(n+sin(lat0)) (1-sin(chi0))]/[(n-sin(lat0)) (1+sin(chi0))]¶¶where:¶sin(chi0) = (w1-1)/(w1+1)¶w1 = (S1.(S2)^e)^n¶S1 = (1+sin(lat0))/(1-sin(lat0))¶S2 = (1-e sin(lat0))/(1+e sin(lat0))¶¶The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from :¶¶chi0 = asin[(w2-1)/(w2+1)]¶¶where S1 and S2 are as above and  w2 = c (S1(S2)^e)^n¶ ¶lambda0  = lon0¶¶For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from ¶lambda = n(lon-lambda0) + lambda0¶chi = asin[(w-1)/(w+1)]¶¶where w = c (Sa (Sb)^e)^n¶Sa = (1+sin(lat))/(1-sin(lat))¶Sb = (1-e.sin(lat))/(1+e.sin(lat))¶ ¶Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)]¶¶N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B¶¶E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B¶¶
The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.¶¶The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) :¶¶chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)]¶¶lambda = j + 2 i + lambda0¶¶where g = 2 R k0 tan(pi/4 - chi0/2)¶h = 4 R k0 tan(chi0) + g¶i = atan[(E-FE) / {h+(N-FN)}]¶j = atan[(E-FE) / (g-(N-FN)] - i¶¶Geodetic longitude lon = (lambda-lambda0 ) / n +  lambda0¶¶Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1-  sin(chi))}] / n¶¶First approximation lat1 = 2 atan(e^psi)  - pi/2  where e=base of natural logarithms.¶¶psii = isometric latitude at lati¶¶where psii= ln[{tan(lati/2 + pi/4}  {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]¶ ¶Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2)¶¶until the change in lat is sufficiently small.¶¶For Oblique Stereographic projections centred on points in the southern hemisphere,  the signs of E, N, lon0, lon,  must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude.¶¶An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed.  The conformal longitude is then always equivalent to the geodetic longitude.  This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different coordinate operation method to that described above.