EPSG Dataset :: v12.053

Mercator (Spherical)

Coordinate Operation Method Details [VALID]

Name:

Mercator (Spherical)

Code:

1026

Operation is Reversible:

Yes

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 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.

Example:

For Projected Coordinate Reference System: World Spherical Mercator (Note: CRS not in EPSG dataset)

Parameters:

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

whence

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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of natural origin	8801	No	The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
Longitude of natural origin	8802	No	The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the longitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0). Sometimes known as "central meridian (CM)".
False easting	8806	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
False northing	8807	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.

Meta Data

Information Source:

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

Data Source:

EPSG

Revision Date:

November 4, 2021

Change ID:

[2008.114] [2009.023] [2021.033]

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.