EPSG:9804

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.

The formulas to derive projected Easting and Northing coordinates are:

E = FE + a*ko(lon - lonO)              
N = FN + a*ko* ln{tan(pi/4 + lat/2)[(1 - esin(lat))/(1 + esin(lat))]^e/2} where symbols are as listed above and logarithms are natural.

The reverse formulas to derive latitude and longitude from E and N values are:

lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi) 
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 +  81e^8/1120) sin(6chi)  + (4279e^8/161280) sin(8chi)

where chi = pi/2 - 2 arctan t
t = B^((FN-N)/(a*ko)) 
B = base of the natural logarithm, 2.7182818...
and  for the 2 SP Case, ko is calculated as for the forward transformation above.
lon  =  ((E - FE)/(a*ko))  + lonO

Note: For the Mercator variant A (1SP) method, in the EPSG Dataset, to be fully transparent about the location of the projection origin, the parameter "latitude of natural origin (LatO)" is included in the defining parameters of the map projection method and map projections. It must have a value of zero because by definition the location of the natural origin for this method is on the equator. However this parameter is not used in the conversion formulas.