# Lambert conformal conic transformations

The mathematical parameters and equations used to convert geodetic datum coordinates to and from a Lambert conformal conic projection.

Projections are used to convert points on a 3-dimensional curved surface of the Earth onto a 2-dimensional flat surface. Unlike geodetic datums, which are used to maintain coordinate accuracy, projections deliberately distort the data for representing a section of the Earth as a map, plan and on-screen visualisation.

Projections

Geodetic datums

One of the more commonly used examples of a Lambert conformal conic projection in New Zealand is the New Zealand Continental Shelf Lambert conformal 2000 (NZCS2000).

New Zealand Continental Shelf Lambert conformal 2000 (NZCS2000)

Lambert conformal conic projection coordinates are usually displayed in terms of northing and easting (N,E) and while geodetic datum coordinates are displayed in terms of latitude and longitude (ϕ, λ).

Coordinate systems used in New Zealand

For most users, Lambert conformal conic transformations can be completed using common spatial software or by using our online coordinate converter.

Online coordinate converter

## Projection parameters

The equations on this page use the following definitions of the parameters, which are common to all projection transformations used in New Zealand.

SymbolParameter
aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
Latitude of first standard parallel
Latitude of second standard parallel
Origin latitude
Origin longitude
False Northing
False Easting
Latitude of computation point
λLongitude of computation point
NNorthing of computation point
EEasting of computation point

## Projection constants

The constants e, n, F, and ρ0 must be calculated before a Lambert conformal projection transformation can be completed.

where:

m1 and m2 are obtained by evaluating m using ϕ1 and ϕ2
t0t1 and tare obtained by evaluating using ϕ0ϕ1, and ϕ2
ρ0, is obtained by evaluating ρ using t0

## Transforming geographic coordinates to Lambert conformal projection coordinates

Determine and ρ at the using the latitude of the computation point (ϕ) and the formulae above, then evaluate θ at the longitude of the computation point (λ):

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

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

## Transforming Lambert conformal projection coordinates to geographic coordinates

Determine N1E1ρ1t1 and θ1:

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

This initial estimate of ϕ is then substituted into:

This value of ϕ should be re-substituted into the above formula until successive values do not change.

The longitude of the computation point (λ) is determined using:

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