Mercator (Spherical)


Data source: EPSG

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

Revision date: 2021-11-04


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 from spherical latitude lat and longitude lon are:

	E = FE   R (lon - lonO)              

	N = FN   R ln[tan(pi/4   lat/2)]                   

where lonO is the longitude of natural origin and FE and FN are false easting and false nothing.

R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the conformal sphere at the projection origin at latitude latO using the formula for Rc given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the projection properties are not preserved.

If latitude lat = 90º, N is infinite. The above formula for N will fail near to the pole, and should not be used poleward of 88º.

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

	D = -(N - FN)/R = (FN - N)/R

	lat = pi/2 -  2 atan(e^D) where e=base of natural logarithms, 2.7182818...

	lon = [(E - FE)/R] + lonO

Note that for the Merctor (Spherical) method, in the EPSG dataset the parameter latitude of natural origin (LatO) is included in the defining parameters of the map projection method. 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 Reference System: World Spherical Mercator (Note: CRS not in EPSG dataset)


Sphere:	R = 6371007.0 metres

Latitude of natural origin (latO) = 0°00'00.000"N = 0.0 rad

Longitude of natural origin (lonO) = 0°00'00.000"E = 0.0 rad

False easting (FE) = 0.00 metres	

False northing (FN) = 0.00 metres	

Forward calculation for: 

Latitude (lat) = 24°22'54.433"N	= 0.425542460 rad

Longitude (lon) = 100°20'00.000"W = -1.751147016 rad


	E = -11 156 569.90 m

	N =   2 796 869.94 m

Reverse calculation for the same point (-11 156 569.90 m E, 2 796 869.94m N) first gives:

	D = -0.438999665

Then	Latitude (lat) = 0.425542460 rad = 24°22'54.433"N

	Longitude (lon) = -1.751147016 rad = 100°20'00.000"W
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