EPSG:1078

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.