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  Semimajor 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 (e, n, F, ρ_{0}). These parameters are constant for a projection.
where:
m_{1} and m_{2} are obtained by evaluating m using ϕ_{1} and ϕ_{2},
t_{0}, t_{1} and t_{2 }are obtained by evaluating t using ϕ_{0}, ϕ_{1}, and ϕ_{2},
ρ_{0}, is obtained by evaluating ρ using t_{0}
Geographic to Lambert conformal projection
The conversion of geographic coordinates (ϕ, λ) to projection coordinates ( N , E ) is achieved in several steps. First, determine t 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:
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 N^{1}, E^{1}, ρ^{1}, t^{1} and θ^{1} 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 ϕ is then substituted into:
This value of ϕ should be resubstituted 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: