EPSG Dataset :: v12.003

Lambert Conic Conformal (2SP Belgium)

Coordinate Operation Method Details [VALID]

Name:

Lambert Conic Conformal (2SP Belgium)

Code:

9803

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.

For the Lambert Conic Conformal (2 SP Belgium), the formulas for the regular two standard parallel case (coordinate operation method code 9802) are used except for easting, northing in the forward formula and lon in the rverse formula.

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:

Easting, E = EF + r sin (theta - alpha)
Northing, N = NF + rF - r cos (theta - alpha)
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 F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin.
theta = n(lon - lon0)
alpha = 29.2985 seconds.

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = ((theta' + alpha)/n) +lon0
where
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n
t' = (r'/(aF))^(1/n)
theta' = atan2 [(E- EF),(rF - (N- NF))]
(see implementation notes in GN7-2 preface for atan2 convention)
alpha = 29.2985 seconds
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.

Example:

For Projected Coordinate System Belge 1972 / Belge Lambert 72

Parameters:
Ellipsoid International 1924, a = 6378388 metres
1/f = 297
then e = 0.08199189 and e^2 = 0.006722670

First Standard Parallel 49°50'00"N = 0.86975574 rad
Second Standard Parallel 51°10'00"N = 0.89302680 rad
Latitude False Origin 90°00'00"N = 1.57079633 rad
Longitude False Origin 4°21'24.983"E = 0.07604294 rad
Easting at false origin EF 150000.01 metres
Northing at false origin NF 5400088.44 metres

Forward calculation for:
Latitude 50°40'46.461"N = 0.88452540 rad
Longitude 5°48'26.533"E = 0.10135773 rad

first gives :
m1 = 0.64628304 m2 = 0.62834001
t = 0.59686306 tF = 0.00000000
t1 = 0.36750382 t2 = 0.35433583
n = 0.77164219 F = 1.81329763
r = 37565039.86 rF = 0.00
alpha = 0.00014204 theta = 0.01953396

Then Easting E = 251763.20 metres
Northing N = 153034.13 metres

Reverse calculation for same easting and northing first gives:
theta' = 0.01939192 r' = 548041.03
t' = 0.35913403
Then Latitude = 50°40'46.461"N
Longitude = 5°48'26.533"E

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of false origin	8821	No	The latitude of the point which is not the natural origin and at which grid coordinate values false easting and false northing are defined.
Longitude of false origin	8822	No	The longitude of the point which is not the natural origin and at which grid coordinate values false easting and false northing are defined.
Latitude of 1st standard parallel	8823	No	For a conic projection with two standard parallels, this is the latitude of one of the parallels of intersection of the cone with the ellipsoid. It is normally but not necessarily that nearest to the pole. Scale is true along this parallel.
Latitude of 2nd standard parallel	8824	No	For a conic projection with two standard parallels, this is the latitude of one of the parallels at which the cone intersects with the ellipsoid. It is normally but not necessarily that nearest to the equator. Scale is true along this parallel.
Easting at false origin	8826	No	The easting value assigned to the false origin.
Northing at false origin	8827	No	The northing value assigned to the false origin.

Meta Data

Remarks:

In 2000 this modification was replaced through use of the regular Lambert Conic Conformal (2SP) method [9802] with appropriately modified parameter values.

Information Source:

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

Data Source:

EPSG

Revision Date:

August 29, 2018

Change ID:

[1999.28] [2017.018] [2017.024]

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.