EPSG

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

2018-08-29

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 K F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin and K is the ellipsoid scaling factor. 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 - 2atan{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'/(aKF))^(1/n) theta' = atan2 [(E- EF),(rF - (N- NF))] (see GN7-2 implementation notes in 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.

For Projected Coordinate System NAD27 / Michigan 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 First Standard Parallel = 44°11'00"N = 0.771144641 rad Second Standard Parallel = 45°42'00"N = 0.797615468 rad Latitude False Origin = 43°19'00"N = 0.756018454 rad Longitude False Origin = 84°20'00"W = -1.471894336 rad Easting at false origin = 2000000.00 US survey feet Northing at false origin = 0.00 US survey feet Ellipsoid scaling factor = 1.0000382 Forward calculation for: Latitude = 43°45'00.00"N = 0.763581548 rad Longitude = 83°10'00.00"W = -1.451532161 rad first gives : m1 = 0.718295175 m2 = 0.699629151 t = 0.429057680 tF = 0.433541026 t1 = 0.424588396 t2 = 0.409053868 n = 0.706407410 F = 1.862317735 r = 21436775.51 rF = 21594768.40 theta = 0.014383991 Then Easting X = 2308335.75 US survey feet Northing Y = 160210.48 US survey feet Reverse calculation for same easting and northing first gives: theta' = 0.014383991 r' = 21436775.51 t' = 0.429057680 Then Latitude = 43°45'00.000"N Longitude = 83°10'00.000"W