EPSG guidance note #7-2, http://www.epsg.org
2017-06-13
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For Projected Coordinate System Trinidad 1903 / Trinidad Grid
Parameters:
Ellipsoid Clarke 1858 a = 20926348 ft = 31706587.88 links
b = 20855233 ft
then 1/f = 294.97870 and e^2 = 0.00676866
Latitude Natural Origin 10°26'30"N = 0.182241463 rad
Longitude Natural Origin 61°20'00"W = -1.07046861 rad
False Eastings FE 430000.00 links
False Northings FN 325000.00 links
Forward calculation for:
Latitude 10°00'00.00" N = 0.17453293 rad
Longitude 62°00'00.00"W = -1.08210414 rad
A = -0.01145876 C = 0.00662550
T = 0.03109120 M = 5496860.24 nu = 31709831.92 M0 = 5739691.12
Then Easting E = 66644.94 links
Northing N = 82536.22 links
Reverse calculation for same easting and northing first gives :
e1 = 0.00170207 D = -0.01145875
T1 = 0.03109544 M1 = 5497227.34
nu1 = 31709832.34 mu1 = 0.17367306
phi1 = 0.17454458 rho1 = 31501122.40
Then Latitude = 10°00'00.000"N
Longitude = 62°00'00.000"W
urn:ogc:def:method:EPSG::9806
Cassini-Soldner
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 formulas to derive projected Easting and Northing coordinates are:
Easting E = FE + nu[A - TA^3/6 -(8 - T + 8C)TA^5/120]
Northing N = FN + M - M0 + nu*tan(lat)*[A^2/2 + (5 - T + 6C)A^4/24]
where A = (lon - lon0)cos(lat)
T = tan^2(lat)
C = e2 cos2*/(1 - e2) nu = a /(1 - esq*sin^2(lat))^0.5
and M, the distance along the meridian from equator to latitude lat, is given by
M = a[(1 - e^2/4 - 3e^4/64 - 5e^6/256 -....)*lat - (3e^2/8 + 3e^4/32 + 45e^6/1024 +....)sin(2*lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4*lat) - (35e^6/3072 + ....)sin(6*lat) + .....]
with lat in radians.
M0 is the value of M calculated for the latitude of the chosen origin. This may not necessarily be chosen as the equator.
To compute latitude and longitude from Easting and Northing the reverse formulas are:
lat = lat1 - (nu1tan(lat1)/rho1)[D2/2 - (1 + 3*T1)D^4/24]
lon = lon0 + [D - T1*D^3/3 + (1 + 3*T1)T1*D^5/15]/cos(lat1)
where lat1 is the latitude of the point on the central meridian which has the same Northing as the point whose coordinates are sought, and is found from:
lat1 = mu1 + (3*e1/2 - 27*e1^3/32 +.....)sin(2*mu1) + (21*e1^2/16 - 55*e1^4/32 + ....)sin(4*mu1)+ (151*e1^3/96 +.....)sin(6*mu1) + (1097*e1^4/512 - ....)sin(8*mu1) + ......
where
e1 = [1- (1 - esq)^0.5]/[1 + (1 - esq)^0.5]
mu1 = M1/[a(1 - esq/4 - 3e^4/64 - 5e^6/256 - ....)]
M1 = M0 + (N - FN)
T1 = tan^2(lat1)
D = (E - FE)/nu1