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
symbol. Latitude of first standard parallel
symbol. Latitude of second standard parallel
symbol. Origin latitude
symbol. Origin longitude
symbol. False Northing
symbol. False Easting
symbol. 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

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

projection constants equation.
projection constants equation.
projection constants equation.
projection constants equation.

where:
projection constants equation.
projection constants equation.

symbol. and symbol. are obtained by evaluating symbol. using symbol. and symbol. ,

symbol. , symbol. and symbol. are obtained by evaluating symbol. using symbol. , symbol. and symbol.

symbol. is obtained by evaluating symbol. using symbol.

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Geographic to Lambert conformal projection

The conversion of geographic coordinates ( symbol. , symbol. ) to projection coordinates ( N , E ) is achieved in several steps. First, determine symbol. and symbol. at the using the latitude of the computation point ( symbol. ) and the formulas above. Then evaluate symbol. at the longitude of the computation point ( symbol. ) using:
geographic to lambert conformal projection equation.

The projection northing ( N ) of the computation point is computed using:
Geographic to Lambert conformal projection equation.

Finally the projection easting ( E ) of the computation point is computed using:
Geographic to Lambert conformal projection equation.

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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 symbol. , symbol. , symbol. , symbol. and symbol. using the following formulas:
Lambert conformal projection to geographic equation.
Lambert conformal projection to geographic equation.
Lambert conformal projection to geographic equation.
Lambert conformal projection to geographic equation.
Lambert conformal projection to geographic equation.

The latitude of the computation point needs to be computed iteratively. The first approximation is obtained from:
Lambert conformal projection to geographic equation.

This initial estimate of symbol. is then substituted into:
Lambert conformal projection to geographic equation.

This value of symbol. 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 ( symbol. ) is determined using:
Lambert conformal projection to geographic equation.

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