Albers Equal Area

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 convert geodetic latitude and longitude (lat, lon) to Easting (E) and Northing (N) are: 
Easting (E)     =  EF + (rho . sin(theta)) 
Northing (N)  =  NF + rhoO – (rho . cos(theta))

where
theta  = n . (lon - lonO)
rho  = [a . (C – n.alpha)^0.5] / n
rhoO = [a . (C – n.alphaO)^0.5] / n
and
C  = m1^2 +  (n . alpha1)
n   = (m1^2 – m2^2) / (alpha2 - alpha1)
m1 = cos lat1 / (1 – e^2 sin^2(lat1))^0.5
m2 = cos lat2 / (1 – e^2 sin^2(lat2))^0.5
alpha  = (1 – e^2) . {[sin(lat) / (1 – e^2 sin^2(lat))] – [1/(2e)] . ln [(1 – e sin(lat)) / (1 + e sin(lat))]}
alphaO  = (1 – e^2) . {[sin(latO) / (1 – e^2 sin^2(latO))] – [1/(2e)] . ln [(1 – e sin(latO)) / (1 + e sin(latO))]}
alpha1  = (1 – e^2) . {[sin(lat1) / (1 – e^2 sin^2(lat1))] – [1/(2e)] . ln [(1 – e sin(lat1)) / (1 + e sin(lat1))]}
alpha2  = (1 – e^2) . {[sin(lat2) / (1 – e^2 sin^2(lat2))] – [1/(2e)] . ln [(1 – e sin(lat2)) / (1 + e sin(lat2))]}

The reverse formulas to derive the geodetic latitude and longitude of a point from its Easting and Northing values are:
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 + (theta / n)

where
C, n and rhoO are as in the forward equations, ß' is the authalic latitude, and:

ß' =  asin(alpha' / {1 – [(1 – e^2) / 2e] . ln [(1 – e) / (1 + e)]})
alpha' =  [C – (rho'^2 . N^2 / a^2)] / n
rho' =  {(E – EF)^2 + [rhoO – (N – NF)]^2 }^0.5
If n is positive, 
theta =  atan2 {(E – EF) , [rhoO – (N – NF)]}
but if n is negative, the signs of both arguments of the atan2 function  must be reversed: 
theta =  atan2 {– (E – EF), – [rhoO – (N – NF)]}
(see implementation notes in GN7-2 preface for atan2 convention).

Example

Example 1 (Northern Hemisphere):
For Projected Coordinate Reference System: NAD83 / Great Lakes Albers (EPSG CRS code 3174)

Parameters:
Ellipsoid: GRS 1980  a = 6378137.00 metres  1/f =  298.2572221
then e = 0.081819191 and e^2 = 0.00669438
			
Latitude of false origin = 45°34'08.3172"N = 0.48578331 rad
Longitude of false origin = 84°27'21.4380"W = -1.72787596 rad
Latitude of 1st standard parallel = 42°07'21.9864"N = 0.49538262 rad
Latitude of 2nd standard parallel = 49°00'54.6480"N = 0.52854388 rad
Easting at false origin	= 1000000.000 metre
Northing at false origin = 1000000.000 metre

Forward calculation for: 
 Latitude = 42°45'00.000"N = 0.746128255 rad
Longitude = 78°45'00.000"W = -1.374446786 rad

first gives :
alpha  = 1.351293970
alpha0 = 1.4218650778
alpha1 = 1.3351453325
alpha2 = 1.5034868510
m1     = 0.7428286888
m2     = 0.6571136237
n      = 0.7128137342
C      = 1.5035043911
rho    = 6577011.9827
rho0   = 6263350.4332
theta  = 0.0709874815				

Then	
 Easting = 1466493.492 metre
Northing =  702903.006 metre

Reverse calculation for same easting and northing first gives:
theta'  = 0.0709874815
alpha'  = 1.3512939695
rho'    = 6577011.983
beta'   = 0.7438962839

Then	
 Latitude = 42°45'00.000"N
Longitude = 78°45'00.000"W

Example 2 (Southern Hemisphere):

Parameters:
Ellipsoid: GRS 1967 Modified  a = 6378160.0 metres  1/f =  298.25
then e =  0.081820180 and e^2 = 0.006694542
			
Latitude of false origin = 32°00'00.000"S = -0.5585053606 rad
Longitude of false origin = 60°00'00.000"W = -1.0471975512 rad
Latitude of 1st standard parallel =  5°00'00.000"S = -0.0872664626 rad
Latitude of 2nd standard parallel = 42°00'00.000"S = -0.7330382858 rad
Easting at false origin	= 0.000 metre
Northing at false origin = 0.000 metre

Forward calculation for: 
 Latitude = 18°30'02.016"S = -0.322895686 rad
Longitude = 46°00'01.538"W = -0.802858912 rad

first gives :
alpha  = -0.630662757
alpha0 = -1.054065016
alpha1 = -0.173150420
alpha2 = -1.331965641
m1     = 0.9962200286
m2     = 0.7442610814
n      = -0.378429434
C      = 1.0579795608
rho    = -15255863.89
rho0   = -13683051.84
theta  = -0.092464933				

Then	
 Easting = 1408623.196 metre
Northing = 1507641.482 metre

Reverse calculation for same easting and northing first gives:
(Since n is negative, we reverse the signs of both atan2 arguments to calculate theta')
theta' = -0.092464933
alpha' = -0.630662757
rho'   = 15255863.888
beta'  = -0.321550075

Then	
 Latitude = 18°30'02.016"S
Longitude = 46°00'01.538"W

EPSG:9822

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