EPSG Dataset :: v12.003

Popular Visualisation Pseudo Mercator

Coordinate Operation Method Details [VALID]

Name:

Popular Visualisation Pseudo Mercator

Code:

1024

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.

This method is utilised by some popular web mapping and visualisation applications. It applies standard Mercator (Spherical) formulas (method code 1026) to ellipsoidal coordinates and the sphere radius is taken to be the semi-major axis of the ellipsoid. This approach only approximates to the more rigorous application of ellipsoidal formulas to ellipsoidal coordinates (as given in EPSG dataset coordinate operation method codes 9804 and 9805). Unlike either the spherical or ellipsoidal Mercator projection methods, this method is not conformal: scale factor varies as a function of azimuth, which creates angular distortion. Despite angular distortion there is no convergence in the meridian.

The formulas to derive projected Easting and Northing coordinates from ellipsoidal latitude (lat) and longitude (lon) first derive the radius of the sphere (R) from:
R = a

Then applying spherical Mercator formulae:

E = FE + R(lon - lonO)
N = FN + R ln[tan(pi/4 + lat/2)]
where FE and FN are false easting and false nothing at the projection origin, other symbols are as listed above and logarithms are natural.

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

If q_alpha is the scale factor at a given azimuth alpha, it is a function of R', the radius of curvature at that azimuth derived from:
R' = rho nu / (nu cos^2alpha + rho sin^2alpha)
q_alpha = R / (R' cos lat)
where rho and nu are the radii of curvature of the ellipsoid at latitude lat in the plane of the meridian and perpendicular to the meridian respectively;
rho = a(1 - e^2)/(1 - e^2 sin^2(lat))^3/2
nu = a /(1 - e^2 sin^2(lat))^1/2

Then when the azimuth is 0º, 180º, 90º or 270º the scale factors in the meridian (h) and on the parallel (k) are:
q_0 = q_180 = h = R / (rho cos(lat))
q_90 = q_270 = k = R / (nu cos(lat))
which demonstrates the non-conformallity of the Pseudo Mercator method.

Maximum angular distortion omega is a function of latitude and is found from:
omega = 2 asin{[ABS(h - k)] / (h + k)}

Example:

For Projected Coordinate Reference System: WGS 84 / Pseudo-Mercator

Parameters:
Ellipsoid: WGS 84 a = 6378137.0 metres 1/f = 298.2572236

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 the same coordinate values as used for the Mercator (1SP) (Spherical) example (method code 9841):
Latitude (lat) = 24°22'54.433"N = 0.425542460 rad
Longitude (lon) = 100°20'00.000"W = -1.751147016 rad

R = 6378137.0
whence
E = -11 169 055.58 m
N = 2 800 000.00 m
and
h = 1.1034264
k = 1.0972914
omega = 0°19'10.01"

Reverse calculation for a point 10km north on the grid (-11 169 055.58 m E, 2 810 000.00m N) first gives:
D = -0.44056752

Then Latitude (lat) = 0.426970023 rad = 24°27'48.889"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

Remarks:

Applies spherical formulas to the ellipsoid. As such does not have the properties of a true Mercator projection.

Information Source:

OGP Guidance Note 7-2

Data Source:

EPSG

Revision Date:

September 22, 2017

Change ID:

[2008.114] [2009.023] [2017.03]

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.