EPSG Dataset :: v12.054

Lambert Conic Near-Conformal

Coordinate Operation Method Details [VALID]

Name:

Lambert Conic Near-Conformal

Code:

9817

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 compute the Lambert Conic Near-Conformal the following formulae are used. First compute constants for the projection:

n = f / (2-f)
A = 1 / (6 rhoO nuO)
A’ = a [ 1- n + 5 (n^2 - n^3 ) / 4 + 81 ( n^4 - n^5 ) / 64]*pi /180
B’ = 3 a [ n - n^2 + 7 ( n^3 - n^4 ) / 8 + 55 n^5 / 64] / 2
C’ = 15 a [ n^2 -n^3 + 3 ( n^4 - n^5 ) / 4 ] / 16
D’ = 35 a [ n^3 - n^4 + 11 n^5 / 16 ] / 48
E’ = 315 a [ n^4 - n^5 ] / 512
r0 = ko nu0 / tan(lat0)
s0 = A’ latO - B’ sin(2 latO) + C’ sin(4 latO) - D’ sin(6 latO) + E’ sin(8 latO) where in the first term latO is in degrees, in the other terms latO is in radians.

Then for the computation of easting and northing from latitude and longitude:

s = A’ lat - B’ sin(2 lat) + C’ sin(4 lat) - D’ sin(6 lat) + E’ sin(8 lat) where in the first term latO is in degrees, in the other terms latO is in radians.
M = s - sO
M = ko ( m + A m^3)
r = rO - M
theta = (lon - lonO) sin(latO)

and
E = FE + r sin(theta)
N = FN + M + r sin(theta) tan(theta/2)

The reverse formulas for latitude and longitude from Easting and Northing are:

theta' = atan2 {(E – FE) , [rO – (N – FN)]} (see GN7-2 implementation notes in preface for atan2 convention)
r' = +/- {(E – FE)^2 + [rO – (N – FN)]}^2}^0.5, taking the sign of latO
M' = rO – r'

If an exact solution is required, it is necessary to solve for m and lat using iteration of the two equations:
m'= m' – [M' – ko m' – ko A (m')^3] / [– ko – 3 ko A (m')^2]
using M' for m' in the first iteration. This will usually converge (to within 1mm) in a single iteration. Then
lat' = lat' +{m' + sO – [A' lat' (180/pi) – B' sin(2 lat') + C' sin(4 lat') – D' sin(6lat') + E' sin(8 lat')]}/A' (pi/180)
first using lat' = latO + m'/A' (pi/180).

However the following non-iterative solution is accurate to better than 0.001" (3mm) within 5 degrees latitude of the projection origin and should suffice for most purposes:
m' = M' – [M' ko M' – ko A (M')^3] / [– ko – 3 ko A (M')^2]
lat' = latO + m'/A' (pi/180)
s' = A ' lat' – B' sin(2 lat') + C' sin(4 lat') – D' sin(6 lat') + E' sin(8 lat')
where in the first term lat' is in degrees, in the other terms lat' is in radians.
Ds' = A'(180 / pi) – 2B' cos(2 lat') + 4C' cos(4 lat') – 6D' cos(6 lat') + 8E' cos(8 lat')
lat = lat' – [(m' + sO – s') / (–ds')] radians

Then after solution of lat using either method above
lon = lonO + theta' / sin(latO) where lonO and lon are in radians.

Example:

For Projected Coordinate System: Deir ez Zor / Levant Zone

Parameters:
Ellipsoid Clarke 1880 (IGN) a = 6378249.2 m 1/f = 293.46602
then b = 6356515.000 n = 0.001706682563

Latitude Natural Origin = 34°39'00"N = 0.604756586 rad
Longitude Natural Origin = 37°21'00"E= 0.651880476 rad
Scale factor at origin ko = 0.99962560
False Eastings FE = 300000.00 m
False Northings FN = 300000.00 m

Forward calculation for:
Latitude of 37°31'17.625"N = 0.654874806 rad
Longitude of 34°08'11.291"E = 0.595793792 rad
first gives
A = 4.1067494 * 10e-15 A’=111131.8633
B’= 16300.64407 C’= 17.38751 D’= 0.02308 E’= 0.000033
so = 3835482.233 s = 4154101.458 m = 318619.225
M = 318632.72 Ms = 30.82262319
q = -0.03188875 ro = 9235264.405 r = 8916631.685

Then Easting E = 15707.96 m (c.f. E = 15708.00 using full formulae)
Northing N = 623165.96 m (c.f. N = 623167.20 using full formulae)

Reverse calculation for the same easting and northing first gives

q' = -0.03188875
r’ = 8916631.685
M’= 318632.72

Latitude = 0.654874806 rad = 37°31'17.625"N
Longitude = 0.595793792 rad = 34°08'11.291"E

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

Remarks:

The Lambert Near-Conformal projection is derived from the Lambert Conformal Conic projection by truncating the series expansion of the projection formulae.

Information Source:

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

Data Source:

EPSG

Revision Date:

August 29, 2018

Change ID:

[1999.811] [2004.61] [2005.39] [2008.029] [2010.024] [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.