EPSG Dataset :: v12.003

Lambert Azimuthal Equal Area

Coordinate Operation Method Details [VALID]

Name:

Lambert Azimuthal Equal Area

Code:

9820

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.

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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of natural origin	8801	No	The latitude 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 latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
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

Remarks:

This is the ellipsoidal form of the projection.

Information Source:

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

Data Source:

EPSG

Revision Date:

June 29, 2023

Change ID:

[2003.35] [2004.43] [2005.075] [2005.29] [2006.2] [2007.005] [2017.018] [2017.024] [2023.004]

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.