Lambert Conformal Conic to Geographic Transformation Formulae

This page explains how to convert Lambert Conformal Conic projection coordinates ( N , E ) to their geographic equivalents and vice versa.

Projection parameters

The equations on this page use the following parameters which are specific to the particular projection that is being converted to or from. The correct values for different New Zealand projections can be found in the projections section.

a	Semi-major axis of reference ellipsoid
f	Ellipsoidal flattening
	Latitude of first standard parallel
	Latitude of second standard parallel
	Origin latitude
	Origin longitude
	False Northing
	False Easting
	Latitude of computation point
λ	Longitude of computation point
N	Northing of computation point
E	Easting of computation point

The following equations are divided into three sections:

• projection constants
• geographic to Lambert Conformal Conic
• Lambert Conformal Conic to geographic.

Projection constants

Several additional parameters need to be computed before transformations can be undertaken (,,,). These parameters are constant for a projection.

where:

 and are obtained by evaluating using  and ,

,  and are obtained by evaluating  using  ,  and ,

  is obtained by evaluating using 

Geographic to Lambert conformal projection

The conversion of geographic coordinates (, ) to projection coordinates ( N , E ) is achieved in several steps. First, determine  and 

at the using the latitude of the computation point () and the formulas above. Then evaluate  at the longitude of the computation point () using:

The projection northing ( N ) of the computation point is computed using:

Finally the projection easting ( E ) of the computation point is computed using:

Lambert conformal projection to geographic

The conversion of Lambert projection coordinates ( N , E ) to geographic coordinates ( f , l ) is achieved in several steps. First, determine  ,  ,

,  and  using the following formulas:

The latitude of the computation point needs to be computed iteratively. The first approximation is obtained from:

This initial estimate of 

is then substituted into:

This value of  should be re-substituted into the above formula until successive values do not change. This is typically achieved after three iterations.
The longitude of the computation point () is determined using: