EPSG Dataset :: v12.053

Lambert Conic Conformal (1SP)

Coordinate Operation Method Details [VALID]

Name:

Lambert Conic Conformal (1SP)

Code:

9801

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.

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon):

E = FE + r sin(theta)
N = FN + r0 - r cos(theta)
where
mO = cos(latO)/(1 – e^2 sin^2(latO))^0.5 where latO is the latitude of natural origin
tO = tan(pi/4 – latO/2)/[(1 – e sin(latO))/(1 + e sin(latO))]^e/2
t = tan(pi/4 – lat/2)/[(1 – e sin(lat))/(1 + e sin(lat))]^e/2
n = sin(latO)
F = mO/(n tO^n)
rO = a F tO^n kO
r = a F t^n kO
theta = n(lon – lonO)
As with other conics, a negative n and r result for projections centered in the Southern Hemisphere.

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 - e sin(lat))/(1 + e sin(lat))]^(e/2)}
lon = theta'/n +lon0
where
n, F, and rO are derived as for the forward calculation
r' = +/-[(E - FE)^2 + {r0 - (N - FN)}^2]^0.5 taking the sign of n
t' = (r'/(a k0 F))^(1/n)
If n is positive, theta' = atan2{(E – FE) , [rO – (N – FN)]}
but if n is negative the signs of both arguments of the atan2 function must be reversed and theta' = atan2{– (E – FE) , – [rO – (N – FN)]}

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 JAD69 / Jamaica National Grid

Parameters:
Ellipsoid: Clarke 1866, a = 6378206.400 m., 1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866

Latitude Natural Origin 18°00'00"N = 0.31415927 rad
Longitude Natural Origin 77°00'00"W = -1.34390352 rad
Scale factor at origin 1.000000
False Eastings FE 250000.00 m
False Northings FN 150000.00 m

Forward calculation for:
Latitude: 17°55'55.80"N = 0.31297535 rad
Longitude: 76°56'37.26"W = -1.34292061 rad
first gives
m0 = 0.95136402 t0 = 0.72806411
F = 3.39591092 n = 0.30901699
r = 19643955.26 r0 = 19636447.86
theta = 0.00030374 t = 0.728965259

Then Easting E = 255966.58 m
Northing N = 142493.51 m

Reverse calculation for the same easting and northing first gives

theta' = 0.000303736
t' = 0.728965259
m0 = 0.95136402
r' = 19643955.26

Then Latitude = 17°55'55.800"N
Longitude = 76°56'37.260"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)".
Scale factor at natural origin	8805	No	The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the natural origin.
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, https://epsg.org

Data Source:

EPSG

Revision Date:

January 13, 2021

Change ID:

[2001.08] [2017.018] [2017.024] [2020.006] [2020.108]

Alias:

Alias	Naming System	Remarks
Lambert Conic Conformal (1SP variant A)	EPSG alias

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.