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 = FE + r sin(theta) N = FN + r0 - 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 t = tan(pi/4 – lat/2)/[(1 – e sin(lat))/(1 + e sin(lat))]^e/2 n = sin(latO) F = mO/(n tO^n) rO = a F tO^n kO r = a F t^n kO 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 +lon0 where n, F, and rO are derived as for the forward calculation r' = +/-[(E - FE)^2 + {r0 - (N - FN)}^2]^0.5 taking the sign of n t' = (r'/(a k0 F))^(1/n) If n is positive, theta' = atan2{(E – FE) , [rO – (N – FN)]} but if n is negative the signs of both arguments of the atan2 function must be reversed and theta' = atan2{– (E – FE) , – [rO – (N – FN)]} 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.

For Projected Coordinate System JAD69 / Jamaica National Grid Parameters: Ellipsoid: Clarke 1866, a = 6378206.400 m., 1/f = 294.97870 then e = 0.08227185 and e^2 = 0.00676866 Latitude Natural Origin 18°00'00"N = 0.31415927 rad Longitude Natural Origin 77°00'00"W = -1.34390352 rad Scale factor at origin 1.000000 False Eastings FE 250000.00 m False Northings FN 150000.00 m Forward calculation for: Latitude: 17°55'55.80"N = 0.31297535 rad Longitude: 76°56'37.26"W = -1.34292061 rad first gives m0 = 0.95136402 t0 = 0.72806411 F = 3.39591092 n = 0.30901699 r = 19643955.26 r0 = 19636447.86 theta = 0.00030374 t = 0.728965259 Then Easting E = 255966.58 m Northing N = 142493.51 m Reverse calculation for the same easting and northing first gives theta' = 0.000303736 t' = 0.728965259 m0 = 0.95136402 r' = 19643955.26 Then Latitude = 17°55'55.800"N Longitude = 76°56'37.260"W