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.

On this page

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-phi-1
Latitude of first standard parallel
symbol-phi-2
Latitude of second standard parallel
symbol-phi-0
Origin latitude
symbol-lambda-0
Origin longitude
symbol-n-0
False Northing
symbol-e-0
False Easting
symbol-phi
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 (e, n, Fρ0). These parameters are constant for a projection.

equation-little-e
equation--little-n
equation-f
equation-rho

where:

equation-m
equation-t

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

Geographic to Lambert conformal projection

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

equation-y

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

equation-n

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

equation-e

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 N1E1ρ1t1 and θ1 using the following formulas:

equation-n-1
equation-e-1
equation-rho-1-new
equation-t-1
equation-y-1

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

equation-phi-first-approx

This initial estimate of ϕ is then substituted into:

equation-phi

This value of ϕ 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 (λ) is determined using:

equation-lambda