EPSG

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

2021-01-13

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. To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon): E = Ef + r sin(theta) N = Nf + rf - r cos(theta) where mO = cos(latO)/(1 – e^2 sin^2(latO))^0.5 where latO is the latitude of natural origin tO = tan(pi/4 – latO/2)/[(1 – e sin(latO))/(1 + e sin(latO))]^e/2 tf = tan(pi/4 – latF/2)/[(1 – e sin(latF))/(1 + e sin(latF))]^e/2 t = tan(pi/4 – lat/2)/[(1 – e sin(lat))/(1 + e sin(lat))]^e/2 n = sin(latO) F = mO/(n tO^n) rf = a F tf^n kO r = a F t^n kO lonF = lonO theta = n(lon – lonO) As with other conics, a negative n and r result for projections centered in the Southern Hemisphere. The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are: lat = pi/2 - 2arctan{t'[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2)} lon = theta'/n +lonO where n, F, rf and lonO are derived as for the forward calculation r' = +/-[(E - Ef)^2 + {rf - (N - Nf)}^2]^0.5 taking the sign of n t' = (r'/(a k0 F))^(1/n) If n is positive, theta' = atan2{(E – Ef) , [rf – (N – Nf)]} but if n is negative the signs of both arguments of the atan2 function must be reversed and theta' = atan2{– (E – Ef) , – [rf – (N – Nf)]} Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat using lat = π/2-2atan(t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.

Parameters: Ellipsoid: GRS 1980, a = 6378137.000 m., 1/f = 298.2572221 then e = 0.081819191 and e^2 = 0.006694380 Latitude of natural origin LatO 44°22'45"N = 0.774562578 rad Scale factor at natural origin ko 1.000000 Latitude of false origin LatF 45°11'00"N = 0.788597934 rad Longitude of false origin LongF 6°49'00"E = 0.118973277 rad Easting at false origin Ef 150000.00 m Northing at false origin Nf 50000.00 m Forward calculation for: Latitude: 47°00'00.000"N = 0.820304748 rad Longitude: 7°00'00.000"E = 0.12217304 rad first gives mO = 0.715900163 tO = 0.422551185 tf = 0.414305398 n = 0.699403505 F = 1.869760448 rf = 6439208.575 t = 0.395846092 r = 6237180.887 theta = 0.002237931 Then Easting E = 163958.366 m Northing N = 252043.307 m Reverse calculation for the same easting and northing first gives r' = 6237180.887 t' = 0.395846092 theta' = 0.002237931 Then Latitude = 47°00'00.000"N Longitude = 7°00'00.000"E