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 here.
|a||Semi-major axis of reference ellipsoid|
||Central meridian scale factor|
||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:
Several additional parameters need to be computed before transformations can be undertaken (b, e2, m0). These parameters are constant for a projection.
m0 is obtained by evaluating m using ϕ0
The conversion of geographic coordinates (ϕ, λ) to projection coordinates ( N, E ) is achieved in several steps. First determine (m, ρ, υ) at the computation point (ϕ, λ):
The projection northing ( N ) of the computation point is determined using:
Finally the projection easting ( E ) of the computation point is determined using:
Transverse Mercator projection to geographic
The conversion of Transverse Mercator projection coordinates ( N, E ) to geographic coordinates (ϕ, λ) is achieved in several steps.
First determine N1, m1, n, G, σ and ϕ1 using:
Then determine ρ1, υ1, Ψ1, t1, E1 and x using:
The latitude of the computation point is then computed using:
Finally the longitude of the computation point is determined using:
Grid convergence & point scale factor
Grid convergence (γ) is the angle at a point between true north and grid (projection) north. The point scale factor (κ) is the scale factor at a point that changes with increasing distance from the central meridian.
Both γ and κ can be evaluated for the coordinates in the Transverse Mercator and Lambert Conformal projections using the respective formulas defined in LINZS25002.