Mercator (variant A)


Data source: EPSG

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2021-11-04

Remarks: Note that in these formulas the parameter latitude of natural origin (latO) is not used. However for this Mercator (variant A) method the EPSG dataset includes this parameter, which must have a value of zero, for completeness in CRS labelling.


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.


For Projected Coordinate System Makassar / NEIEZ

Ellipsoid   Bessel 1841  a = 6377397.155 m   1/f = 299.15281
then e = 0.08169683

Latitude of natural origin = 00°00'00"N  = 0.0000000 rad
Longitude of natural origin = 110°00'00"E  = 1.91986218 rad
Scale factor at natural origin ko = 0.997
False Eastings FE = 3900000.00 m
False Northings FN = 900000.00 m

Forward calculation for: 
Latitude = 3°00'00.00"S   = -0.05235988 rad
Longitude = 120°00'00.00"E   =  2.09439510 rad
Easting  E  =  5009726.58 m
Northing N  =   569150.82 m

Reverse calculation for same easting and northing first gives :
t    = 1.0534121
chi = -0.0520110

Latitude   =   3°00'00.000"S
Longitude  = 120°00'00.000"E
MapTiler banner