EPSG

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

2018-08-29

This is the ellipsoidal form of the projection.

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)

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