Transverse mercator transformation formulae

This page explains how to convert Transverse Mercator 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 here.

a Semi-major axis of reference ellipsoid
f Ellipsoidal flattening
symbol-phi-a
Origin latitude
symbol-lambda-a
Origin longitude
symbol-n-a
False Northing
symbol-e-a
False Easting
symbol-k-a
Central meridian scale factor
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 (b, e2m0).  These parameters are constant for a projection.

equation-b
equation-e-2
equation-e-2

m0 is obtained by evaluating m using ϕ0

Geographic to Transverse Mercator projection

The conversion of geographic coordinates (ϕ, λ) to projection coordinates ( N, E ) is achieved in several steps. First determine (m, ρ, υat the computation point (ϕ, λ):

equation-rho (1)
equation-nu
equation-psi
equation-t (1)
equation-phi (1)

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

equation-n (1)

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

equation-e (1)

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 N1m1n, G, σ and ϕ1 using:

equation-n-1 (1)
equation-m-1
equation-little-n
equation-g
equation-phi-1

Then determine ρ1υ1Ψ1t1E1 and x using:

equation-rho-1
equation-nu-1
equation-psi-1
equation-t-1 (1)
equation-e-1 (1)
equation-x

The latitude of the computation point is then computed using:

equation-little-phi (1)

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.