EPSG Dataset :: v12.003

Equal Earth

Coordinate Operation Method Details [VALID]

Name:

Equal Earth

Code:

1078

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 Equal Earth projection is an equal-area pseudocylindrical projection. It is appropriate for mapping global phenomena or for any other thematic world map that requires areas at their correct relative sizes. Its key features are its resemblance to the Robinson projection and continents with a visually pleasing appearance similar to those found on a globe.

Forward:
To derive the projected coordinates of a point, geodetic latitude (lat) is converted to authalic latitude (ß). The formulae to convert geodetic latitude and longitude (lat,lon) to Easting (E) and Northing (N) are:

Easting, E = FE + Rq 2 (lon – lonO) cos(θ) / {√3 [1.340264 – 0.243318 θ^2 + θ^6 (0.006251 + 0.034164 θ^2)]}
Northing, N = FN + Rq θ [1.340264 – 0.081106 θ^2 + θ^6 (0.000893 + 0.003796 θ^2)]

where
θ = asin [sin(ß) √3 / 2]
Rq = a (qP / 2)^0.5
ß = asin(q / qP)
q = (1 – e^2) ({sin(lat)/ [1 – e^2 sin^2(lat)]} – 1/(2e) ln{[1 – e sin(lat)] / [1 + e sin(lat)]})

qP = (1 – e^2) ({sin(latP)/ [1 – e^2 sin^2(latP)]} – 1/(2e) ln{[1 – e sin(latP)] / [1 + e sin(latP)]})

where latP = π/2 radians, thus

qP = (1 – e^2) ([1 / (1 – e^2)] – {[1/(2e)] ln [(1 – e) / (1 + e)]})

Reverse:
The reverse conversion from easting and northing to latitude and longitude requires iteration of northing (N) equation to obtain θ. θ0 = (N–FN) / Rq is used as the first trial θ. A correction Δθn is calculated and subtracted from the trial θn value to obtain the next trial θn+1. n is the suffix number of iteration.

θ0 = (N–FN) / Rq
Δθ0 = { θ0 [1.340264 – 0.081106 θ0^2 + θ0^6 (0.000893 + 0.003796 θ0^2)] – [N–FN] / Rq} /
{1.340264 – 0.243318 θ0^2 + θ0^6 [0.006251 + 0.034164 θ0^2]}
θ1 = θ0 – Δθ0

Δθ1 = {θ1 [1.340264 – 0.081106 θ1^2 + θ1^6 (0.000893 + 0.003796 θ1^2)] – [N–FN] / Rq} /
{1.340264 – 0.243318 θ1^2 + θ1^6 [0.006251 + 0.034164 θ1^2]}
θ2 = θ1 – Δθ1

etc.

The calculation is repeated until Δθn is less than a predetermined convergence value. Then, using the final θn+1 as θ, the geodetic latitude and longitude of a point are determined as follows:

lat = ß + {[(e^2/3 + 31e^4/180 + 517e^6/5040) sin(2ß)] + [(23e^4/360 + 251e^6/3780) sin(4ß)]
+ [(761e^6/45360) sin(6ß)]}
lon = lonO + √3 (E–FE) {1.340264 – 0.243318 θ^2 + θ^6 (0.006251 + 0.034164 θ^2)} / {2 Rq cos(θ) }

where ß = asin{ 2 sin(θ) / √3} and Rq is defined as in the forward equations.

Sphere:
For the spherical form of the projection, ß = lat and Rq = R, where R is the radius of the sphere.

Example:

Parameters:
Ellipsoid: WGS 1984 a = 6378137.0 metres 1/f =298.257223563
then e = 0.08181919084262

Longitude of natural origin (lonO) = 90°00'00.00"W = -1.5707963268 rad
False easting (FE) = 0.000 m
False northing (FN) = 0.000 m

Forward calculation for:
Latitude = 34°03'27.169" N = 0.5944163293 rad
Longitude = 117°11'48.349" W = -2.0454693977 rad

First gives
ß = 0.5923399644 Rq = 6371007.181
θ = 0.5046548375

whence
E = -2390749.042 m
N = 4242849.758 m

Reverse calculation for the same Easting and Northing (-2390749.042 E, 4242849.758 N) starts with an iteration to obtain θ:

(N–FN) / Rq = 0.665962169
Step 1: θ0 = 0.665962169 Δθ0 = 0.164312383
Step 2: θ1 = 0.501649786 Δθ1 = -0.003004201
Step 3: θ2 = 0.504653987 Δθ2 = -8.512742e-07
Step 4: θ3 = 0.504654838 Δθ3 = -6.859963e-14
θ = θ4 = 0.504654838

This gives:
ß = 0.592339965

Then Latitude = 34°03'27.169" N
Longitude = 117°11'48.349" W

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
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:

Šavrič et al, International Journal of Geographical Iinformation Science, August 2018.

Data Source:

EPSG

Revision Date:

May 3, 2019

Change ID:

[2018.048] [2019.018]

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.