# EPSG:9820

## Lambert Azimuthal Equal Area

### Attributes

Data source: EPSG

Information source: USGS Professional Paper 1395, "Map Projections - A Working Manual" by John P. Snyder.

Revision date: 2018-08-29

Remarks: This is the ellipsoidal form of the projection.

### 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.

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)```

### Example

```For Projected Coordinate Reference System: ETRS89 / ETRS-LAEA

Parameters:
Ellipsoid:GRS 1980  a = 6378137.0 metres    1/f = 298.2572221
then e = 0.081819191

Latitude of natural origin (latO): 52°00'00.000"N = 0.907571211  rad
Longitude of natural origin (lonO): 10°00'00.000"E = 0.174532925  rad
False easting (FE): 4321000.00 metres
False northing (FN) 3210000.00 metres

Forward calculation for:
Latitude (lat) =  50°00'00.000"N = 0.872664626 rad
Longitude(lon) = 5°00'00.000"E = 0.087266463 rad

First gives
qP = 1.995531087
qO = 1.569825704
q = 1.525832247
Rq = 6371007.181
betaO = 0.905397517
beta = 0.870458708
D = 1.000425395
B = 6374393.455

whence
E = 3962799.45  m
N = 2999718.85  m

Reverse calculation for the same Easting and Northing (3962799.45 E, 2999718.85  N) first gives:

rho = 415276.208
C = 0.065193736
beta' = 0.870458708

Then Latitude = 50°00'00.000"N
Longitude = 5°00'00.000"E```