Lambert Conic Conformal (2SP)

Name Lambert Conic Conformal (2SP)
EPSG Code 9802
GeoTIFF Code CT_LambertConfConic_2SP (9)
CT_LambertConfConic (9)
OGC WKT Name Lambert_Conformal_Conic_2SP
Supported By EPSG, GeoTIFF, PROJ.4, OGC WKT

Projection Parameters

Name EPSG # GeoTIFF ID OGC WKT Units Notes
Latitude of false origin 1 FalseOriginLat latitude_of_origin Angular
Longitude of false origin 2 FalseOriginLong central_meridian Angular
Latitude of first standard parallel 3 StdParallel1 standard_parallel_1 Angular
Latitude of second standard parallel 4 StdParallel2 standard_parallel_2 Angular
Easting of false origin 6 FalseOriginEasting false_easting Linear
Northing of false origin 7 FalseOriginNorthing false_northing Linear

PROJ.4 Organization

  +proj=lcc   +lat_1=Latitude of first standard parallel
              +lat_2=Latitude of second standard parallel
              +lat_0=Latitude of false origin 
              +lon_0=Longitude of false origin
              +x_0=False Origin Easting
              +y_0=False Origin Northing

EPSG Notes

Conical projections with one standard parallel are normally considered to maintain the nominal map scale along the parallel of latitude which is the line of contact between the imagined cone and the ellipsoid. For a one standard parallel Lambert the natural origin of the projected coordinate system is the intersection of the standard parallel with the longitude of origin (central meridian). See Figure 5. To maintain the conformal property the spacing of the parallels is variable and increases with increasing distance from the standard parallel, while the meridians are all straight lines radiating from a point on the prolongation of the ellipsoid's minor axis.

Sometimes however, although a one standard parallel Lambert is normally considered to have unity scale factor on the standard parallel, a scale factor of slightly less than unity is introduced on this parallel. This is a regular feature of the mapping of some former French territories and has the effect of making the scale factor unity on two other parallels either side of the standard parallel. The projection thus, strictly speaking, becomes a Lambert Conic Conformal projection with two standard parallels. From the one standard parallel and its scale factor it is possible to derive the equivalent two standard parallels and then treat the projection as a two standard parallel Lambert conical conformal, but this procedure is seldom adopted. Since the two parallels obtained in this way will generally not have integer values of degrees or degrees and minutes it is instead usually preferred to select two specific parallels on which the scale factor is to be unity, - as for several State Plane Coordinate systems in the United States.

The choice of the two standard parallels will usually be made according to the latitudinal extent of the area which it is wished to map, the parallels usually being chosen so that they each lie a proportion inboard of the north and south margins of the mapped area. Various schemes and formulas have been developed to make this selection such that the maximum scale distortion within the mapped area is minimised, e.g. Kavraisky in 1934, but whatever two standard parallels are adopted the formulas for the projected coordinates are the same.

For territories with limited latitudinal extent but wide longitudinal width it may sometimes be preferred to use a single projection rather than several bands or zones of a Transverse Mercator. If the latitudinal extent is also large there may still be a need to use two or more zones if the scale distortion at the extremities of the one zone becomes too large to be tolerable.

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (*,*) the formulas for the two standard parallel case are:

	Easting, E = EF + r sin *
	Northing, N = NF + rF - r cos *	 

where 	m = cos*/(1 - e2sin2*)1/2     for m1, *1, and m2, *2 where *1 and *2  are the latitudes 
of 			the standard parallels
	t  = tan(*/4 - */2)/[(1 - e sin*)/(1 + e sin*)]e/2   for t1, t2, tF and t using *1,*2,*
F and * 			respectively
	n = (loge m1 - loge m2)/(loge t1 - loge t2)
	F = m1/(nt1n)
	r =  a F tn         for rF and r, where rF is the radius of the parallel of latitude of the 
false origin
	* = n(* - *0)

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

	* = */2 - 2arctan{t'[(1 - esin*)/(1 + esin*)]e/2}
	* = *'/n +*0
	r' = *[(E - EF)2 + {rF - (N - NF)}2]1/2 , taking the sign of n
	t' = (r'/aF)1/n
	*' = arctan [(E- EF)/(rF - (N- NF))]
and n, F, and rF are derived as for the forward calculation.

With minor modifications these formulas can be used for the single standard parallel 
case. Then
	E = FE + r sin*
	N = FN + r0 - r cos*,  using the natural origin rather than the false origin.
	n = sin *0
	r = aFtn k0     	for r0, and r
	t is found for  t0, *0 and t, * and m, F, and * are found as for the two standard 
parallel case
	The reverse formulas for * and * are as for the two standard parallel case above, 
with n, F and r0 as before and

	*' = arctan[(E - FE)/{r0 -(N - FN)}]
	r' = *[(E - FE)2 + {r0 - (N - FN)}2]1/2
	t' = (r'/ak0F)1/n","For Projected Coordinate System NAD27 / Texas South Central

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

First Standard Parallel          28o23'00""N  =   0.49538262 rad
Second Standard Parallel    30o17'00""N  =   0.52854388 rad
Latitude False Origin            27o50'00""N  =   0.48578331 rad
Longitude False Origin         99o00'00""W = -1.72787596 rad
Easting at false origin           2000000.00  US survey feet
Northing at false origin          0.00  US survey feet

Forward calculation for: 
Latitude       28o30'00.00""N  =  0.49741884 rad
Longitude    96o00'00.00""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     	= 28o30'00.000""N
         Longitude   = 96o00'00.000""W