Previous Section	Projection Formula Table of Contents	Next Section

3.4.4.1 Lambert Conic Conformal

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 3-2 below. 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 minimized, 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 (j,l) the formulas for the two standard parallel case are:

Easting:

Northing:

where , for m₁, j₁, and m₂, j₂, where j₁ and j₂ are the latitudes of the standard parallels.

, for t₁, t₂, t_F and t using j₁, j₂, j_F and j respectively.

, for r_F and r, where r_F is the radius of the parallel of latitude of the false origin.

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

where , taking the sign of n

and n, F, and r_F are derived as for the forward calculation.

With minor modifications these formulas can be used for the single standard parallel case. Then

using the natural origin rather than the false origin.

where

for r₀ and r.

t is found for t₀, j₀ and t, j and m, F, and q are found as for the two standard parallel case.

The reverse formulas for j and l are as for the two standard parallel case above, with n, F, and r₀ as before and

Since 1972 a modified form of the two standard parallel case has been used in Belgium. For the Lambert Conic Conformal (2SP Belgium), the formulas for the two standard parallel case given above are used except for:

Easting,

Northing,

and for the reverse formulas:

where a = 29.2985 seconds.

Example:

1. Lambert Conic Conformal (2SP)

For Projected Coordinate System NAD27 / Texas South Cen.

Parameters:

Note: Since the false easting and false northing are given in US Survey Feet (ftUS), and since the final easting and northing are required in US Survey Feet, it is necessary to perform the calculations in these units. The ellipsoid semimajor axis is given in metres. Either the calculations using this value must be converted to US Survey Feet, or else the original semimajor axis value can be given converted to US Survey Feet.

Forward calculation for:

first gives

Then:

Reverse the calculations. Use the same parameters and use the E and N values to calculate the latitude and longitude:

Then

2. Lambert Conic Conformal (1SP)

For Projected Coordinate System JAD69 / Jamaica National Grid

Parameters:

Forward calculation for:

first gives

Then:

Reverse the calculations. Use the same parameters and use the E and N values to calculate the latitude and longitude:

Then

3. Lambert Conic Conformal (2SP Belgium)

For Projected Coordinate System Belge l972 / Belge Lambert 72

Parameters:

Forward calculation for:

first gives

Then:

Reverse the calculations. Use the same parameters and use the E and N values to calculate the latitude and longitude:

Then

Previous Section	Projection Formula Table of Contents	Next Section