This page explains how to convert NZMG projection coordinates ( N , E ) to their NZGD1949 geographic equivalents and vice versa.

Projection parameters

aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
Origin latitudeOrigin latitude
Origin longitudeOrigin longitude
False NorthingFalse Northing
False EastingFalse Easting
Latitude of computation pointLatitude of computation point
Longitude of computation pointLongitude of computation point
NNorthing of computation point
EEasting of computation point

The following equations are divided into three sections:

NZGD1949 geographic to NZMG projection

The conversion of NZGD1949 geographic coordinates (Theta symbol,Longitude of computation point) to NZMG projection coordinates ( N , E ) is achieved in several steps.

The conversion requires the latitude and longitude expressed in decimal degrees (north and east are positive). The constants are provided on the NZMG page and coefficients used in these formulae are listed below.

Note: NZMG applies to coordinates referenced to NZGD1949. Coordinates from other datums, such as NZGD2000 or WGS84, must be converted to NZGD1949 before these formulae can be applied.

Calculatesymbol-delta-psi andsymbol-delta-lambda using:

equation-delta-phi-psi-lambda

Calculate a complex polynomial function of the complex numberequation-theta using the formula:

equation-z

Express the complex number z in terms of its real and imaginary partsequation-imaginary-parts and calculate the northing N and easting E from x and y as:

equation-n-e

 

NZMG projection to NZGD1949 geographic

The conversion from NZMG to latitude and longitude involves an iterative approximation to reverse step 2 above. This starts with the easting E and northing N. The steps to calculate latitude and longitude are as follows. Note that the constants and coefficients used in these formulae are listed below. Derive the complex number z as the following steps.

Note: NZMG applies to coordinates referenced to NZGD1949. Coordinates from other datums, such as NZGD2000 or WGS84, must be converted to NZGD1949 before these formulae can be applied.

equation-z2

Determine the complex number θ as a series of approximations θ0 , θ1 , θ2 and so on. The first approximation is:

equation-theta-0

Successive approximations are obtained by applying the formula:

equation-successive-approximation

Two iterations of this formula will give millimetre transformation accuracy.
ExpressTheta symbolin terms of its real and imaginary partsequation-real-and-imaginary and calculate the latitude and longitude as follows

equation-lat-long

 

 

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