# New Zealand Map Grid transformations

The mathematical parameters and equations used to convert New Zealand Map Grid coordinates to other datums and projections.

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

New Zealand Map Grid (NZMG) was the national projection of New Zealand until it was replaced by the New Zealand Transverse Mercator projection. Coordinates in either of these projections are usually displayed in terms of Northing and Easting (N,E), while geodetic datum coordinates are displayed in terms of Latitude and Longitude (ϕ, λ).

New Zealand Map Grid (NZMG)

New Zealand Transverse Mercator 2000

Coordinate systems used in New Zealand

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

Online coordinate converter

## Converting from NZMG to NZTM2000

A three-step process is used to convert coordinates from New Zealand Map Grid (NZMG) to New Zealand Transverse Mercator 2000 (NZTM2000).

1. De-project NZMG coordinates to NZGD1949 (described below)
2. Convert NZGD1949 geographic coordinates to NZGD2000
3. Project NZGD2000 geographic coordinates onto NZTM2000

This process is used by our online coordinate converter and downloadable software.

## Transforming coordinates between NZMG and NZGD1949

### 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.

SymbolParameter
aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
Origin latitude
Origin longitude
False Northing
False Easting
Latitude of computation point
Longitude of computation point
NNorthing of computation point
EEasting of computation point

## Transformations from NZGD1949 to NZMG

Note that these conversions require the latitude and longitude to be expressed in decimal degrees, where north and east are positive.

i) Calculate ΔΨ and Δλ:

ii) Calculate a complex polynomial function of the complex number θ = ΔΨ + iΔλ:

iii) Calculate the northing N and easting E from x and y by expressing the complex number z in terms of its real and imaginary parts z = x + iy:

## Transformations from NZMG to NZGD1949

The conversion from NZMG to NZGD1949 geographic coordinates (θ, λ)  involves an iterative approximation (which is used to reverse step ii) above).

i) Derive the complex number z:

ii) Determine the complex number θ as a series of approximations (θ0 , θ1 , θ2 and so on). Two iterations of this formula will give millimetre transformation accuracy. The first approximation is:

iii) Successive approximations are obtained by applying the formula:

iv) Calculate the latitude and longitude, by expressing θ in terms of its real and imaginary parts θ = ΔΨ + iΔλ:

## NZMG coefficients

CoefficientReal PartImaginary Part
A10.6399175073
A2-0.1358797613
A30.063294409
A4-0.02526853
A50.0117879
A6-0.0055161
A70.0026906
A8-0.001333
A90.00067
A10-0.00034
B10.75578532280
B20.2492046460.003371507
B3-0.0015417390.041058560
B4-0.101629070.01727609
B5-0.26623489-0.36249218
B6-0.6870983-1.1651967
C11.32312704390
C2-0.577245789-0.007809598
C30.508307513-0.112208952
C4-0.150947620.18200602
C51.014181791.64497696
C61.96605492.5127645
D11.5627014243
D20.5185406398
D3-0.03333098
D4-0.1052906
D5-0.0368594
D60.007317
D70.01220
D80.00394
D9-0.0013
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