Lambert Conic Conformal (2SP)

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) the formulas for the one standard parallel case are:

E = EF + r sin(theta)
N = NF + rF - r cos(theta)
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)

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'/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)
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

Example 1 (Northern Hemisphere) 
For Projected Coordinate System NAD27 / Texas South Central

Parameters:
Ellipsoid  Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet
                      1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866

Latitude False Origin     = 27°50'00"N = 0.48578331 rad
Longitude False Origin    = 99°00'00"W = -1.72787596 rad
First Standard Parallel   = 28°23'00"N = 0.49538262 rad
Second Standard Parallel  = 30°17'00"N = 0.52854388 rad
Easting at false origin   = 2000000.00 US survey feet
Northing at false origin  =       0.00 US survey feet

Forward calculation for: 
Latitude     28°30'00.00"N =  0.49741884 rad
Longitude    96°00'00.00"W = -1.67551608 rad

first gives :
m1 = 0.88046050
m2 = 0.86428642
t = 0.59686306
tF = 0.60475101
t1 = 0.59823957
t2 = 0.57602212
n = 0.48991263
F = 2.31154807
r = 37565039.86
rF = 37807441.20
theta = 0.02565177

Then
 Easting E = 2963503.91 US survey feet
Northing N =  254759.80 US survey feet

Reverse calculation for same easting and northing first gives:
theta' = 0.025651765     
r'     = 37565039.86
t'     = 0.59686306

Then 
 Latitude = 28°30'00.000"N
Longitude = 96°00'00.000"W

Example 2 (Southern Hemisphere):

For Projected Coordinate Reference System: AGD66 / Vicgrid66 (EPSG CRS code 3110)

Parameters:
Ellipsoid: Australian National Spheroid  a = 6378160.0 metres  1/f = 298.25
then:
e = 0.081820179996 and e^2 = 0.00669454185
			
Latitude of false origin = 37°00'00.000"S = -0.64577182 rad
Longitude of false origin = 145°00'00.000"E = 2.530727415 rad
Latitude of 1st standard parallel = 36°00'00.000"S = -0.62831853 rad
Latitude of 2nd standard parallel = 38°00'00.000"S = -0.66322512 rad
Easting at false origin	= 2500000.000 metres
Northing at false origin = 4500000.000 metres

Forward calculation for: 
Latitude  =  37°45'00.000"S = -0.658861793 rad
Longitude = 144°45'00.000"E = 2.5438173848 rad

first gives :
m1    = 0.809954211
m2    = 0.789012446
t     = 2.030654821
tF    = 1.997618776
t1    = 1.954896966
t2    = 2.0418636298
n     = -0.6018461050
F     = -2.0145905003
r     = -8389432.783
rF    =	-8472661.322
theta = 0.002626049				

Then	
 Easting E = 2477968.963 metres
Northing N = 4416742.535 metres

Reverse calculation for same easting and northing first gives:
(Since n is negative, we reverse the signs of both atan2 arguments to calculate theta')
theta' = 0.002626049
t'     = 2.030654821
r'     = -8389432.783

Then	
 Latitude =  37°45'00.000"S
Longitude = 144°45'00.000"E

EPSG:9802

Lambert Conic Conformal (2SP)

Attributes

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Formula

Example