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# 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

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: