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.

aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
symbol-phi-aOrigin latitude
symbol-lambda-aOrigin longitude
symbol-n-aFalse Northing
symbol-e-aFalse Easting
symbol-k-aCentral meridian scale factor
symbol-phiLatitude of computation point
λLongitude of computation point
NNorthing of computation point
EEasting 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 (symbol-b,symbol-e-2,symbol-m-0).  These parameters are constant for a projection.




symbol-m-0is obtained by evaluating symbol-musing symbol-phi-0

Geographic to Transverse Mercator projection

The conversion of geographic coordinates (symbol-phi ,symbol-lambda ) to projection coordinates ( N, E ) is achieved in several steps. First determine symbol-m,symbol-rho,symbol-nuat the computation point (symbol-phi,symbol-lambda):

equation-rho (1)



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 (symbol-phi ,symbol-lambda) is achieved in several steps.

First determine symbol-n-1,symbol-m-1,symbol-little-n,symbol-g,symbol-sigmaand using:

equation-n-1 (1)





Then determine symbol-rho-1,symbol-nu-1,symbol-psi-1,symbol-t-1,symbol-e-1and symbol-xusing:




equation-t-1 (1)

equation-e-1 (1)


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 (symbol-upsilon ) is the angle at a point between true north and grid (projection) north. The point scale factor (symbol-k) is the scale factor at a point that changes with increasing distance from the central meridian.

Both symbol-upsilon and symbol-k can be evaluated for the coordinates in the Transverse Mercator and Lambert Conformal projections using the respective formulas defined in LINZS25002.

Last Updated: 13 April 2015