Mercator (variant B)

Formula

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:

ko = cos(latSP1)/(1 - e^2*sin^2(latSP1))^0.5 
where latSP1 is the absolute value of the first standard parallel (i.e. positive).

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 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 ko is calculated as for the forward transformation above.
lon  =  ((E - FE)/(a*ko)) + lonO

Example

For Projected Coordinate System Pulkovo 1942 / Caspian Sea Mercator

Parameters:
Ellipsoid  Krassowski 1940   a = 6378245.00m   1/f = 298.300
then e = 0.08181333 and e^2 = 0.00669342

Latitude of first SP = 42°00'00"N = 0.73303829 rad
Longitude of natural origin = 51°00'00"E = 0.89011792 rad
False Eastings FE = 0.00 m
False Northings (at equator) FN = 0.00 m
Forward calculation for: 
Latitude = 53°00'00.00"N = 0.9250245 rad
Longitude = 53°00'00.00"E  = 0.9250245 rad

gives 
  
ko = 0.744260894
Easting E = 165704.29 m 
Northing N = 5171848.07 m

Reverse calculation for same easting and northing first gives :
t = 0.336391288
chi = 0.921795958
Latitude = 53°00'00.000"N
Longitude = 53°00'00.000"E

EPSG:9805

Mercator (variant B)

Attributes

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Formula

Example