EPSG:9820

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.

Oblique aspect
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 and Northing are:

Easting, E   = FE + {(B . D) . [cos ß . sin(lon – lonO)]}
Northing, N = FN + (B / D) . {(cos ßO . sin ß) –  [sin ßO . cos ß . cos(lon – lonO)]}

where
B = Rq . (2 / {1 + sin ßO . sin ß + [cos ßO . cos ß . cos(lon – lonO)]})^0.5
D = a . [cos latO / (1 – e2 sin2 latO)^0.5] / (Rq . cos ßO)
Rq = a . (qP  / 2)^0.5
ß = asin (q / qP)
ßO = asin (qO / qP)
q = (1 – e^2) . ([sin(lat) / (1 – e^2 sin^2(lat))] – {[1/(2e)] . ln [(1 – e sin(lat)) / (1 + e sin(lat))]})
qO = (1 – e^2) . ([sin(latO) / (1 – e^2 sin^2(latO))] – {[1/(2e)] . ln [(1 – e sin(latO)) / (1 + e sin(latO))]})
qP = (1 – e^2) . ([sin(latP) / (1 – e^2 sin^2(latP))] – {[1/(2e)] . ln [(1 – e sin(latP)) / (1 + e sin(latP))]})
where *P = p/2 radians, thus
qP = (1 – e^2) . ([1 / (1 – e^2)] – {[1/(2e)] . ln [(1 – e) / (1 + e)]})

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 + atan2 {(E-FE) . sin C , [D. rho . cos ßO . cos C – D^2. (N-FN) . sin ßO . sin C]}  (see implementation notes in GN7-2 preface for atan2 convention)
where
ß' = asin{(cosC . sin ßO) + [(D . (N-FN) . sinC . cos ßO) / rho]}
C = 2 . asin(rho / 2 . Rq)
rho = {[(E-FE)/D]^2 + [D . (N –FN)]^2}^0.5

and D, Rq, and ßO are as in the forward equations.

Polar aspect
For the polar aspect of the Lambert Azimuthal Equal Area projection, some of the above equations are indeterminate. Instead, for the forward case from latitude and longitude (lat, lon) to Easting (E) and Northing (N):

For the north polar case:
	Easting, E   = FE + [rho  sin(lon – lonO)]
	Northing, N = FN –  [rho  cos(lon – lonO)]
where
rho = a (qP  – q)^0.5
and qP  and q are found as for the general case above.

For the south polar case:
	Easting, E   = FE + [rho . sin(lon – lonO)]
	Northing, N = FN +  [rho . cos(lon – lonO)]
where
rho = a (qP  + q)^0.5
and qP  and q are found as for the general case above.

For the reverse formulas to derive the geodetic latitude and longitude of a point from its Easting and Northing:
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']
as for the oblique case, but where
ß' = ±asin [1– rho^2 / (a^2{1– [(1– e^2)/2e)) ln[(1-e)/(1+ e)]})], taking the sign of  latO
and rho = {[(E –FE)]^2 + [(N – FN)]^2}^0.5
Then
lon = lonO + atan2 [(E – FE)] , (N – FN)] for the south pole case
and
lon = lonO + atan2 [(E – FE)] , –(N – FN)] for the north pole case.
(see implementation notes in GN7-2 preface for atan2 convention)