EPSG:9801

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.

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon):

E = FE + r sin(theta)
N = FN + r0 - r cos(theta)
where
	mO = cos(latO)/(1 – e^2 sin^2(latO))^0.5 where latO is the latitude of natural origin
	tO = tan(pi/4 – latO/2)/[(1 – e sin(latO))/(1 + e sin(latO))]^e/2
	t = tan(pi/4 – lat/2)/[(1 – e sin(lat))/(1 + e sin(lat))]^e/2
	n = sin(latO)
	F = mO/(n tO^n)
	rO = a F tO^n kO  
	r =  a F t^n kO 
	theta = n(lon – lonO)
As with other conics, a negative n and r result for projections centered in the Southern Hemisphere.

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

lat = pi/2 - 2arctan{t'[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2)}
lon = theta'/n +lon0
where
n, F, and rO are derived as for the forward calculation
r' = +/-[(E - FE)^2 + {r0 - (N - FN)}^2]^0.5  taking the sign of n
t' = (r'/(a k0 F))^(1/n)
If n is positive, theta' = atan2{(E –  FE) , [rO – (N –  FN)]}
but if n is negative the signs of both arguments of the atan2 function  must be reversed and theta' = atan2{– (E –  FE) , – [rO – (N –  FN)]}

Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat using 
lat = π/2-2atan(t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.