# Transverse Mercator transformations

The mathematical parameters and equations used to convert geodetic datum coordinates to and from a transverse Mercator projection.

Projections are used to convert points on a 3-dimensional curved surface of the Earth onto a 2-dimensional flat surface. Unlike geodetic datums, which are used to maintain coordinate accuracy, projections deliberately distort the data for representing a section of the Earth as a map, plan and on-screen visualisation.

Projections

Geodetic datums

Transverse Mercator projection coordinates are usually given in terms of Northing and Easting (N,E) while geodetic datum coordinates are in terms of Latitude and Longitude (ϕ, λ).

Coordinate systems used in New Zealand

New Zealand Transverse Mercator 2000

For most users, transverse Mercator transformations can be completed using common spatial software or by using our online coordinate converter.

Online coordinate converter

The mathematical formulae which are used to calculate a transverse Mercator projection coordinate transformation are defined in the Standard for New Zealand Geodetic Datum 2000 Projections.

Standard for New Zealand Geodetic Datum 2000 Projections: Version 2

## Projection parameters

The equations on this page use the following definitions of the parameters, which are common to all projection transformations used in New Zealand.

SymbolParamater
aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
Origin latitude
Origin longitude
False Northing
False Easting
Central meridian scale factor
Latitude of computation point
λLongitude of computation point
NNorthing of computation point
EEasting of computation point

## Projection constants

Before a transverse Mercator projection transformation can be completed, several additional parameters need to be computed: the constants b, e2 and m0.

m0 is obtained by evaluating m using ϕ0

## Converting geographic to transverse Mercator coordinates

First determine (m, ρ, υat the computation point (ϕ, λ):

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

The projection easting (E) of the computation point is determined last:

## Converting transverse Mercator projection to geographic coordinates

First determine N1m1n, G, σ and ϕ1:

Then determine ρ1υ1Ψ1t1E1 and x:

The latitude of the computation point is then computed:

Finally the longitude of the computation point is determined:

## Grid convergence and 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 Standard for New Zealand Geodetic Datum 2000 Projections.

Standard for New Zealand Geodetic Datum 2000 Projections: Version 2 - LINZS25002

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