# EPSG:9806

## Cassini-Soldner

### Attributes

Data source: EPSG

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

Revision date: 2017-06-13

### Formula

```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) + .....]

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```

### Example

```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

Forward calculation for:
Latitude       10°00'00.00" N = 0.17453293 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