EPSG Dataset :: v12.003

Cassini-Soldner

Coordinate Operation Method Details [VALID]

Name:

Cassini-Soldner

Code:

9806

Operation is Reversible:

Yes

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) + .....]
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

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
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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of natural origin	8801	No	The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
Longitude of natural origin	8802	No	The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the longitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0). Sometimes known as "central meridian (CM)".
False easting	8806	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
False northing	8807	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.

Meta Data

Information Source:

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

Data Source:

EPSG

Revision Date:

June 13, 2017

Change ID:

[2017.018]

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.