EPSG

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

2021-12-30

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) the formulas for the one standard parallel case are: E = EF + r sin(theta) N = NF + rF - r cos(theta) where m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the two standard parallels. t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively. n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2)) F = m1/(n t1^n) r = a F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin. theta = n(lon - lon0) 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 - esin(lat))/(1 + esin(lat))]^(e/2)} lon = theta'/n +lon0 where r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n t' = (r'/(aF))^(1/n) theta' = atan2 [(E- EF),(rF - (N- NF))] (see implementation notes in GN7-2 preface for atan2 convention) and n, F, and rF are derived as for the forward calculation. 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.

Example 1 (Northern Hemisphere) For Projected Coordinate System NAD27 / Texas South Central Parameters: Ellipsoid Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet 1/f = 294.97870 then e = 0.08227185 and e^2 = 0.00676866 Latitude False Origin = 27°50'00"N = 0.48578331 rad Longitude False Origin = 99°00'00"W = -1.72787596 rad First Standard Parallel = 28°23'00"N = 0.49538262 rad Second Standard Parallel = 30°17'00"N = 0.52854388 rad Easting at false origin = 2000000.00 US survey feet Northing at false origin = 0.00 US survey feet Forward calculation for: Latitude 28°30'00.00"N = 0.49741884 rad Longitude 96°00'00.00"W = -1.67551608 rad first gives : m1 = 0.88046050 m2 = 0.86428642 t = 0.59686306 tF = 0.60475101 t1 = 0.59823957 t2 = 0.57602212 n = 0.48991263 F = 2.31154807 r = 37565039.86 rF = 37807441.20 theta = 0.02565177 Then Easting E = 2963503.91 US survey feet Northing N = 254759.80 US survey feet Reverse calculation for same easting and northing first gives: theta' = 0.025651765 r' = 37565039.86 t' = 0.59686306 Then Latitude = 28°30'00.000"N Longitude = 96°00'00.000"W Example 2 (Southern Hemisphere): For Projected Coordinate Reference System: AGD66 / Vicgrid66 (EPSG CRS code 3110) Parameters: Ellipsoid: Australian National Spheroid a = 6378160.0 metres 1/f = 298.25 then: e = 0.081820179996 and e^2 = 0.00669454185 Latitude of false origin = 37°00'00.000"S = -0.64577182 rad Longitude of false origin = 145°00'00.000"E = 2.530727415 rad Latitude of 1st standard parallel = 36°00'00.000"S = -0.62831853 rad Latitude of 2nd standard parallel = 38°00'00.000"S = -0.66322512 rad Easting at false origin = 2500000.000 metres Northing at false origin = 4500000.000 metres Forward calculation for: Latitude = 37°45'00.000"S = -0.658861793 rad Longitude = 144°45'00.000"E = 2.5438173848 rad first gives : m1 = 0.809954211 m2 = 0.789012446 t = 2.030654821 tF = 1.997618776 t1 = 1.954896966 t2 = 2.0418636298 n = -0.6018461050 F = -2.0145905003 r = -8389432.783 rF = -8472661.322 theta = 0.002626049 Then Easting E = 2477968.963 metres Northing N = 4416742.535 metres Reverse calculation for same easting and northing first gives: (Since n is negative, we reverse the signs of both atan2 arguments to calculate theta') theta' = 0.002626049 t' = 2.030654821 r' = -8389432.783 Then Latitude = 37°45'00.000"S Longitude = 144°45'00.000"E