EPSG

EPSG guidance note #7-2, http://www.epsg.org

2013-05-20

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 forward conversion from latitude and longitude to easting and northing is: E = FE + A * nu * cos(lat) * (lon – lonO) N = FN + G * rhoO * [(lat – latO) + {B *(lon – lonO)^2 * nu^2 * cos^2(lat)}] where latO, lonO and hO are the latitude, longitude and height of the projection origin, rho is the radius of curvature of the ellipsoid in the plane of the meridian at latitude lat, rhoO at latitude latO, and rhoM at mid-latitude (latO + lat)/2; rho = a(1 – e^2)/(1 – e^2 * sin^2(lat))^1.5 for lat, latO and latM nu is the radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat with nuO at latitude latO; nu = a /(1 – e^2 * sin^2(lat))^0.5 for lat and latO e is the eccentricity of the ellipsoid and e^2 = (a^2 – b^2)/a^2 = 2f – f^2 a is the ellipsoidal semi-major and 1/f is the inverse flattening A = 1 + hO/nuO B = tan(latO) / 2 * rhoO * nuO G = 1 + hO/latM The reverse conversion from easting and northing to latitude and longitude is: lat = latO + [(N – FN) / D] – B [(E–FE) / C]^2 lon = lonO + (E – FE) / C *nu * cos(lat) where B is as in the forward conversion, C = 1 + hO/a D = rhoO * (1 + hO/a(1-e^2))

For Projected Coordinate Reference System: MAGNA-SIRGAS / Bogota urban grid Parameters: Ellipsoid: GRS 1980 a = 6378137.0 metres 1/f = 298.2572221 then e^2 = 0.006694380 Latitude of natural origin = 4°40'49.75"N = 0.081689893 rad Longitude of natural origin = 74°08'47.73"W = -1.294102154 rad False easting = 92334.879 metres False northing = 109320.965 metres Height of projection origin = 2550.000 metres Forward calculation for: Latitude = 4°48'00.000"N = 0.083775804 rad Longitude = 74°15'00.000"W = -1.295906970 rad First calculate constants for the projection: rho_O = 6335862.944 nu_O = 6378279.154 A = 1.000399794 B = 1.012970E-15 C = 1.000399803 D = 6338413.114 Then nu = 6378286.489 G = 1.000402470 and then E = 80859.033m N = 122543.174m Reverse calculation for the same Easting and Northing (80859.033E, 122543.174N) first gives: Latitude = 4°48'00.000"N then nu = 6378286.489 and Longitude = 74°15'00.000"W