# Cassini-Soldner

Name | Cassini-Soldner
EPSG Code | 9806
GeoTIFF Code | CT_CassiniSoldner (18)
OGC WKT Name | Cassini_Soldner
Supported By | EPSG, GeoTIFF, OGC WKT

### Projection Parameters

Name | EPSG # | GeoTIFF ID | OGC WKT | Units | Notes
--- | --- | --- | --- | --- | ---
Latitude of natural origin | 1 | NatOriginLat | latitude_of_origin | Angular |
Longitude of natural origin | 2 | NatOriginLong | central_meridian | Angular |
False Easting | 6 | FalseEasting | false_easting | Linear |
False Northing | 7 | FalseNorthing | false_northing | Linear |

### Notes

The Intergraph sample files include a non 1.0 scale at the natural origin; however, this isn't currently an official part of the spec. However, if you can handle a scale parameter, you should look for ScaleAtNatOrigin.

### PROJ.4 Organization

      +proj=cass +lat_0=Latitude of natural origin
                 +lon_0=Longitude of natural origin
                 +x_0=False Easting
                 +y_0=False Northing

There does not appear to be any scaling support with PROJ.4's Cassini. The documentation for PROJ.4 just calls the projection Cassini, but does indicate support for ellipical as well as spherical forms so I presume it is the same as Cassini-Soldner.

### EPSG Notes

Cassini-Soldner Formula

The Cassini-Soldner projection is the ellipsoidal version of the Cassini projection for the sphere. It is not conformal but as it is relatively simple to construct it was extensively used in the last century and is still useful for mapping areas with limited longitudinal extent. It has now largely been replaced by the conformal Transverse Mercator which it resembles. Like this, it has a straight central meridian along which the scale is true, all other meridians and parallels are curved, and the scale distortion increases rapidly with increasing distance from the central meridian.

The formulas to derive projected Easting and Northing coordinates are:

    	Easting, E = FE + *[A - TA3/6 -(8 - T + 8C)TA5/120]

    	Northing, N = FN + M - M0 + *tan*[A2/2 + (5 - T + 6C)A4/24]

    where	A = (* - *0)cos*
    	T = tan2*
    	C = e2 cos2*/(1 - e2)
    and M, the distance along the meridian from equator to latitude *, is given by
    	M = a[1 - e2/4 - 3e4/64 - 5e6/256 -....)* - (3e2/8 + 3e4/32 + 45e6/1024 +....)sin2*
    		+ (15e4/256 + 45e6/1024 +.....)sin4* - (35e6/3072 + ....)sin6* + .....]
    with * 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:
    	* = *1 - (*1tan*1/*1)[D2/2 - (1 + 3T1)D4/24]
    	* =  *0 + [D - T1D3/3 + (1 + 3T1)T1D5/15]/cos*1

    where	*1 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:

    	*1 = *1 + (3e1/2 - 27e13/32 +.....)sin2*1 + (21e12/16 - 55e14/32 + ....)sin4*1
    		+ (151e13/96 +.....)sin6*1 + (1097e14/512 - ....)sin8*1 + ......
    where
    	e1 = [1- (1 - e2)1/2]/[1 + (1 - e2)1/2]
    	*1 = M1/[a(1 - e2/4 - 3e4/64 - 5e6/256 - ....)]
    	M1 = M0 + (N - FN)
    	T1 = tan2*1
    	D = (E - FE)/*1","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       10o26'30""N  =  0.182241463 rad
    Longitude Natural Origin    61o20'00""W = -1.07046861 rad
    False Eastings FE              430000.00 links
    False Northings FN            325000.00 links

    Forward calculation for:
    Latitude       10o00'00.00"" N = 0.17453293 rad
    Longitude    62o00'00.00""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     = 10o00'00.000""N
             Longitude  =  62o00'00.000""W